Do higher-order spectral-function identities determine the renormalization constants $Z_2$, $Y_2$, and $Z_3$ in the pseudoscalar meson theory after $Z_1=0$?

Nambu's 1956 paper 'Renormalization Constants' analyzed Dyson's renormalization constants in the symmetrical pseudoscalar meson-nucleon theory (interaction $H_{\mathrm{int}}=g\,i\bar{\psi}\gamma_5\tau_i\psi\varphi_i$ , no meson self-coupling) by expressing them through the spectral functions of the exact nucleon, meson, and vertex Green's functions. Using only the spectral functions and equations of motion of the first two orders in the coupling, the paper shows that the charge renormalization constant must vanish ( $Z_1=0$ ), while the remaining constants — the nucleon wave-function constant $Z_2$ , its scalar-part companion $Y_2$ , and the meson wave-function constant $Z_3$ — are left completely undetermined, because with $Z_1=0$ the relevant spectral integrals become indeterminate. The paper explicitly asks whether the unused portion of the spectral-function system (higher spectral functions and the remaining equations of motion) yields more information about these constants; it closes with the remark that its results 'need not necessarily be restricted to the specific model' and that in quantum electrodynamics the Ward identity $Z_1Z_2^{-1}=1$ 'would be an additional relation which must be taken into account', but it poses no separate question about other theories. This problem asks for a resolution of the paper's question: determine whether more information on $Z_2$ , $Y_2$ , $Z_3$ can be obtained by further considerations within the spectral-function framework, or establish by reasoned argument that the framework's spectral relations leave them undetermined.

Background

In quantum field theory, the bare fields and coupling constants appearing in the Lagrangian are related to physical quantities through Dyson's renormalization constants. In a meson-nucleon theory the relevant ones are $Z_2$ , the nucleon wave-function renormalization constant (the residue of the single-nucleon pole in the interacting nucleon propagator, relative to the free one); $Z_3$ , the analogous meson wave-function renormalization constant; $Z_1$ , the charge (coupling) renormalization constant (the correction to the lowest-order meson-nucleon vertex); and $Y_2$ , the companion of $Z_2$ associated with the scalar part of the nucleon propagator, the part proportional to the Dirac mass term rather than to $\gamma\cdot p$ .

Nambu's 1955 papers I and II developed a non-perturbative framework in which the exact two- and three-point Green's functions are represented through spectral functions, in the spirit of the Källén-Lehmann representation (the exact representation of a Green's function as an integral, over all invariant masses squared, of free-particle propagators weighted by real spectral densities). The nucleon propagator is written in terms of densities $\nu_\lambda(u)$ with $\lambda=0$ (scalar) and $\lambda=1$ (vector) Dirac structure, the meson propagator in terms of a density $\nu(w)$ , and the vertex function in terms of densities $\nu_{\lambda\lambda'}(u,v,w)$ depending on three invariant masses. Positivity of the Hilbert-space norm gives convergence conditions and inequalities such as $\nu(w)\ge 0$ ; the one-particle poles (the point spectrum) of the densities define the observed nucleon mass $\kappa$ , the observed meson mass $\mu$ , and the renormalized coupling $g$ , and express the renormalization constants as spectral sum rules (Eq. (2.8) of the 1956 paper):

Z_2^{-1}=\int\nu_1(u)\,du\ge 1,\qquad Y_2^{-1}=\int\nu_0(u)\,du=-\kappa_0 Z_2^{-1},\qquad Z_3^{-1}=\int\nu(w)\,dw\ge 1,

where $\kappa_0$ is the bare (mechanical) nucleon mass. The framework further supplies equations of motion: integral equations, derived from the field equations, that relate the continuous parts of the two-point spectral functions to integrals over the three-point spectral functions, with kinematic weight functions fixed by phase space.

The 1956 paper applies this framework to the symmetrical pseudoscalar meson theory: nucleons interacting with a triplet of equal-mass pseudoscalar mesons through $H_{\mathrm{int}}=g\,i\bar{\psi}\gamma_5\tau_i\psi\,\varphi_i$ , where $\tau_i$ are the isospin Pauli matrices and $\gamma_5$ the Dirac matrix coupling a pseudoscalar meson to the nucleon axial current, with no direct meson-meson self-coupling ( $\lambda=0$ ). Using the spectral functions and equations of motion of only the first two orders in the coupling, Nambu derives (Eq. (3.5))

Z_2^{-1}=1+\frac{3g}{8\pi^2}J_1^{(1)},\qquad Y_2^{-1}=-\kappa+\frac{3g}{8\pi^2}J_0^{(1)},\qquad Z_3^{-1}=1+\frac{3g}{2\pi^2}J^{(1)},

where the $J$ integrals are explicit multiple integrals over the spectral functions. Estimating the integrand of $J^{(1)}$ , he finds it does not decay, so the integral diverges unless the charge renormalization constant vanishes: to avoid contradiction one must demand $Z_1\to 0$ . The same conclusion follows from the asymptotic behavior of the renormalized full vertex function, $\Gamma_5(pqk)\to Z_1\gamma_5\tau_i$ as the meson momentum squared $k^2\to\infty$ (Eq. (4.4)). With $Z_1=0$ , however, the integrals $J^{(1)}$ and $J_\lambda^{(1)}$ become indeterminate (of the form $0\times\infty$ ), so these relations determine nothing about $Z_2$ , $Y_2$ , $Z_3$ .

The question belongs to a lineage of non-perturbative results on renormalization constants: Källén (1953) showed in quantum electrodynamics (QED) that not all renormalization constants can be finite simultaneously, and Gotô and Imamura (1955) showed that if the constants and certain higher spectral moments are all finite, the Green's functions reduce to those of a free field. In QED, gauge invariance additionally imposes the Ward identity, an exact relation between the vertex function and the propagator that at the level of constants reads $Z_1Z_2^{-1}=1$ . Nambu's $Z_1=0$ is a sharper statement in the same spirit: rather than an inconsistency among finite constants, the framework forces one constant to vanish, and the fate of the remaining ones is the open question. The 1956 paper closes with the declarative remark that its results 'need not necessarily be restricted to the specific model we have chosen' and that in quantum electrodynamics the Ward identity $Z_1Z_2^{-1}=1$ 'would be an additional relation which must be taken into account'; this is a statement about the generality of the approach, not a second open question posed by the paper.

Problem Statement

Nambu (1956) poses, in the Discussion of 'Renormalization Constants', an open question that the paper leaves unresolved.

In the symmetrical pseudoscalar meson theory with $H_{\mathrm{int}}=g\,i\bar{\psi}\gamma_5\tau_i\psi\,\varphi_i$ and $\lambda=0$ , the analysis of the first two orders of the spectral functions and their equations of motion forces $Z_1=0$ and thereby yields no information on $Z_2$ , $Y_2$ , $Z_3$ . The paper states: 'Since we have used so far only part of the whole variety of the spectral functions and their equations of motion, it is yet to be seen whether more information can be obtained by further considerations.' The question: can more information on the remaining constants $Z_2$ , $Y_2$ , $Z_3$ be obtained by further considerations within the spectral-function framework?

References

Y. Nambu, "Renormalization Constants", Physical Review 101, 1183-1186 (1956). DOI: 10.1103/physrev.101.1183, https://doi.org/10.1103/physrev.101.1183
Y. Nambu, "Structure of Green's Functions in Quantum Field Theory", Physical Review 100, 394-411 (1955). DOI: 10.1103/PhysRev.100.394, https://doi.org/10.1103/PhysRev.100.394
Y. Nambu, "Structure of Green's Functions in Quantum Field Theory. II", Physical Review 101, 459-467 (1956). DOI: 10.1103/PHYSREV.101.459, https://doi.org/10.1103/PHYSREV.101.459
G. Källén, "Paper [1953c]: On the Magnitude of the Renormalization Constants in Quantum Electrodynamics", originally published as Mat.-Fys. Medd. Dan. Vid. Selsk. 27, No. 12 (1953); reproduced in C. Jarlskog (ed.), Portrait of Gunnar Källén, Springer, Cham, pp. 537-554 (published online 2013, print 2014). DOI: 10.1007/978-3-319-00627-7_92, https://doi.org/10.1007/978-3-319-00627-7_92
T. Gotô and T. Imamura, "Note on the Non-Perturbation-Approach to Quantum Field Theory", Progress of Theoretical Physics 14, 396-397 (1955). DOI: 10.1143/ptp.14.396, https://doi.org/10.1143/ptp.14.396
N. Nakanishi, "On Lehmann's method of renormalization", Il Nuovo Cimento 5, 520-522 (1957). DOI: 10.1007/BF02855260, https://doi.org/10.1007/BF02855260
K. Hiida and M. Sawamura, "Some relations among Green's functions", Il Nuovo Cimento 5, 896-906 (1957). DOI: 10.1007/BF02903212, https://doi.org/10.1007/BF02903212
N.V. Krasnikov, "On the ultraviolet behavior of the invariant charge in quantum electrodynamics", arXiv:2603.24092 [hep-th] (2026). DOI: 10.48550/arXiv.2603.24092, https://doi.org/10.48550/arXiv.2603.24092

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Do higher-order spectral-function identities determine the renormalization constants $Z_2$ , $Y_2$ , and $Z_3$ in the pseudoscalar meson theory after $Z_1=0$ ?

Background

Problem Statement

References