Finiteness of the bare charge in quantum electrodynamics: the strong-coupling asymptotics of the photon spectral weight function and the Gell-Mann-Low function

The photon propagation function of pure quantum electrodynamics (QED) admits a spectral representation whose positive weight function $s(M^2/m^2, e^2)$ determines an effective coupling $e_M^2$ . Schwinger (1974) showed that the Gell-Mann-Low function of the renormalization group is $\psi(x) = x^2 \sigma(x)$ , where $\sigma$ is the spectral weight reparametrized as a function of the running coupling, and posed the resulting ultraviolet question: is the limiting coupling $e_\infty^2$ (the 'bare charge' of renormalized operator field theory) finite - a first positive zero of $\sigma$ , amounting to an eigenvalue condition on the electric charge - or infinite, as the aggressive source-theory interpretation insists? He also derived a dichotomy for the phase of the propagation function at timelike momenta and left the detailed asymptotic behavior of $\sigma$ , of the phase, and of the propagation function undetermined, judging the question 'likely to remain moot for some time to come'. Half a century of later work - rigorous inequalities, lattice evidence for the zero-charge (triviality) picture, resummation studies claiming an exact strong-coupling law, and a very recent direct contradiction over whether the Landau pole exists - has still not produced a rigorous determination. The problem is to determine the strong-coupling asymptotics of $\sigma(x)$ , equivalently of the Gell-Mann-Low function, in pure QED, and thereby settle whether the bare charge is finite or infinite.

Background

In quantum electrodynamics (QED), the photon propagation function $D(k)$ - the momentum-space Green function of the electromagnetic field, with $k$ the four-momentum and $e$ the physical electron charge - can be written as a spectral representation

k^2 e^2 D(k) = 1 - k^2 e^2 \int_{m^2}^{\infty} \frac{dM^2\, s(M^2/m^2, e^2)}{k^2 + M^2},

where $m$ is the electron mass and $s$ is the spectral weight function, fixed by the imaginary part of the propagation function on the timelike axis ( $k^2 = -M^2$ ). Positivity of $s$ ( $s > 0$ for all finite $M^2 > 0$ ) is a physical input, tied to the existence of photon-photon scattering states (light-by-light scattering). Schwinger developed this analysis within source theory - his reformulation of particle physics in which radiative corrections are computed from emission and absorption sources, deliberately avoiding the renormalized operator fields of conventional field theory - and contrasted it with renormalized operator field theory, the standard formulation whose ultraviolet structure is nominally controlled by a 'bare' (unrenormalized) charge. The spectral representation defines an effective, scale-dependent coupling $e_M^2$ . Reparametrizing the spectral weight along the resulting renormalization-group flow, $\sigma(e_M^2) = s(M^2/m^2, e^2)$ , one obtains the flow equation $d \log M^2 = dx/(x^2 \sigma(x))$ with $x = e_M^2$ . Schwinger's footnote observation was that this is exactly of the Gell-Mann-Low form: the Gell-Mann-Low function $\psi(x)$ - introduced in 1954 to describe how the effective electromagnetic coupling changes with momentum scale - is $\psi(x) = x^2 \sigma(x)$ , a connection he noted 'has escaped notice for two decades'. The limiting value $e_\infty^2 = e^2/\left(1 - e^2 \int_{m^2}^{\infty} (dM^2/M^2)\, s(M^2/m^2, e^2)\right)$ plays the role of the bare charge of renormalized operator field theory. If $e_\infty^2$ is finite, $\sigma$ must have its first positive zero there - an eigenvalue condition on $e^2$ , the possibility explored early on by Baker and Johnson; if infinite, $\sigma(x)$ must decay to zero as $x \to \infty$ and the continuum theory is 'trivial' in the zero-charge sense of Landau (the renormalized charge vanishes for any finite bare charge as the cutoff is removed), with or without a Landau pole - a divergence of the effective coupling at finite momentum scale. Continuing the modified propagation function $k^2 e^2 D$ to timelike momenta yields a complex quantity whose phase $\varphi(M^2)$ is fixed by $s$ ; positivity of $s$ implies $\sin\varphi > 0$ . The established constraints on $\sigma$ are: (i) $\sigma(x) > 0$ ; (ii) the crude bound $\sigma(x) < 1/x$ ; (iii) initially rising behavior $\sigma(x) = \sigma_0 + \sigma_1 x + \cdots$ with $\sigma_0$ fixed by the one-loop vacuum polarization (the lowest-order screening of the charge by virtual electron-positron pairs); (iv) $\sigma$ must vanish suitably at $x = e_\infty^2$ ; and (v) the dichotomy that the timelike phase either approaches a non-zero limit as $M^2 \to \infty$ , or approaches zero 'essentially no more rapidly than' $\varphi(M^2) \to 1/(\log M^2 \, \log\log M^2)$ . A companion paper (Schwinger, December 1974) exhibited the explicit deviations between these asymptotic functions - the spectral weight function $\sigma(x)$ , the phase $\varphi$ , and the Gell-Mann-Low function - and proved the existence of a transformation of variable that produces the Gell-Mann-Low function $\psi(x) = x^2 \sigma(x)$ from the spectral weight function, so these statements apply to all three functions at once. The bound $0 \le \beta(g) < g$ was later rederived from analyticity and spectral representations by Krasnikov and by Yamagishi, and confirmed perturbatively through five loops by Kataev and Larin. Lattice studies of cutoff QED (e.g. Kim, Kogut and Lombardo) provide numerical evidence for the triviality (infinite bare charge) picture, and analytic resummation work by Suslov claims an exact strong-coupling law, while the most recent convergent-series analysis by Komarova and Nalimov claims the opposite conclusion on the Landau pole. No rigorous determination of the strong-coupling behavior exists.

Problem Statement

In pure quantum electrodynamics with one charged lepton of mass $m$ and physical charge $e$ , consider the spectral representation of the photon propagation function,

k^2 e^2 D(k) = 1 - k^2 e^2 \int_{m^2}^{\infty} \frac{dM^2\, s(M^2/m^2, e^2)}{k^2 + M^2},

with $s > 0$ for all finite $M^2 > 0$ . Along the renormalization-group flow this defines the effective coupling $e_M^2$ and the spectral weight function in coupling form, $\sigma(e_M^2) = s(M^2/m^2, e^2)$ , related by $d \log M^2 = dx/(x^2 \sigma(x))$ with $x = e_M^2$ , so that the Gell-Mann-Low function is $\psi(x) = x^2 \sigma(x)$ . Determine the asymptotic behavior of $\sigma(x)$ as $x \to \infty$ , and thereby decide whether the limiting coupling

e_\infty^2 = \frac{e^2}{1 - e^2 \int_{m^2}^{\infty} \frac{dM^2}{M^2}\, s(M^2/m^2, e^2)}

is finite or infinite. If $e_\infty^2$ is finite, it is the first positive zero of $\sigma$ - the quantity that plays the role of the bare charge of renormalized operator field theory, and an eigenvalue condition on $e^2$ . If $e_\infty^2$ is infinite, determine the decay law of $\sigma(x)$ as $x \to \infty$ and the corresponding asymptotic form of the propagation function and of its timelike phase $\varphi(M^2)$ , including which branch of the dichotomy holds (a non-zero limit of $\varphi$ , or decay no faster than of order $1/(\log M^2 \log\log M^2)$ ). The established constraints on $\sigma$ are: positivity of the spectral weight, the bound $\sigma(x) < 1/x$ , the initially rising one-loop expansion of $\sigma$ , the flow relation, and the phase dichotomy.

References

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