Nontrivial zeros of the QED β-function (ultraviolet fixed points)

Determine whether the Gell-Mann–Low β-function of quantum electrodynamics possesses a nontrivial zero at finite coupling, thereby establishing or excluding the existence of a nontrivial ultraviolet fixed point for QED beyond perturbation theory.

Background

Gell-Mann and Low introduced the renormalisation group framework and highlighted the special role of zeros of the β-function as fixed points that govern high-energy behavior. They conjectured that the fine structure constant might correspond to such a fixed point.

However, beyond perturbation theory, the global behavior of the QED β-function is unknown, and in particular it is unknown whether it has any nontrivial (nonzero) zeros. Resolving this would clarify the ultraviolet completeness or triviality of QED.

References

Obviously, as long as perturbation theory is our only guide, we cannot say anything about the properties of the β(λ) for arbitrary λ and we do not know whether it has any non-trivial zeros.

— From Many Models to ONE THEORY  (2501.10233 - Iliopoulos, 17 Jan 2025) in Subsection: The renormalisation group

Four-loop matching reveals a complementary possibility: with the same denominator, the interacting UV branch persists beyond the three-loop merger boundary, with no UV--IR merger found over the color and flavor ranges studied. Within the chosen rational family, this continuation extends the candidate domain of asymptotic safety to larger flavor numbers. At five-loop matching, an IR partner reappears alongside the UV zero at the $SU(3)$ three-loop merger flavor, $N_f\simeq6.379$. The matching-order comparison therefore exposes the sensitivity of the merger scenario while identifying how higher-order information can alter the extent of the UV branch.

— Asymptotic Safety in Gauge Theories beyond Four Dimensions: Perturbative Tests and Fixed-Point Mergers  (2610.12225 - Deandrea et al., 8 Oct 2026) in Conclusion, final discussion of four-loop and five-loop matching