Establish whether the five-loop ultraviolet and infrared zeros merge

Determine whether the five-loop ultraviolet and infrared fixed points of the five-dimensional SU(3) fixed-denominator rational beta function merge as the number of fundamental fermion flavors is varied.

Background

At the flavor value corresponding to the three-loop merger, the paper finds that four-loop matching produces a single ultraviolet zero, while five-loop matching produces both an ultraviolet and an infrared zero. This demonstrates that the existence and arrangement of the fixed points depend sensitively on the matching order.

The authors explicitly leave unresolved whether the two zeros ultimately coalesce at some other value of N_f. Resolving this requires following both branches as the flavor number changes, rather than evaluating the fixed points only at the three-loop merger flavor.

References

An IR partner thus reappears at five-loop matching for the same $(D,N,N_f)$ at which four-loop matching gives a single UV zero. This comparison concerns the fixed-denominator rational family; determining whether the five-loop UV and IR zeros merge requires tracking both branches as $N_f$ varies.

— Asymptotic Safety in Gauge Theories beyond Four Dimensions: Perturbative Tests and Fixed-Point Mergers  (2610.12225 - Deandrea et al., 8 Oct 2026) in Appendix, Section “Matching formulas, merger conditions, and sensitivity tests,” subsection “Dependence on the matching order”