Primitive asymptotics in $ϕ^4$ vector theory

A longstanding conjecture in $ϕ^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in $N$ depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer $N$, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.

💡 Research Summary

The paper tackles a long‑standing conjecture in four‑dimensional scalar φ⁴ theory: that primitive (i.e. overall‑divergent but without subdivergences) Feynman graphs dominate the asymptotic behaviour of the minimal‑subtraction (MS) β‑function at large loop order L. To gain additional control, the authors extend the scalar field to an O(N)‑symmetric vector model, thereby introducing a second expansion parameter 1/N. This allows them to study the N‑dependence of graph symmetry factors, denoted T(G,N), which are polynomials in N and factorise under graph insertions.

The analysis proceeds in two parallel settings. First, a zero‑dimensional version of the theory is considered, where the path integral reduces to an ordinary integral. By expanding the integral in powers of the coupling ℏ, the authors obtain exact generating functions that enumerate vacuum, connected, 1‑particle‑irreducible (1PI) and primitive graphs, each weighted by the symmetry factor 1/|Aut(G)| and the O(N) factor T(G,N). Algebraic transformations (logarithm, exponential) separate connected from disconnected contributions, while a further “Martin invariant” condition isolates primitive graphs. The main result is an explicit asymptotic formula for the sum p_L(N) of primitive graphs at loop order L:

p_L(N) ∼ C(N)·L!·L^{(3N+5)/2}·