Large-order asymptotics of perturbation theory in the Yukawa model: fermionic Green functions, controlled fermion-determinant treatment, and the four-dimensional case

Parisi's 1977 Physics Letters B paper estimated the $n$-th order of perturbation theory for a Yukawa interaction among fermions, producing the first large-order asymptotics for a fermionic quantum field theory: the estimates concern Green functions of the scalar (bosonic) field and are extracted from a saddle point of the effective action in which the fermion determinant is treated in a quasi-local (relativistic Thomas-Fermi) approximation. Completing that program remains open. No later work has derived the large-order behavior of the fermionic Green functions of the Yukawa model; the quasi-local approximation, its derivative corrections, and the constants in the large-order formula were never controlled from first principles (only partial results exist: the determinant's growth order, and exact symmetric-sector determinants via inverse scattering); and the four-dimensional renormalizable case — where the induced one-loop $-g^4\sigma^4\ln(g\sigma)$ term signals non-convergence of the scalar functional integral — has become the triviality question for four-dimensional Yukawa and Higgs-Yukawa models, which has strong lattice and functional-renormalization-group support but no rigorous proof. The problem is to complete the large-order program for Yukawa theory to the controlled standard described below, including the fermionic Green functions and the four-dimensional case.

Background

Perturbation expansions in quantum field theory are divergent. Since Dyson's 1952 argument, the coefficients $a_n$ of the expansion in a coupling $g$ are expected to grow factorially at large order, typically as $C\,n!\,(-Rg)^n\,n^\alpha$, so the series has zero radius of convergence and must be resummed — most commonly by Borel summation, meaning one resums the transformed series of $a_n/n!$, which has a finite radius of convergence. Knowing the constants in the large-order growth is what makes such resummation quantitatively useful, since it fixes the analytic structure of the Borel transform. In 1977, Lipatov explained the factorial growth for scalar field theories: at negative (or complex) coupling the Euclidean functional integral is dominated by a non-perturbative saddle point — an 'instanton', a nontrivial finite-action solution of the classical field equations in the complexified coupling regime — and the Gaussian fluctuation determinant around it, together with the integration over the saddle's collective coordinates (its 'zero modes', i.e. the position and scale symmetries broken by the saddle), yields the $n!$ growth and the constants. Brézin, Le Guillou and Zinn-Justin then systematized the method for $\phi^4$-type scalar theories, including their renormalization. Extending the method to theories with fermions raises new problems, which Parisi (1977) attacked for the Yukawa model — a Dirac fermion $\psi$ interacting with a scalar field $\sigma$ through the coupling term $g\bar\psi\sigma\psi$. Integrating out the fermions multiplies the functional integral by $\det(\partial+g\sigma)$, the fermion determinant, a nonlocal functional of $\sigma$. Parisi treated this determinant in a quasi-local approximation, also called the relativistic Thomas-Fermi approximation by analogy with the atomic Thomas-Fermi model (in which a slowly varying external potential is treated as locally constant): for field configurations varying slowly on the fermion Compton scale, $g\sigma\gg(\partial\sigma)/\sigma$, the determinant is replaced by a local functional of $\sigma$, with omitted derivative corrections of order $g^{D-1}\partial_\mu\sigma^{D-2}$. The resulting bosonic effective action has saddle points from which the large-order behavior of the bosonic (scalar-field) Green functions was estimated. In four spacetime dimensions, however, evaluating the determinant on a constant field produces a one-loop term $-g^4\sigma^4\ln(g\sigma)$: an induced attractive quartic self-interaction that grows logarithmically and makes the effective potential unbounded, so the scalar functional integral no longer converges. This is the counterpart of the classic zero-charge (Landau-pole) problem, and in modern language the mechanism behind the 'triviality' of four-dimensional Yukawa and Higgs-Yukawa models — the statement that their only continuum limit is a free, Gaussian theory. A theory is superrenormalizable when only finitely many divergent diagrams exist (as for the Yukawa model in fewer than four dimensions) and just renormalizable in $D=4$, where divergences accumulate at every order and the analysis of non-perturbative saddle contributions must be intertwined with renormalization. Within this landscape the source paper established a heuristic large-order estimate for the bosonic Green functions of the Yukawa model; what was left open — the fermionic Green functions, the control of the quasi-local approximation and its constants, and the careful renormalization-level treatment in $D=4$ — is precisely the open problem.

Problem Statement

Complete the large-order program for the Yukawa model that the source paper initiated, to the standard that the fluctuation determinant and the constants in the large-order formula are computed independently from the effective action and are not quoted from or matched to the source paper, and that the approximations underlying the derivation are justified. For the Euclidean Yukawa theory of a Dirac fermion coupled to a scalar by $g\bar\psi\sigma\psi$, in general spacetime dimension $D$ with the renormalizable case $D=4$ explicitly included, the program left open by the source covers the following facets of one objective — the controlled theory of the large-order behavior of Yukawa perturbation series, of which the 1977 bosonic estimates are the heuristic first step. The three facets are each a substantive part of the objective in its own right. First, derive the explicit leading large-order behavior of the perturbation-expansion coefficients of the fermion two-point function $\langle\psi(x)\bar\psi(y)\rangle$ and of general fermionic correlation functions, where the source paper derived estimates only for the bosonic (scalar) Green functions after integrating out the fermions and indicated that similar techniques should apply without carrying out the derivation. Second, make the derivation controlled: the constants in the large-order formula are to be computed independently from the effective action rather than quoted from or matched to the source paper, and the approximations on which the derivation rests are to be justified — in the source's approach, the quasi-local (relativistic Thomas-Fermi) treatment of the fermion determinant, valid for slowly varying fields with $g\sigma\gg(\partial\sigma)/\sigma$, and the size of the derivative corrections of order $g^{D-1}\partial_\mu\sigma^{D-2}$ omitted along the relevant saddle. Third, address the renormalizable case $D=4$ in the framing the source's companion Cargèse lectures give it: when the interaction is renormalizable, new difficulties arise for the large-order estimates that were 'not completely understood at the present moment'; there the fermion-induced one-loop $-g^4\sigma^4\ln(g\sigma)$ term makes the effective potential unbounded and signals non-convergence of the scalar functional integral; the $D=4$ facet is to determine the $n$-th-order asymptotics of the relevant coefficients together with a precise statement of which large-order assertion survives the non-convergence of the scalar integral, in whichever consistent formulation of the regulated theory is adopted (finite cutoff, renormalized continuum, or another controlled framework).

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