Finite-$N$ fate of the negative-coupling loophole: is four-dimensional scalar theory with negative quartic coupling non-trivial and asymptotically free at small $N$?

Two no-go results shape the conventional wisdom that four-dimensional scalar field theories cannot be interacting and ultraviolet-complete: the Coleman–Gross theorem on asymptotic freedom and the Aizenman–Duminil-Copin proof of quantum triviality of the critical 4D Ising and $\phi^4$ models. Both proofs assume a positive scalar self-coupling. Romatschke (2023) showed that in the large- $N$ limit the four-dimensional $O(N)$ model with negative bare quartic coupling — a classically unbounded, upside-down potential — has well-behaved observables (real bounded free energy, sensible propagators and cross sections), a Landau pole only in the infrared, and a negative beta function in the ultraviolet, i.e. asymptotic freedom; this was independently confirmed by Weller. This exposes the loophole but leaves the crucial question open: does the phenomenon survive at finite, especially small, $N$ (e.g. $N=1,2,4$ , the Higgs value)? Answering it requires a controlled non-perturbative definition of the finite- $N$ negative-coupling theory (e.g. a complex-contour-deformed lattice formulation); closely related open issues concern which analytic continuations in complex coupling space define the same theory — singularities and instanton/Stokes structures are known to obstruct equivalence in quantum-mechanical, zero- and one-dimensional cases — and the subleading $1/N$ structure of the large- $N$ starting point (whether the NLO scalar bound state at $m_2\simeq 1.84m$ acquires a finite width at NNLO). A proof that no unitary non-trivial finite- $N$ continuation exists would equally resolve the question.

Background

Asymptotic freedom means a quantum field theory's dimensionless couplings vanish logarithmically at short distances, so the ultraviolet (UV) is weakly coupled. Coleman and Gross (1973) proved that in four dimensions only non-abelian gauge theories can be asymptotically free — under the assumption that scalar self-couplings are positive, so that the classical potential is bounded from below. Quantum triviality is the complementary statement for scalars: Aizenman and Duminil-Copin (2021) proved that the scaling limits of the critical four-dimensional Ising and positively-coupled $\phi^4$ models are marginally trivial, i.e. the renormalized quartic coupling vanishes in the continuum, so no interacting continuum theory results — again only for classically stable (positive) couplings.

Already in 1973 Symanzik suggested that classically unstable potentials might still define sensible quantum field theories. In quantum mechanics this is understood: Bender and Böttcher (1998) showed that non-Hermitian Hamiltonians with $\mathcal{PT}$ symmetry (invariant under combined parity and time reversal), such as $p^2-(ix)^4$ , can have entirely real, positive, bounded spectra despite the upside-down classical potential; Jones and Mateo (2006) showed the $-x^4$ case is isospectral to an equivalent Hermitian Hamiltonian. Ai, Bender and Sarkar (2022) formulated the $-g\varphi^4$ field theory as a $\mathcal{PT}$ -symmetric path integral and conjectured that its Euclidean partition function is the boundary-value analytic continuation of the Hermitian $\lambda\varphi^4$ ( $\lambda>0$ ) theory across $\lambda\to -g$ .

Romatschke (2022–2023) made this concrete in four dimensions: solving the massless $O(N)$ model with $N$ real scalar fields $\vec\phi$ and interaction $-g(\vec\phi\cdot\vec\phi)^2$ in the large- $N$ limit (the $1/N$ expansion in which resummed chains of 'bubble' diagrams control the dynamics, with the Hubbard–Stratonovich auxiliary-field method giving an exact saddle point at $N=\infty$ ), the theory has a negative beta function in the UV — asymptotic freedom — with the coupling running through a Landau pole (a finite scale where perturbative coupling diverges; here in the infrared, playing a role analogous to $\Lambda_{\overline{\rm MS}}$ in QCD, as explored in 'Life at the Landau pole'). Observables (free energy, propagators, two-particle scattering cross sections) come out real, bounded and interacting, including a stable scalar bound state of mass $m_2\simeq 1.84m$ at next-to-leading order (NLO) in $1/N$ , a state known from 1970s large- $N$ analyses. Weller (2023) independently confirmed non-triviality, asymptotic freedom and an infrared Landau pole at large $N$ with fields on a $\mathcal{PT}$ -symmetric complex domain, matching the thermal partition function of the positive-coupling theory.

This construction evades both no-go theorems only through the sign of the coupling, hence 'loophole'. However, at finite $N$ none of the standard tools applies directly: the Euclidean action is unbounded below, so Monte Carlo importance sampling has no positive measure, and perturbation theory fails near the Landau pole. A lattice definition is possible by contour deformation — rotating the field integration from the real axis into the complex plane, generalizing the $\mathcal{PT}$ -symmetric quantum-mechanical contour — but the resulting lattice action is non-polynomial and the integrals are complex, so direct (non-Monte-Carlo) integration has only been feasible on very small lattices to date. Rigorous forebears exist but are incomplete: Gawedzki and Kupiainen (1985) rigorously constructed a non-trivial, asymptotically free negative-coupling $\phi^4$ model in four dimensions within the hierarchical approximation (a block-local model capturing the same renormalization flow), with the extension to the full model possibly violating reflection positivity (the Osterwalder–Schrader positivity condition on Euclidean correlators needed to reconstruct a unitary Lorentzian quantum theory); Ito (1992) gave a similar construction via field rotation; but Hosaka (2006) found that hierarchical models with small negative coupling are in fact trivial, so even the hierarchical rigorous status is contested, and no construction establishes a unitary finite- $N$ four-dimensional theory.

Problem Statement

Determine whether the four-dimensional $O(N)$ -symmetric scalar quantum field theory with negative bare quartic coupling, $V(\vec\phi)=\frac{m^2}{2}\vec\phi^2-\frac{g}{4}(\vec\phi\cdot\vec\phi)^2$ with $g>0$ , admits a non-trivial, asymptotically free continuum limit at finite $N$ — in particular at the small values $N=1,2,4$ relevant for single-component scalars and the Standard Model Higgs doublet — where the large- $N$ construction of Romatschke and Weller no longer applies. An answer covering the small values $N=1,2,4$ suffices to resolve the question; a result valid for all finite $N$ is welcome but not required. The objective is a yes-or-no characterization of the finite- $N$ theory's continuum existence:

An affirmative resolution consists of a controlled non-perturbative definition or construction of the finite- $N$ theory (for example, a lattice regularization with complex contour deformation of the field variables, or a rigorous constructive analysis) and a demonstration that the theory is unitary (its Euclidean correlators satisfying reflection/Osterwalder–Schrader positivity) and that physical observables — free energy, propagators, mass spectrum, scattering amplitudes — are real, bounded and interacting in the continuum limit, with the ultraviolet behavior asymptotically free.
Equally, a negative resolution — a demonstration (rigorous or controlled-numerical) that no non-trivial unitary finite- $N$ continuation of the negative-coupling theory exists — would answer the question on its own.

Two related open issues are context for, rather than components of, this yes-or-no target: the analytic structure in complex coupling space (singularities, branch cuts, instanton/Stokes obstructions) that determines when the candidate definitions — the $\mathcal{PT}$ -symmetric path integral, analytic continuation of the positive-coupling theory, and the contour-deformed lattice — define the same continuum theory at finite $N$ (this delineation is already known to obstruct equivalence in quantum-mechanical, zero- and one-dimensional cases); and the subleading $1/N$ structure of the four-dimensional model, in particular whether the scalar bound state of mass $m_2\simeq 1.84m$ found at NLO acquires the expected finite width at next-to-next-to-leading order (NNLO) and whether unforeseen infrared or analytic-structure singularities appear at subleading orders. Downstream Higgs phenomenology is likewise context, not part of the completion boundary, since the source itself conditions it on first establishing the finite- $N$ theory.

References

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Finite- $N$ fate of the negative-coupling loophole: is four-dimensional scalar theory with negative quartic coupling non-trivial and asymptotically free at small $N$ ?

Background

Problem Statement

References