In the strict large-N limit, the 3+1-dimensional O(N) model with quartic interaction has a Landau pole where the running coupling diverges, yet physical observables (pressure, scattering amplitude, thermal pole mass) remain finite, and a PT-symmetric analytic continuation of the coupling through the pole renders the theory asymptotically free and UV complete. The source work (Romatschke, 'Life at the Landau pole') explicitly leaves open whether this benign behavior survives the approximations used to establish it: subleading 1/N corrections are uncomputed; the real-part analytic-continuation prescription is a conjecture checked only in d=1 quantum mechanics and not justified for the interacting 3+1d continuum theory at finite temperature; the proposed phenomenological application to QCD (continuing the strong coupling beyond the Lambda scale of the MS-bar scheme) is purely qualitative; and parameter regimes where the real saddle points turn complex were excluded from the analysis. This record asks whether the Landau pole of the 3+1d O(N) model is genuinely benign once these gaps are closed.

Background

A Landau pole is the finite momentum scale ΛLP\Lambda_{LP} at which the running coupling of a renormalizable quantum field theory with a marginally irrelevant interaction, such as λϕ4\lambda\phi^4 theory in 3+1 dimensions, diverges according to the renormalization group. Since Landau's work in the 1950s, such a pole has been regarded as fatal, and for scalar theories with positive coupling the conclusion is reinforced by rigorous proofs of quantum triviality — the theorem (proved for one- and two-component fields by Aizenman and Duminil-Copin) that no interacting continuum limit exists for 4d ϕ4\phi^4 theory with positive coupling.

The O(N) model is the theory of an N-component real scalar field ϕ⃗\vec\phi with quartic self-interaction (ϕ⃗⋅ϕ⃗)2(\vec\phi\cdot\vec\phi)^2, normalized so that the limit N→∞N\to\infty is finite. At leading order in the 1/N1/N expansion (introduce an auxiliary field by a Hubbard-Stratonovich transformation, integrate out ϕ⃗\vec\phi, and take the saddle point), the model becomes exactly solvable in any dimension. Romatschke (arXiv:2211.15683) showed that in 3+1 dimensions the large-N renormalized coupling runs as λR(μˉ)=4π2/ln⁡(ΛLP2/μˉ2)\lambda_R(\bar\mu)=4\pi^2/\ln(\Lambda_{LP}^2/\bar\mu^2), so a Landau pole appears at μˉ=ΛLP\bar\mu=\Lambda_{LP}, yet observables — the thermodynamic pressure, the s-wave scattering amplitude, the thermal pole mass — stay finite and well-behaved at the pole.

To make sense of the theory beyond the pole, where naive continuation flips the sign of the coupling, Romatschke adopted a conjecture of Ai, Bender and Sarkar from PT-symmetric field theory. PT symmetry is invariance under the combined operation of parity reflection and time reversal; Bender and Boettcher showed that non-Hermitian Hamiltonians with this symmetry can have entirely real energy spectra, giving 'wrong-sign' potentials a unitary quantum interpretation. The Ai-Bender-Sarkar (ABS) conjecture states that the partition function of the flipped-coupling theory is obtained by ln⁡ZPT(g,β)=Re ln⁡Z(λ=−g+i0+,β)\ln Z_{PT}(g,\beta)=\mathrm{Re}\,\ln Z(\lambda=-g+i0^+,\beta). With the flipped sign, the renormalized coupling gR(μˉ)=4π2/ln⁡(μˉ2/ΛLP2)g_R(\bar\mu)=4\pi^2/\ln(\bar\mu^2/\Lambda_{LP}^2) decreases at high energies: the theory is asymptotically free, evading the Coleman-Gross classification and the triviality proofs, whose hypotheses assume a positive UV coupling.

In 'Life at the Landau pole' (arXiv:2212.03254), Romatschke showed that deforming the model by relevant operators (a mass term) and irrelevant ones (a sextic coupling, characterized by a renormalization-group-invariant scale MM via αR/64gR3=1/M2\alpha_R/64g_R^3=1/M^2) does not qualitatively change the benign behavior near the pole. At finite temperature the pressure remains real and continuous across the scale where complex saddle-point solutions first appear (a second-order transition at Tc≃0.616 ΛLPT_c\simeq 0.616\,\Lambda_{LP} in the massless case). As a phenomenological aside, the paper compares the model to QCD by identifying the Landau pole with the scale ΛMS‾\Lambda_{\overline{\mathrm{MS}}} at which the perturbative strong coupling αs\alpha_s diverges, suggesting that αs\alpha_s might be analytically continued to negative or complex values in the infrared without adverse physical consequences — explicitly a qualitative speculation.

The paper itself acknowledges what remains unresolved. First, the entire construction holds at strict N=∞N=\infty: whether subleading 1/N1/N corrections preserve the finiteness of observables at and beyond the pole is not established. Second, the ABS prescription is a conjecture whose checks existed only in d=1d=1 (quantum mechanics) at zero temperature, per the paper's own footnote; its status for the interacting 3+1d continuum theory at finite temperature is open. Third, for sextic scale M≲84 ΛLPM\lesssim 84\,\Lambda_{LP} all real saddle-point solutions except the trivial one become complex-valued, and these regimes were excluded from the analysis. The status of the ABS prescription has since been refined: it is expected to fail in general (Lawrence, Weller, Peterson and Romatschke, arXiv:2303.01470), while an exact identity equating the finite-volume wedge-contour partition function to the real part of the analytically continued Hermitian partition function at negative quartic coupling is established in arbitrary dimension (Hatsuda and Kikuchi, arXiv:2608.04387).

Problem Statement

Establish whether the Landau pole of the 3+1-dimensional O(N) model is genuinely benign — i.e., whether the finiteness, reality, and asymptotic freedom found at strict N=infinity survive once the supporting approximations are removed. The common objective uniting the requested results is a controlled definition of the continued theory at finite N and a test of its observables at and beyond the pole scale Lambda_LP. Specifically:

  1. Subleading 1/N corrections. Compute (or rigorously bound) the O(1/N) corrections to physical observables — pressure, s-wave scattering amplitude, thermal pole mass — of the quartic (and sextic-deformed) O(N) model continued through its Landau pole, and determine whether they remain finite and real at and beyond mu = Lambda_LP, or whether the benign large-N behavior is an artifact of the strict large-N limit. Note that the two candidate definitions of the continued theory (the PT-symmetric 'cone' contour and the wedge contour of the analytic continuation, distinguished in Lawrence, Weller, Peterson and Romatschke, arXiv:2303.01470) are known to coincide only at leading order in 1/N, so this computation also fixes which theory is being defined at finite N.

  2. Analytic continuation in 3+1d at finite temperature. Justify rigorously, or refute, the real-part prescription ln Z = Re ln Z(lambda = -g + i0^+) for the interacting 3+1-dimensional continuum theory at finite temperature, including the role of non-perturbative (bounce/instanton) contributions and the relation between integration contours in the complex field plane at finite N (the existing exact identity is proved at finite volume on the lattice; the interacting continuum, finite-T case is open).

  3. Complete the parameter map. Determine the phase structure and observables in the regimes excluded from the original analysis, where the sextic renormalized scale satisfies M <~ 84 Lambda_LP and the real saddle-point solutions disappear. (Later work has substantially mapped this regime at leading large N; the unresolved part is its behavior beyond leading order in 1/N.)

Any resolution must go beyond the strict large-N saddle-point approximation in which the source results were obtained.

References

  1. Paul Romatschke, 'Life at the Landau pole', AppliedMath 4(1), 55-69 (2024), DOI: 10.3390/appliedmath4010003, arXiv:2212.03254, https://arxiv.org/abs/2212.03254
  2. Paul Romatschke, 'A solvable quantum field theory with asymptotic freedom in 3+1 dimensions', Int. J. Mod. Phys. A 38, 2350157 (2023), DOI: 10.1142/S0217751X23501579, arXiv:2211.15683, https://arxiv.org/abs/2211.15683
  3. Wen-Yuan Ai, Carl M. Bender, Sarben Sarkar, 'PT-symmetric -g phi^4 theory', Phys. Rev. D 106, 125016 (2022), DOI: 10.1103/PhysRevD.106.125016, arXiv:2209.07897, https://arxiv.org/abs/2209.07897
  4. Scott Lawrence, Ryan Weller, Christian Peterson, Paul Romatschke, 'Instantons, analytic continuation, and PT-symmetric field theory', Phys. Rev. D 108, 085013 (2023), DOI: 10.1103/PhysRevD.108.085013, arXiv:2303.01470, https://arxiv.org/abs/2303.01470
  5. Ryan D. Weller, 'Can negative bare couplings make sense? The phi^4 theory at large N', arXiv:2310.02516 (2023), https://arxiv.org/abs/2310.02516
  6. Paul Romatschke, 'Negative Coupling phi^4 on the Lattice', PoS LATTICE2023, 367 (2023), DOI: 10.22323/1.453.0367, arXiv:2310.03815, https://arxiv.org/abs/2310.03815
  7. Leqian Chen, Sarben Sarkar, 'Phases of quartic scalar theories and PT symmetry', Phys. Rev. D 111, 025001 (2025), DOI: 10.1103/PhysRevD.111.025001, arXiv:2409.05439, https://arxiv.org/abs/2409.05439
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  9. Yasuyuki Hatsuda, Kengo Kikuchi, 'Tensor Network Formulation of PT-Symmetric Quantum Field Theory', arXiv:2608.04387 (2026), https://arxiv.org/abs/2608.04387
  10. Paul Romatschke, 'On the negative coupling O(N) model in 2d at high temperature', JHEP 04, 012 (2025), DOI: 10.1007/JHEP04(2025)012, arXiv:2412.10496, https://arxiv.org/abs/2412.10496
  11. Paul Romatschke, 'Phase Diagram and Finite Temperature Properties of Negative Coupling Scalar Field Theory', arXiv:2603.19877 (2026), https://arxiv.org/abs/2603.19877
  12. Paul Romatschke, 'On self-dualities for scalar phi^4 theory', arXiv:2602.18286 (2026), https://arxiv.org/abs/2602.18286
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