[Paper Review] Inconsistencies of higgsplosion

Alexander Monin|arXiv (Cornell University)|Aug 17, 2018

Quantum Mechanics and Applications24 references17 citations

TL;DR

This paper demonstrates that the higgsplosion scenario—characterized by exponentially growing N-particle spectral densities in the λφ⁴ scalar field theory—is inconsistent with the principles of local quantum field theory (QFT). Using the Weinberg theorem and analyticity constraints, it shows that such exponential growth violates the polynomial boundedness of spectral densities required by locality and unitarity, rendering the proposed UV behavior unphysical and implying a surprisingly low effective cutoff scale Λ ≳ |m log² ε_max|, contrary to expectations from perturbative analysis.

ABSTRACT

It is shown that higgsplosion scenario is impossible within local QFT framework.

Motivation & Objective

• To assess the consistency of the higgsplosion scenario—where spectral density grows exponentially with N—within local quantum field theory.
• To identify the physical and mathematical inconsistencies in the claimed exponential decay of the Feynman propagator and growth of the spectral density at high energies.
• To demonstrate that the assumptions underlying the higgsplosion proposal violate fundamental theorems of QFT, particularly the Weinberg theorem on spectral density behavior.
• To clarify the implications for UV completion of the λφ⁴ model, showing that it cannot be UV complete under the proposed dynamics.

Proposed method

• Derives the Källén–Lehmann spectral representation for the two-point Wightman function to analyze the analytic structure and boundedness of spectral densities in local QFT.
• Applies the Weinberg theorem to show that spectral densities cannot grow faster than polynomially at infinity, which rules out the exponential growth proposed in higgsplosion.
• Uses analyticity and unitarity arguments to show that exponential decay of the Feynman propagator in Minkowski space contradicts the behavior of the spectral density and leads to exponential growth in Euclidean space.
• Analyzes the asymptotic behavior of the N-particle matrix element ⟨N|φ(x)|0⟩ using non-perturbative methods, comparing perturbative results (factorial growth) with exact non-perturbative results (exponential in N³/²).
• Applies Borel summation techniques to asymptotic series in perturbation theory to illustrate the breakdown of perturbation theory for large N, and contrasts this with non-perturbative saddle-point methods.
• Uses an analogy with a one-dimensional integral to illustrate the transition from perturbative to non-perturbative regimes via saddle-point approximation, mirroring the method used in the field theory computation.

Experimental results

Research questions

• RQ1 Can the higgsplosion scenario, with its exponentially growing spectral density, be consistently embedded in local quantum field theory?
• RQ2 Does the claimed exponential decay of the Feynman propagator in Minkowski space satisfy the analyticity and unitarity constraints of QFT?
• RQ3 What are the implications of the spectral density growth ∼ exp(˜c N³/² / √λ) for the UV behavior and cutoff scale of the λφ⁴ theory?
• RQ4 Is the proposed effective cutoff scale Λ ≳ |m log² ε_max| physically viable, and how does it depend on the regime of validity of the non-perturbative computation?
• RQ5 Does the non-perturbative result for the spectral density (6) remain valid for finite ε, or is it restricted to extremely small kinetic energies per particle?

Key findings

• The spectral density ρ(E, N) ∼ ε³/² N exp(˜c N³/² / √λ) for E ≈ Nm(1 + ε) and ε ≪ 1 violates the Weinberg theorem, which requires spectral densities to grow at most polynomially at infinity.
• Exponential growth of the spectral density implies that the theory cannot be UV complete, as it leads to a breakdown of the polynomial boundedness required by locality and tempered distribution behavior.
• The claimed exponential decay of the Feynman propagator in Minkowski space contradicts analyticity and unitarity, as it would imply exponential growth in the Euclidean region, which is inconsistent with the spectral density behavior.
• The effective cutoff scale is found to be unexpectedly low, Λ ≳ |m log² ε_max|, depending on the domain of validity of the non-perturbative result, rather than being ∼ m exp(16π² / λ) as in standard perturbative estimates.
• The non-perturbative result (6) is likely only valid for extremely small ε, pushing the effective cutoff higher, and its validity cannot be confirmed without alternative derivations, such as lattice computations.
• The paper concludes that the higgsplosion scenario is inconsistent with local QFT, as it violates fundamental theorems on spectral functions and analyticity, rendering the proposed UV behavior unphysical.