Effective Potential in Subleading Logarithmic Approximation in Arbitrary Non-renormalizable Scalar Field Theory

Following the previously developed approach to the calculation of quantum corrections to the effective potential in arbitrary scalar field theories in the leading logarithmic approximation, we extended it to the next-to-leading order. Based on Bogoliubov-Parasiuk-Hepp-Zimmerman renormalization procedure and the Bogoliubov-Parasiuk theorem, we construct recurrence relations and renormalization group equations that allow one to sum up the leading and subleading logarithms in all orders of perturbation theory. The formalism is applicable to an arbitrary scalar potential, renormalizable or not. To verify the results, we compare them with a renormalizable model treated within the standard renormalization group approach.

💡 Research Summary

The paper develops a systematic method to compute the effective potential of an arbitrary scalar field theory, extending previous work that summed leading logarithms (LLA) to include next‑to‑leading logarithms (NLLA). The authors start from the Bogoliubov‑Parasiuk‑Hepp‑Zimmermann (BPHZ) renormalization framework, separating the R‑operation into the divergent‑part extractor K and the incomplete R′‑operation that subtracts subdivergences. By exploiting the Bogoliubov‑Parasiuk theorem, they derive explicit relations among the pole coefficients A(n), B(n), C(n) that appear at order 1/εⁿ, 1/εⁿ⁻¹, 1/εⁿ⁻² respectively. Crucially, these relations allow all higher‑order poles to be expressed through a small set of one‑, two‑, and three‑loop diagrams.

For a generic scalar Lagrangian L = ½(∂ϕ)² − g V₀(ϕ), the effective potential is written as a power series V_eff = g ∑ₙ(−g)ⁿ Vₙ. In the leading logarithmic approximation the authors obtain a simple recurrence relation (13) that connects the n‑th order leading pole V_Aₙ with lower‑order terms via a second derivative operator D² = ∂²/∂ϕ². Introducing the generating function Σ_A(z,ϕ)=∑ₙ(−z)ⁿ V_Aₙ with z = g/ε, the recurrence becomes the differential equation ∂_z Σ_A = −¼ D²Σ_A. Solving this equation and substituting z → g log(μ²/m²) reproduces the standard RG‑improved effective potential, confirming that the leading‑log sum is scheme‑independent.

The core contribution of the paper is the extension to NLLA. Here two‑loop vacuum diagrams (bubble and sunset topologies) and the appearance of higher‑derivative operators D³ and D⁴ must be taken into account. The authors compute the 1/ε² and 1/ε poles of these diagrams (eqs. 19‑22), adopt the minimal subtraction scheme (c₁ = 0), and construct a much more intricate recurrence relation (23) for the subleading coefficients V_Bₙ and for the derivative‑term coefficients V_Gₙ. The structure of (23) reflects the rule that each term contains exactly one subleading object (either B or G) while the remaining factors are leading. Multiplying by (−z)ⁿ and summing over n yields two coupled nonlinear RG equations for the generating functions Σ_B and Σ_G:

• ∂_z Σ_B = −¼ D²Σ_B + (1/8)