Effective sextic field theory for tricritical-critical crossover

Abstract: Effective field theories provide a suitable framework for both particle physics and statistical physics. We delve deeper into the study of the effective three-dimensional scalar field theory for its application to statistical physics, especially considering the role of the sextic coupling in the tricritical-to-critical crossover. The three-loop renormalization of the mass and the two coupling constants that we perform allows us to obtain, for the first time, the complete renormalization group flow of the couplings in that order. We analyze what universality means in this problem and how we can recover non-universal terms from the renormalization group beta functions. The crossover is realized by the convergence of the renormalization group flow towards the line connecting the tricritical and critical fixed points.

• The paper presents a novel three-loop renormalization analysis of the (λφ⁴+gφ⁶)₃ model, capturing explicit RG flows between tricritical and critical fixed points.
• The study computes beta functions free of scheme-dependent constants, thereby establishing strong universality in the crossover behavior.
• Methodologies including the background-field approach and analytic Feynman integrals yield precise separatrix expansions and bridge perturbative with nonperturbative insights.

Three-Loop Effective Sextic Field Theory for Tricritical-Critical Crossover

Introduction

The paper "Effective sextic field theory for tricritical-critical crossover" (2605.09631) presents a rigorous treatment of three-dimensional scalar field theory incorporating both quartic and sextic interactions, aiming to elucidate the renormalization structure and renormalization group (RG) flows governing crossover phenomena between tricritical and critical points. The work leverages effective field theory frameworks, pushing the analysis to three-loop order. This allows, for the first time, the full three-loop renormalization group flow for mass and couplings, and a careful inspection of universality and scheme dependence in this sector.

Formulation and Renormalization

The central model is the $(\lambda\phi^4+g\phi^6)_3$ theory, whose interaction potential $U(\phi)$ consists of both quartic $(\lambda\phi^4)$ and sextic $(g\phi^6)$ terms. The motivation arises from the distinct universality classes encoded in the RG fixed points: the quartic theory captures Ising-type criticality with a single relevant parameter, while the combined theory accommodates tricriticality, where two parameters become independently relevant.

The renormalization procedure is formulated in the language of effective field theory, maintaining a physical cutoff $\Lambda$ and avoiding the formal removal of divergences. All relevant divergences are regularized and subtracted, primarily through a normal-ordering prescription which isolates positive powers of $\Lambda$ and absorbs them in counterterms. Logarithmic divergences are kept as they encode RG flow. Results from both cutoff and dimensional regularization schemes are carefully compared, revealing universal RG flows unaffected by scheme-dependent finite terms when mass is well below the cutoff.

The three-loop calculation is accomplished using the background-field method, extending previous two-loop results and utilizing analytic expressions for vacuum Feynman graph integrals from Rajantie (2605.09631) and Kudlis and Pikelner. Additional combinatorial and field renormalization contributions are included, culminating in explicit expressions for the renormalized mass and couplings as functions of the physical parameters and initial (bare) values.

Renormalization Group Analysis and Universality

Upon renormalization, the theory is described by three parameters: mass $m$, quartic coupling $\lambda$, and sextic coupling $g$. The RG beta functions for the dimensionless couplings $u=\lambda/m$ and $U(\phi)$0 are computed to three-loop order. Importantly, the beta functions are shown to be devoid of scheme-dependent constants and positive powers of the cutoff, establishing strong universality at this order. Beta functions are contrasted with those obtained via dimensional regularization in $U(\phi)$1, which are only valid in the tricritical regime and cannot describe the crossover in fixed $U(\phi)$2.

The RG flow in the $U(\phi)$3-plane displays a saddle-node structure at the origin (tricritical fixed point), characterized by one null and one negative Jacobian eigenvalue. Dynamical systems theory is invoked to rigorously classify the flow, with separatrix calculations revealing the manifold connecting the tricritical point to the expected Wilson-Fisher critical fixed point. Explicit three-loop expressions for these separatrices are derived, providing analytic detail of the crossover boundary.

Strong numerical results include the derived expansions for the flow on separatrices:

• For the critical connecting line: $U(\phi)$4
• For the marginal separatrix associated with pure sextic theory: $U(\phi)$5

These expansions show convergence with results from higher-loop studies and pseudo-$U(\phi)$6 expansions.

Crossover and Nonperturbative Insights

The RG flows produce a crossover from tricritical behavior ($U(\phi)$7 relevant) to critical behavior ($U(\phi)$8 becomes a function of $U(\phi)$9, marginal). The transition is governed by the approach and departure from the origin in parameter space as system parameters are tuned. The paper discusses several schemes to define the crossover, including geometric loci in $(\lambda\phi^4)$0-space, and observes the critical exponents associated with crossover, notably $(\lambda\phi^4)$1.

Nonperturbative aspects are addressed by comparison to exact RG results, particularly the Wegner-Houghton equation. Eigenvectors at the origin reveal that the critical surface is tangent to the scaling dimensions, and numerical agreement between perturbative and nonperturbative flows is quantified, demonstrating high precision in the perturbative domain and highlighting limitations in the treatment of derivative couplings.

Implications and Directions for Future Research

The work establishes the three-loop effective field theory framework as a robust tool for describing tricritical-critical crossover in scalar field theories. The universality of RG flows at three loops and the reduction of all non-universal scheme dependence to logarithms represent key technical advances. Integration with nonperturbative RG supports the reliability of the effective theory approach within its domain of validity.

Practical implications extend to statistical mechanics models exhibiting multicriticality, such as fluid mixtures and magnetic systems. Theoretical implications concern the extension of these methods to more complex field theories (e.g., multi-component models, gauge theories, gravitational systems). The author explicitly points to potential generalizations via inclusion of additional (possibly non-renormalizable) couplings within the effective field theory framework, exploiting its flexibility for fine-tuning and symmetry breaking studies.

Conclusion

This paper rigorously develops the three-loop effective field theory for $(\lambda\phi^4)$2 scalar models, providing analytic expressions for renormalized parameters and renormalization group beta functions. Universality is demonstrated at three-loop order, and the RG flow structure is thoroughly classified via dynamical systems theory, yielding explicit descriptions of separatrices and crossover phenomena. The approach bridges perturbative and nonperturbative perspectives, delivers precise numerical results, and clarifies scheme dependence in the renormalization process. The work lays a foundation for further explorations of multicriticality and crossover effects in both statistical and quantum field theoretic contexts.