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arXiv:2606.12269

Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O($N$) model in $d=3$ and $N=\infty$

Shunsuke Yabunaka, Bertrand Delamotte

Published Jun 10, 2026• 0 citations
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Abstract

Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical $O(N)$ model in three dimensions in the limit $N\to\infty$. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent $ν$ which takes the value $ν= 1/2$ along the regular part of the BMB line, that is, for $0 \leq λ< λ_{\rm BMB}$, jumps to $ν= 1/3$ on the singular part of this line with the BMB FP, corresponding to $λ= λ_{\rm BMB}$, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.

DOI
10.48550/arxiv.2606.12269

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Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O(NNN) model in d=3d=3d=3 and N=∞N=\inftyN=∞
Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical O(N)O(N)O(N) model in three dimensions in the limit N→∞N\to\inftyN→∞. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent ννν which takes the value ν=1/2ν= 1/2ν=1/2 along the regular part of the BMB line, that is, for 0≤λ<λBMB0 \leq λ< λ_{\rm BMB}0≤λ<λBMB​, jumps to ν=1/3ν= 1/3ν=1/3 on the singular part of this line with the BMB FP, corresponding to λ=λBMBλ= λ_{\rm BMB}λ=λBMB​, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.
We show that at $N=\ensuremath{\infty}$ and below its upper critical dimension, d<dupd<{d}_{\mathrm{up}}d<dup​, the critical and tetracritical behaviors of the O(N)\mathrm{O}(N)O(N) models are associated with the same renormalization group fixed point (FP) potential. Only their derivatives make them different with the subtleties that taking their $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the $\ensuremath{\epsilon}$---and the 1/N1/N1/N---expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at $N=\ensuremath{\infty}$ and d=dupd={d}_{\mathrm{up}}d=dup​ can be understood from a finite-NNN analysis.
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Dynamical renormalization group analysis of O(n)O(n)O(n) model in steady shear flow
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Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
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