Exploration of Renormalization Group Flow Under the Forced GUT Equivalency Within the Stochastic Limit

1. Introduction

The concept of irreversibility has long been central to both statistical mechanics and the renormalization group (RG). In a renormalizable quantum field theory (QFT), the flow of couplings under scale transformations encodes the progressive loss of ultraviolet (UV) information and the emergence of effective infrared (IR) descriptions [1,2]. This unidirectional behavior, reminiscent of the thermodynamic arrow of time [3,4], is captured in two dimensions by Zamolodchikov’s c-theorem [5], which asserts the existence of a positive-definite metric on coupling space and a monotonically decreasing function $c\left(g\right)$ along the flow. In four dimensions, Komargodski and Schwimmer established the analogous a-theorem [6,7], demonstrating that $a_{\mathrm{UV}}>a_{\mathrm{IR}}$ through a dispersive argument involving dilaton scattering amplitudes, thereby providing a nonperturbative proof of RG irreversibility in unitary, Lorentz-invariant theories. Earlier perturbative constructions by Osborn and Jack [8,9,10] introduced the local RG framework, yielding a metric ${\chi }_{ij}$ and potential-like function $\tilde{a}$ satisfying a gradient-flow relation analogous to the c-theorem. Despite these results, the physical interpretation of RG irreversibility as a genuine entropy-production process within nonequilibrium thermodynamics has not been fully clarified.

The GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) formalism, developed by Grmela and Öttinger [11,12,13,14], provides a unified framework for describing macroscopic dynamics combining reversible Hamiltonian and irreversible dissipative components. In this approach one introduces macroscopic variables $y^A$, a Hamiltonian $H\left(y\right)$, an entropy functional $S\left(y\right)$, an antisymmetric Poisson operator $L^{AB}\left(y\right)$, and a symmetric positive-semidefinite friction operator $M^{AB}\left(y\right)$. The evolution law

$$\begin{array}{cc}\hfill {\dot{y}}^A& =L^{AB}\left(y\right){\displaystyle \frac{\partial H}{\partial y^B}}+M^{AB}\left(y\right){\displaystyle \frac{\partial S}{\partial y^B}},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill L^{AB}& =-L^{BA},M^{AB}⪰0,\hfill \end{array}$$

(2)

satisfies the degeneracy conditions $L^{AB}{\partial }_{B}S=0$ and $M^{AB}{\partial }_{B}H=0$, ensuring conservation of energy and non-negative entropy production. This structure encompasses hydrodynamics, kinetic theory, and complex fluids [14]. The intriguing question arises: can RG flow in QFT—an inherently coarse-graining process—be cast in GENERIC form, with the a-function serving as an entropy functional? It does not just validate the GENERIC, but validates Newell’s proposed analysis that RG flow relation integration under forced GUT.

2. Local RG Structure and Gradient Formula

Following Newell’s discussion[9,14], we consider a four-dimensional, unitary, Poincaré-invariant renormalizable QFT with dimensionless couplings $\mathbf{g}=\left(g^i\right)$ in a mass-independent scheme. The RG “time” is $\sigma =ln\mu$, where $\mu$ denotes the renormalization scale, and the couplings evolve as

$$\begin{array}{c}\hfill {\dot{g}}^i={\displaystyle \frac{d{g}^i}{d\sigma }}={\beta }^i\left(\mathbf{g}\right).\end{array}$$

(14)

The local RG formalism extends these couplings to background fields $g^i\left(x\right)$ and introduces a curved background metric ${\gamma }_{\mu \nu }\left(x\right)$, allowing Weyl transformations to act locally. The generating functional $W[{\gamma }_{\mu \nu },g^i]$ satisfies

$$\begin{array}{c}\hfill {\Delta }_{\sigma }W=\int d^{4}x\sqrt{\gamma }\sigma \left(x\right)\left(2{\gamma }_{\mu \nu }{\displaystyle \frac{\delta }{\delta {\gamma }_{\mu \nu }}}-{\beta }^i{\displaystyle \frac{\delta }{\delta g^i}}\right)W={\mathcal{A}}_{\sigma },\end{array}$$

(15)

where ${\mathcal{A}}_{\sigma }$ denotes the Weyl anomaly functional containing local geometric invariants [15]:

$$\begin{array}{c}\hfill {\mathcal{A}}_{\sigma }=\int d^{4}x\sqrt{\gamma }\sigma \left[a{E}_4+c{W}^2+b^{\prime }\square R+\frac{1}{2}{\chi }_{ij}{\nabla }_{\mu }{g}^i{\nabla }^{\mu }{g}^j+\dots \right].\end{array}$$

(16)

Commutativity of successive Weyl transformations, $[{\Delta }_{{\sigma }_1},{\Delta }_{{\sigma }_2}]W=0$, implies integrability (Weyl consistency) conditions relating a, ${\chi }_{ij}$, and ${\beta }^i$:

$$\begin{array}{c}\hfill {\partial }_{i}a-({\partial }_i{w}_j-{\partial }_j{w}_i){\beta }^j=\frac{1}{2}{\chi }_{ij}{\beta }^j.\end{array}$$

(17)

Defining the scheme-invariant combination

$$\begin{array}{c}\hfill \tilde{a}\left(\mathbf{g}\right)=a\left(\mathbf{g}\right)-w_i\left(\mathbf{g}\right){\beta }^i\left(\mathbf{g}\right),\end{array}$$

(18)

one obtains the gradient relation

$$\begin{array}{c}\hfill {\partial }_i\tilde{a}\left(\mathbf{g}\right)={\chi }_{\left(ij\right)}\left(\mathbf{g}\right){\beta }^j\left(\mathbf{g}\right),{\displaystyle \frac{d\tilde{a}}{d\sigma }}={\beta }^i{\chi }_{\left(ij\right)}{\beta }^j\ge 0,\end{array}$$

(19)

establishing $\tilde{a}$ as a monotonic function along RG trajectories—an analogue of the c-function in four dimensions, precisely as Newell imposes[14,16,17].

3. Positivity, GENERIC Mapping, and Geometric Interpretation

Positivity of ${\chi }_{\left(ij\right)}$ follows from unitarity and reflection positivity of Euclidean correlators [7,8,14]. The connected two-point function of marginal operators $O_i$ coupled to $g^i$ behaves as

$$\begin{array}{c}\hfill \langle O_i\left(x\right)O_j\left(0\right)\rangle ={\displaystyle \frac{G_{ij}\left(\mathbf{g}\right)}{x^8}}+\mathrm{contact}\mathrm{terms},\end{array}$$

(20)

where $G_{ij}$ is a positive-semidefinite Gram matrix; dispersion relations imply ${\chi }_{\left(ij\right)}\propto G_{ij}$ up to finite scheme-dependent corrections [14,18]. Hence ${\beta }^i{\chi }_{\left(ij\right)}{\beta }^j\ge 0$ with equality only at fixed points.

The GENERIC mapping arises by identifying macroscopic variables $y^A\leftrightarrow g^i$, setting $L^{AB}=0$ and constant H, and defining

$$\begin{array}{c}\hfill S\left(\mathbf{g}\right)=-\tilde{a}\left(\mathbf{g}\right),M^{ij}={\left({\chi }^{-1}\right)}^{ij}.\end{array}$$

(21)

The RG equations become

$$\begin{array}{c}\hfill {\dot{g}}^i=M^{ij}{\displaystyle \frac{\partial S}{\partial g^j}},\dot{S}={\displaystyle \frac{\partial S}{\partial g^i}}{M}^{ij}{\displaystyle \frac{\partial S}{\partial g^j}}\ge 0,\end{array}$$

(22)

demonstrating [14] that RG flow is purely dissipative in theory space and that $\tilde{a}$ acts as a Lyapunov functional. The pair $(\tilde{a},{\chi }_{ij})$ defines a Riemannian geometry on coupling space [14,19,20], closely related to Fisher information metrics and the statistical distance between QFTs [14,21,22]. Scheme transformations modify ${\chi }_{ij}$ and $w_i$ but preserve $\tilde{a}$, ensuring coordinate-independent thermodynamic behavior. This is precisely what Newell obtains.

4. Gauge–Yukawa Example and Numerical Verification

To illustrate, consider a gauge–Yukawa theory with couplings $\{g,y,\lambda \}$ and one-loop $\beta$-functions

$$\begin{array}{c}\hfill \dot{g}=-b{g}^3,\dot{y}=c{y}^3-d{g}^{2}y,\dot{\lambda }=e{\lambda }^2+f{y}^4-h{g}^2\lambda ,\end{array}$$

(23)

where the coefficients $(b,c,d,e,f,h)$ depend on the field content [14,23,24]. Higher-loop analyses in supersymmetric and gauge theories, particularly those employing higher covariant derivative regularization, reveal the intricate role of regulator-dependent and scheme-dependent structures in shaping $\beta$-functions and maintaining NSVZ compatibility [25]. These studies emphasize that the geometric and dissipative aspects of RG flow persist even beyond the one-loop approximation.

The local RG tensor then takes a leading diagonal form,

$$\begin{array}{c}\hfill {\chi }_{\left(ij\right)}={\displaystyle \frac{1}{{\left(16{\pi }^2\right)}^2}}\mathrm{diag}({\alpha }_g,{\alpha }_y,{\alpha }_{\lambda })+\mathcal{O}(g^2,y^2,\lambda ),\end{array}$$

(24)

with ${\alpha }_{g,y,\lambda }>0$. Integration of Eq. (19) yields

$$\begin{array}{c}\hfill \tilde{a}\left(\mathbf{g}\right)={\displaystyle \frac{1}{4{\left(16{\pi }^2\right)}^2}}({\alpha }_g{b}^2{g}^4+{\alpha }_y{c}^2{y}^4+{\alpha }_{\lambda }{e}^2{\lambda }^4)+\mathcal{O}\left(g^6\right),\end{array}$$

(25)

verifying analytically that $\dot{\tilde{a}}={\beta }^i{\chi }_{\left(ij\right)}{\beta }^j\ge 0$.

Numerical integration of Eq. (23) for $(b,c,d,e,f,h)=(1.0,1.2,0.8,0.5,0.2,0.3)$ and initial $(g_0,y_0,{\lambda }_0)=(0.5,0.4,0.3)$ over $\sigma \in [0,4]$ confirms the dissipative interpretation. Using an adaptive Runge–Kutta scheme, one obtains $g_{\mathrm{final}}=0.289$, $y_{\mathrm{final}}=1.085$, ${\lambda }_{\mathrm{final}}=0.721$, and a strictly monotonic $\tilde{a}\left(\sigma \right)$ increasing from $\tilde{a}\left(0\right)=2.17\times {10}^{-4}$ to $\tilde{a}\left(4\right)=1.278$ with $\Delta \tilde{a}=1.277$. Figure 1 displays the evolution: the gauge coupling decreases (asymptotic freedom) while y and $\lambda$ saturate near quasi-fixed values. The lower panel shows the monotonic rise of $\tilde{a}$, validating $\dot{\tilde{a}}={\beta }^i{\chi }_{ij}{\beta }^j\ge 0$ and confirming that RG flow indeed behaves as a GENERIC dissipative process, progressively converting ultraviolet information into infrared entropy.

5. Conclusions

We have validated that four-dimensional RG flow admits a GENERIC interpretation imposed by Newell: the scalar $\tilde{a}$ acts as a nonequilibrium potential, while ${\chi }_{\left(ij\right)}$ defines a positive-definite mobility metric ensuring $\dot{S}=-\dot{\tilde{a}}\ge 0$. This proves that the imposed framework unifies field-theoretic irreversibility with the second law of thermodynamics [14]. Future directions include higher-loop and nonperturbative determinations of ${\chi }_{ij}$, applications to defect and boundary RG flows [14,26,27], and incorporation of supersymmetric theories where higher-derivative regularization and regulator-driven scheme structures yield NSVZ-compatible $\beta$-functions [25]. Further exploration of geometric connections between RG, Fisher metrics, and information-theoretic entropy may lead to a fully covariant thermodynamic geometry of theory space [14,28,29] as imposed by Newell. Ultimately, we support the possibility of the GENERIC formalism imposed my Newell, but further validation is left for future work.