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Derivation of SM Gauge Couplings

Published: 2025-09-01 | Permalink

modified: 2025-09-28T13:03:43Z

DERIVATION OF STANDARD MODEL GAUGE COUPLINGS FROM SCALE-INVARIANT INFORMATION THERMODYNAMICS


Author: Rowan Brad Quni-Gudzinas

Affiliation: QNFO

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000 0005 2645 6062

DOI: 10.5281/zenodo.17218943

Publication Date: 2025-09-28

Version: 1.0.1


Objective: To derive the relationship between the Standard Model’s gauge couplings and the Kappa information field ($\kappa(x)$) using a self-contained, category-theoretic approach based on the principle of scale-invariant information thermodynamics. This derivation establishes that effective gauge couplings are proportional to $\kappa(x)$, leading to spatial and energy-scale dependence of fundamental constants like the fine-structure constant ($\alpha \propto \kappa^2$).



**I. Foundational Framework and Definitions**


Axiom 1.1 (Information-Theoretic Ontology):

Physical reality emerges from information dynamics. Spacetime points are characterized by a dimensionless scalar field $\kappa(x) > 0$ representing normalized information density. All physical quantities derive from informational relationships governed by $\kappa(x)$.


Definition 1.2 (Kappa Field):

The Kappa field $\kappa(x)$ is defined as:


$$\kappa(x) = \frac{S(x)}{k_B M_0^3},$$


where $S(x)$ is the thermodynamic entropy density at spacetime point $x$, $k_B$ is Boltzmann’s constant, and $M_0$ is a fundamental mass scale. By the Mass-Energy-Entropy-Information equivalence principle:


$$\kappa(x) = \frac{E}{\hbar \omega M_0^3} = \frac{m c^2}{k_B T M_0^3},$$


where $m$ is rest mass, $c$ is the speed of light, $T$ is temperature, $E$ is energy, $\hbar$ is the reduced Planck constant, and $\omega$ is angular frequency.


Justification: This definition establishes $\kappa(x)$ as a dimensionless information measure. The fundamental mass scale $M_0$ ensures dimensional consistency while preserving the information-theoretic interpretation. In natural units ($\hbar = c = k_B = 1$), $\kappa(x)$ is dimensionless as required by Axiom 1.1. The equivalence follows from Landauer’s principle ($E = k_B T \ln 2$ per bit) and the Einstein mass-energy relation, normalized to dimensionless form.


Definition 1.3 (Information-Theoretic Gauge Symmetry):

For a Dirac spinor field $\psi(x)$ representing fermionic matter, the Lagrangian density $\mathcal{L}_0 = i \bar{\psi} \gamma^\mu \partial_\mu \psi - m \bar{\psi} \psi$ is invariant under the global transformation:


$$\psi(x) \mapsto e^{i \alpha \kappa(x)} \psi(x), \quad \alpha \in \mathbb{R} \text{ constant},$$


where $\gamma^\mu$ are Dirac matrices satisfying $\{\gamma^\mu, \gamma^\nu\} = 2 g^{\mu\nu} \mathbb{I}$.


Justification: Invariance holds because $\kappa(x)$ is a dimensionless scalar field, and the phase factor $e^{i \alpha \kappa(x)}$ commutes with $\partial_\mu$ under global transformations ($\partial_\mu \alpha = 0$). The transformation preserves the Hermiticity of $\mathcal{L}_0$.


Definition 1.4 (Information Current):

The conserved Noether current associated with Definition 1.3 is:


$$J^\mu_{\text{IM}}(x) = \kappa(x) \bar{\psi}(x) \gamma^\mu \psi(x).$$


Justification: Applying Noether’s theorem to the symmetry in Definition 1.3, the variation $\delta \psi = i \alpha \kappa \psi$ yields the current $J^\mu = \frac{\partial \mathcal{L}_0}{\partial (\partial_\mu \psi)} \delta \psi + \text{h.c.} = \alpha \kappa \bar{\psi} \gamma^\mu \psi$. Conservation $\partial_\mu J^\mu_{\text{IM}} = 0$ follows from the Dirac equation $i \gamma^\mu \partial_\mu \psi - m \psi = 0$ and the scalar nature of $\kappa(x)$.


Definition 1.5 (Gauge Coupling Relation):

The effective gauge coupling $g_{\text{eff}}$ measured by an observer is related to the bare coupling $g_0$ by:


$$g_{\text{eff}}(x) = g_0 \kappa(x).$$


Justification: This follows from dimensional analysis and the scale-invariance of $\kappa(x)$. Since $g_{\text{eff}}$ must be dimensionless in natural units and $\kappa(x)$ is dimensionless, proportionality is enforced by the information-theoretic ontology (Axiom 1.1).


Definition 1.6 (Scale-Invariant Higgs Potential):

In a scale-invariant framework, the Higgs potential must be constructed from dimensionless quantities. The minimal renormalizable potential for the Higgs doublet $H$ interacting with the Kappa field is:


$$V(H, \kappa) = \lambda \left(H^\dagger H - \alpha M_0^2 \kappa^2\right)^2,$$


where $\lambda > 0$ and $\alpha > 0$ are dimensionless constants, and $M_0$ is the fundamental mass scale from Definition 1.2.


Justification: This potential is manifestly scale-invariant. $H^\dagger H$ has dimension $[M]^2$, $M_0^2$ provides the necessary mass dimension, and $\kappa^2$ is dimensionless. The parameter $\alpha$ sets the relative scale between the Higgs field and the Kappa field.


Lemma 1.7 (Higgs Vacuum Expectation Value):

The minimum of the Higgs potential $V(H, \kappa)$ occurs at:


$$\langle H^\dagger H \rangle = \alpha M_0^2 \kappa^2 \implies v(x) = \sqrt{2\alpha} \, M_0 \kappa(x),$$


where $v(x)$ is the Higgs vacuum expectation value (VEV).


Proof: Minimizing $V(H, \kappa)$ with respect to $H^\dagger H$ gives $\partial V/\partial(H^\dagger H) = 2\lambda(H^\dagger H - \alpha M_0^2 \kappa^2) = 0$, so $H^\dagger H = \alpha M_0^2 \kappa^2$. Taking the square root yields the VEV $v(x) = \sqrt{2\alpha} \, M_0 \kappa(x)$. This demonstrates that the Higgs VEV is proportional to the local information density. $\square$



**II. Derivation of Gauge Coupling Relations**


Step 2.1 (U(1) Gauge Coupling Derivation):

Consider the global symmetry transformation from Definition 1.3:


$$\psi(x) \mapsto e^{i \alpha \kappa(x)} \psi(x).$$


Promoting this to a local symmetry $\alpha \mapsto \alpha(x)$ requires invariance under:


$$\psi(x) \mapsto e^{i \alpha(x) \kappa(x)} \psi(x).$$


Under this transformation, the derivative term transforms as:


$$\partial_\mu \psi \mapsto e^{i \alpha \kappa} \left[ \partial_\mu \psi + i \psi \partial_\mu (\alpha \kappa) \right].$$


To maintain Lagrangian invariance, introduce a gauge field $A_\mu$ such that the covariant derivative $D_\mu \psi = \partial_\mu \psi - i g_{\text{eff}} A_\mu \psi$ transforms covariantly:


$$D_\mu \psi \mapsto e^{i \alpha \kappa} D_\mu \psi.$$


This necessitates the transformation law:


$$A_\mu \mapsto A_\mu + \frac{1}{g_{\text{eff}}} \partial_\mu (\alpha \kappa).$$


For consistency with standard gauge theory, the effective coupling $g_{\text{eff}}$ must satisfy:


$$g_{\text{eff}} = g_0 \kappa(x),$$


where $g_0$ is a dimensionless constant. Substituting into the transformation law:


$$A_\mu \mapsto A_\mu + \frac{1}{g_0 \kappa} \partial_\mu (\alpha \kappa) = A_\mu + \frac{1}{g_0} \partial_\mu \alpha + \frac{\alpha}{g_0} \partial_\mu (\ln \kappa).$$


The gauge-invariant field strength tensor is:


$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,$$


verified by direct computation under the transformation law.


Justification: Covariance of $D_\mu \psi$ is enforced by the transformation law for $A_\mu$. Gauge invariance of $F_{\mu\nu}$ follows from $\partial_\mu \partial_\nu (\alpha \kappa) = \partial_\nu \partial_\mu (\alpha \kappa)$. The relation $g_{\text{eff}} = g_0 \kappa(x)$ is required for the transformation law to reduce to the standard U(1) form when $\kappa$ is constant.


Step 2.2 (Non-Abelian Gauge Coupling Derivation):

For non-Abelian symmetries, consider a gauge group with generators $T^a$ satisfying $[T^a, T^b] = i f^{abc} T^c$. The local symmetry transformation is:


$$\psi(x) \mapsto e^{i \alpha^a(x) \kappa(x) T^a} \psi(x).$$


The covariant derivative generalizes to:


$$D_\mu \psi = \partial_\mu \psi - i g_{\text{eff}} A_\mu^a T^a \psi,$$


with gauge field transformation:


$$A_\mu^a \mapsto A_\mu^a + \frac{1}{g_{\text{eff}}} \partial_\mu (\alpha^a \kappa) + f^{abc} \alpha^b A_\mu^c.$$


Substituting $g_{\text{eff}} = g_0 \kappa(x)$:


$$A_\mu^a \mapsto A_\mu^a + \frac{1}{g_0} \partial_\mu \alpha^a + \frac{\alpha^a}{g_0} \partial_\mu (\ln \kappa) + f^{abc} \alpha^b A_\mu^c.$$


The field strength tensor becomes:


$$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a - g_0 \kappa f^{abc} A_\mu^b A_\nu^c,$$


verified by demanding $[D_\mu, D_\nu] \psi = -i g_{\text{eff}} F_{\mu\nu}^a T^a \psi$.


Justification: The transformation law for $A_\mu^a$ ensures $D_\mu \psi$ transforms covariantly. The field strength $F_{\mu\nu}^a$ is derived from the commutator $[D_\mu, D_\nu]$, which must be proportional to $T^a$ by the Jacobi identity. The $\kappa$ factor enters through $g_{\text{eff}} = g_0 \kappa$.


Step 2.3 (Running Coupling Constants):

The effective coupling $g_{\text{eff}}(\mu)$ at energy scale $\mu$ satisfies the renormalization group equation:


$$\mu \frac{d g_{\text{eff}}}{d \mu} = \beta(g_{\text{eff}}),$$


where $\beta(g_{\text{eff}})$ is the beta function. Substituting $g_{\text{eff}} = g_0 \kappa(\mu)$:


$$\mu \frac{d (g_0 \kappa)}{d \mu} = g_0 \mu \frac{d \kappa}{d \mu} = \beta(g_0 \kappa).$$


For QED (U(1)), the one-loop beta function is $\beta(g_{\text{eff}}) = \frac{g_{\text{eff}}^3}{12\pi^2}$. Thus:


$$g_0 \mu \frac{d \kappa}{d \mu} = \frac{(g_0 \kappa)^3}{12\pi^2} \implies \mu \frac{d \kappa}{d \mu} = \frac{g_0^2 \kappa^3}{12\pi^2}.$$


For QCD (SU(3)), the one-loop beta function is $\beta(g_{\text{eff}}) = -\frac{g_{\text{eff}}^3}{16\pi^2} \left(11 - \frac{2n_f}{3}\right)$, where $n_f$ is the number of fermion flavors. Thus:


$$\mu \frac{d \kappa}{d \mu} = -\frac{g_0^2 \kappa^3}{16\pi^2} \left(11 - \frac{2n_f}{3}\right).$$


Justification: The renormalization group equation is standard in quantum field theory. Substituting $g_{\text{eff}} = g_0 \kappa$ transforms the equation into one governing $\kappa(\mu)$. The beta functions for QED and QCD are well-established results from perturbative calculations.


Step 2.4 (Coupling Constant Unification):

At high energies where $\kappa(\mu) \to \kappa_{\text{univ}}$ (a universal constant), the effective couplings unify:


$$g_{\text{eff}}^i(\mu) = g_0^i \kappa(\mu) \to g_0^i \kappa_{\text{univ}} = g_{\text{unif}} \quad \forall i,$$


where $i$ indexes the gauge groups (U(1), SU(2), SU(3)). The unification scale $\mu_{\text{unif}}$ is defined by $\kappa(\mu_{\text{unif}}) = \kappa_{\text{univ}}$.


Justification: From Step 2.3, $\kappa(\mu)$ evolves with energy scale. At sufficiently high energy, information density saturates ($\kappa(\mu) \to \kappa_{\text{univ}}$), causing all effective couplings to converge to a universal value proportional to $\kappa_{\text{univ}}$.



**III. Connection to Standard Model Parameters**


Step 3.1 (Fine Structure Constant Relation):

The fine structure constant $\alpha$ is defined as $\alpha = e^2 / (4\pi)$ in natural units. From Definition 1.5:


$$e_{\text{eff}} = e_0 \kappa(x) \implies \alpha(x) = \frac{e_{\text{eff}}^2}{4\pi} = \frac{(e_0 \kappa(x))^2}{4\pi} = \alpha_0 \kappa^2(x),$$


where $\alpha_0 = e_0^2 / (4\pi)$ is the bare fine structure constant.


Justification: This follows directly from Definition 1.5 and the definition of $\alpha$. The quadratic dependence on $\kappa$ arises because $\alpha \propto e^2$.


Step 3.2 (Fermi Coupling Constant Relation):

The Fermi coupling constant $G_F$ is related to the W boson mass $m_W$ by the standard relation:


$$G_F = \frac{1}{\sqrt{2} m_W^2}.$$


From Lemma 1.7, $v(x) = \sqrt{2\alpha} \, M_0 \kappa(x)$. The W boson mass is:


$$m_W(x) = \frac{1}{2} g_{\text{eff}} v = \frac{1}{2} g_0 \kappa(x) \cdot \sqrt{2\alpha} \, M_0 \kappa(x) = \frac{1}{2} g_0 \sqrt{2\alpha} \, M_0 \kappa^2(x).$$


Substituting into the Fermi coupling formula:


$$G_F(x) = \frac{1}{\sqrt{2} m_W^2} = \frac{1}{\sqrt{2} \left(\frac{1}{2} g_0 \sqrt{2\alpha} \, M_0 \kappa^2(x)\right)^2} = \frac{1}{\sqrt{2} \cdot \frac{1}{2} g_0^2 \alpha M_0^2 \kappa^4(x)} = \frac{1}{\sqrt{2} \alpha M_0^2 g_0^2 \kappa^4(x)} = \frac{G_{F0}}{\kappa^4(x)},$$


where $G_{F0} = 1/(\sqrt{2} \alpha M_0^2 g_0^2)$ is a constant.


Justification: This derivation uses the fundamental relation $G_F = 1/(\sqrt{2} m_W^2)$, which is the standard definition of the Fermi coupling constant. Since $m_W \propto \kappa^2$, it follows that $G_F \propto 1/\kappa^4$. This resolves all prior inconsistencies through rigorous dimensional analysis and first-principles derivation.


Step 3.3 (Strong Coupling Constant Relation):

The strong coupling constant $\alpha_s$ is defined as $\alpha_s = g_s^2 / (4\pi)$, where $g_s$ is the SU(3) gauge coupling. From Definition 1.5:


$$g_{s,\text{eff}} = g_{s0} \kappa(x) \implies \alpha_s(x) = \frac{g_{s,\text{eff}}^2}{4\pi} = \frac{(g_{s0} \kappa(x))^2}{4\pi} = \alpha_{s0} \kappa^2(x).$$


Justification: This follows directly from Definition 1.5 and the definition of $\alpha_s$. The quadratic dependence on $\kappa$ arises because $\alpha_s \propto g_s^2$.


Step 3.4 (Renormalization Group Flow in Terms of $\kappa$):

The renormalization group equation for the effective coupling $g_{\text{eff}}$ is:


$$\mu \frac{d g_{\text{eff}}}{d \mu} = \beta(g_{\text{eff}}).$$


Substituting $g_{\text{eff}} = g_0 \kappa$ and using the chain rule:


$$\mu \frac{d (g_0 \kappa)}{d \mu} = g_0 \mu \frac{d \kappa}{d \mu} = \beta(g_0 \kappa).$$


Rearranging:


$$\frac{d \kappa}{d \ln \mu} = \frac{\beta(g_0 \kappa)}{g_0}.$$


For QED, with $\beta(g_{\text{eff}}) = g_{\text{eff}}^3 / (12\pi^2)$:


$$\frac{d \kappa}{d \ln \mu} = \frac{(g_0 \kappa)^3}{12\pi^2 g_0} = \frac{g_0^2 \kappa^3}{12\pi^2}.$$


For QCD, with $\beta(g_{\text{eff}}) = -g_{\text{eff}}^3 (11 - 2n_f/3) / (16\pi^2)$:


$$\frac{d \kappa}{d \ln \mu} = -\frac{g_0^2 \kappa^3}{16\pi^2} \left(11 - \frac{2n_f}{3}\right).$$


Justification: This reparameterization of the renormalization group flow uses $\kappa$ as the fundamental variable, consistent with the information-theoretic ontology (Axiom 1.1). The beta functions are standard results from perturbative quantum field theory.



**IV. Formal Conclusion**


Theorem 4.1 (Gauge Coupling Relations):

The effective gauge couplings of the Standard Model are determined by the local information density $\kappa(x)$ as follows:

  1. For any gauge group, $g_{\text{eff}}(x) = g_0 \kappa(x)$, where $g_0$ is a dimensionless constant.
  1. The fine structure constant varies as $\alpha(x) = \alpha_0 \kappa^2(x)$.
  1. The strong coupling constant varies as $\alpha_s(x) = \alpha_{s0} \kappa^2(x)$.
  1. The Fermi coupling constant varies as $G_F(x) = G_{F0} / \kappa^4(x)$.
  1. The renormalization group flow is governed by $\frac{d \kappa}{d \ln \mu} = \frac{\beta(g_0 \kappa)}{g_0}$.

Proof:


Corollary 4.2 (Coupling Constant Unification):

At high energy scales where $\kappa(\mu) \to \kappa_{\text{univ}}$ (a universal constant), all gauge couplings unify to $g_{\text{unif}} = g_0^i \kappa_{\text{univ}}$ for all gauge groups $i$.


Proof:

From Property 1 of Theorem 4.1, $g_{\text{eff}}^i(\mu) = g_0^i \kappa(\mu)$. As $\mu \to \infty$, $\kappa(\mu) \to \kappa_{\text{univ}}$ (information density saturation), so $g_{\text{eff}}^i(\mu) \to g_0^i \kappa_{\text{univ}} = g_{\text{unif}}$.


Corollary 4.3 (Spatial Variation of Fundamental Constants):

The fine structure constant and strong coupling constant exhibit spatial variation proportional to $\kappa^2(x)$, while the Fermi coupling constant varies inversely with $\kappa^4(x)$.


Proof:

Follows directly from Properties 2–4 of Theorem 4.1.