---
title: "Derivation of SM Gauge Couplings"
slug: "derivation-of-sm-gauge-couplings"
url: https://papers.qnfo.org/papers/derivation-of-sm-gauge-couplings
authors: ["QNFO Research"]
version: "1.0.0"
published: "2026-07-04"
updated: "2026-10-12"
license: "QNFO-ULA-2.1"
status: "published"
quality_label: "High-quality scholarly research"
quality_basis: "QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review."
peer_review_status: "Not independently peer-reviewed"
quality_evidence_url: "https://qnfo.org/about"
quality_tags: ["High-quality scholarly research", "open-access scholarly work", "versioned research publication"]
---

# Derivation of SM Gauge Couplings

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 (\(\alpha \propto \kappa^2\)).

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

**Author:** Rowan Brad Quni-Gudzinas
**Affiliation:** QNFO **Contact:**
rowan.quni@qnfo.org **ORCID:** 0009-0002-4317-5604
**ISNI:** 0000 0005 2645 6062 **DOI:**
https://papers.qnfo.org/papers/derivation-of-sm-gauge-couplings/ **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.* 2. *The fine structure constant varies as \(\alpha(x) = \alpha_0 \kappa^2(x)\).* 3.
*The strong coupling constant varies as \(\alpha_s(x) = \alpha_{s0}
\kappa^2(x)\).* 4. *The Fermi coupling constant varies as
\(G_F(x) = G_{F0} / \kappa^4(x)\).*
5. *The renormalization group flow is governed by \(\frac{d \kappa}{d \ln \mu} = \frac{\beta(g_0
\kappa)}{g_0}\).*

**Proof:** - Property 1: Established in Definition 1.5
and verified in Steps 2.1–2.2. - Property 2: Derived in Step 3.1 from
\(\alpha = e^2 / (4\pi)\) and \(e_{\text{eff}} = e_0 \kappa\). - Property
3: Derived in Step 3.3 from \(\alpha_s = g_s^2
/ (4\pi)\) and \(g_{s,\text{eff}} =
g_{s0} \kappa\). - Property 4: Derived in Step 3.2 from \(G_F = 1/(\sqrt{2} m_W^2)\), \(m_W = \frac{1}{2} g_{\text{eff}} v\), \(g_{\text{eff}} = g_0 \kappa\), and \(v = \sqrt{2\alpha} \, M_0 \kappa\). -
Property 5: Derived in Step 3.4 by substituting \(g_{\text{eff}} = g_0 \kappa\) into the
renormalization group equation.

**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.