Derivation of SM Gauge Couplings

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:

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 \(x\), \(kB\) is Boltzmann’s constant, and \(M0\) 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}{kB T M0^3},\]

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) = g0

\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

M0^2 \kappa^2 \implies v(x) = \sqrt{2\alpha} \, M0

\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}} = g0

\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}{g0} \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}} = g0

\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}} = g0

\kappa(x)\):

\[A\mu^a \mapsto A\mu^a + \frac{1}{g_0}

\partial\mu \alpha^a + \frac{\alpha^a}{g0} \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 - g0 \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 \(g{\text{eff}} = g0 \kappa(\mu)\):

\[\mu \frac{d (g0 \kappa)}{d \mu} = g0

\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{2nf}{3}\right)\), where \(nf\) 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 \(g{\text{eff}} = g0 \kappa\) 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) = g0^i

\kappa(\mu) \to g0^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}} = e0 \kappa(x) \implies

\alpha(x) = \frac{e{\text{eff}}^2}{4\pi} = \frac{(e0

\kappa(x))^2}{4\pi} = \alpha_0 \kappa^2(x),\]

where \(\alpha0 = e0^2 / (4\pi)\)

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:

\[mW(x) = \frac{1}{2} g{\text{eff}} v =

\frac{1}{2} g0 \kappa(x) \cdot \sqrt{2\alpha} \, M0 \kappa(x) =

\frac{1}{2} g0 \sqrt{2\alpha} \, M0 \kappa^2(x).\]

Substituting into the Fermi coupling formula:

\[GF(x) = \frac{1}{\sqrt{2} mW^2} =

\frac{1}{\sqrt{2} \left(\frac{1}{2} g0 \sqrt{2\alpha} \, M0

\kappa^2(x)\right)^2} = \frac{1}{\sqrt{2} \cdot \frac{1}{2} g_0^2 \alpha

M0^2 \kappa^4(x)} = \frac{1}{\sqrt{2} \alpha M0^2 g_0^2 \kappa^4(x)} =

\frac{G_{F0}}{\kappa^4(x)},\]

where \(G{F0} = 1/(\sqrt{2} \alpha M0^2

g_0^2)\) is a constant.

Justification: This derivation uses the fundamental relation

\(GF = 1/(\sqrt{2} mW^2)\), 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 \(\alphas = gs^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 \alphas(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}} = g0

\kappa\) and using the chain rule:

\[\mu \frac{d (g0 \kappa)}{d \mu} = g0

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

Rearranging:

\[\frac{d \kappa}{d \ln \mu} =

\frac{\beta(g0 \kappa)}{g0}.\]

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 g0} = \frac{g0^2 \kappa^3}{12\pi^2}.\]

For QCD, with \(\beta(g_{\text{eff}}) =

-g{\text{eff}}^3 (11 - 2nf/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, \(g{\text{eff}}(x) = g0 \kappa(x)\), 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 \(\alphas(x) = \alpha{s0}

\kappa^2(x)\). 4. The Fermi coupling constant varies as

\(GF(x) = G{F0} / \kappa^4(x)\).*

1. *The renormalization group flow is governed by \(\frac{d \kappa}{d \ln \mu} = \frac{\beta(g_0

\kappa)}{g_0}\).*

Proof: - 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 \(e{\text{eff}} = e0 \kappa\). - Property

3: Derived in Step 3.3 from \(\alphas = gs^2

/ (4\pi)\) and \(g_{s,\text{eff}} =

g{s0} \kappa\). - Property 4: Derived in Step 3.2 from \(GF = 1/(\sqrt{2} mW^2)\), \(mW = \frac{1}{2} g{\text{eff}} v\), \(g{\text{eff}} = g0 \kappa\), and \(v = \sqrt{2\alpha} \, M0 \kappa\). -

Property 5: Derived in Step 3.4 by substituting \(g{\text{eff}} = g0 \kappa\) 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}} = g0^i

\kappa_{\text{univ}}\) for all gauge groups \(i\).*

Proof: From Property 1 of Theorem 4.1, \(g{\text{eff}}^i(\mu) = g0^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.