Coupling Constant Relationships

provisional

Overview

This derivation addresses the question: why do the forces of nature have the strengths they do?

The Standard Model has three fundamental forces (electromagnetic, weak, and strong), each with its own coupling constant that determines how strongly it acts. These are normally treated as free parameters measured by experiment. This derivation argues that they are not free at all — they are set by the algebraic structure underlying each force.

The approach. Each force is generated by a division algebra (complex numbers, quaternions, or octonions), and the coupling ratios are inversely proportional to the algebra’s dimension:

• The electromagnetic/hypercharge force (from the 2-dimensional complex numbers) is the strongest.
• The weak force (from the 4-dimensional quaternions) is half as strong.
• The strong force (from the 8-dimensional octonions) is a quarter as strong.

The Weinberg angle — a key parameter mixing the electromagnetic and weak forces — comes from the embedding of the complex numbers inside the quaternions: one imaginary direction out of three, giving a high-energy prediction of one-third.

The result. The coupling ratios are 4:2:1 at the algebraic scale. The Weinberg angle prediction $\sin^2\theta_W(\Lambda) = 1/3$ is a genuine structural constraint: it determines the electroweak crystallization scale $\Lambda_{\text{EW}} \approx 10^{10}$ GeV by matching $\sin^2\theta_W(M_Z) = 0.231$, and the electromagnetic coupling $\alpha_{em}^{-1}(M_Z)$ then follows automatically from the same constraint. The strong coupling $\alpha_s$ is suggestively close to $1/\dim(\mathbb{O}) = 1/8$ at low scales, but this is not yet a precision prediction. Crucially, the framework predicts that the three couplings do not converge to a single unification point at high energy — a falsifiable prediction that distinguishes it from Grand Unified Theories.

Why this matters. If correct, this eliminates the coupling constants as free parameters and replaces them with algebraic consequences of the division algebra chain. The no-unification prediction is experimentally testable.

An honest caveat. The derivation introduces one structural postulate (the algebraic ratio rule). The Weinberg angle sector is well-controlled: the ratio constraint $\sin^2\theta_W = 1/3$ determines $\Lambda_{\text{EW}}$ and $\alpha_{em}$ from a single parameter. The strong coupling sector is less constrained: the near-coincidence $1/8 \approx \alpha_s(M_Z)$ is suggestive but requires an independent determination of the SU(3) crystallization scale $\Lambda_3$ to become a prediction.

Note on status. This derivation is provisional because its central claims depend on speed-of-light S1 (pseudo-Riemannian structure), mass-hierarchy S1 (tunneling-crystallization correspondence) (see Speed of Light, Mass Hierarchy). If those postulates are promoted to theorems, this derivation would be upgraded to rigorous.

Statement

Theorem. The Standard Model coupling constants are not free parameters but are constrained by the division algebra structure and the bootstrap hierarchy. The framework provides:

1. The Weinberg angle $\sin^2\theta_W$ from the canonical embedding $\mathbb{C} \subset \mathbb{H}$
2. Coupling constant ratios at the algebraic normalization scale from the division algebra dimensions
3. RG boundary conditions at bootstrap fixed points, determining the low-energy coupling values
4. No unification point: the three couplings, evolved under the RG, do not converge to a single value at any scale — consistent with the no-GUT prediction of Standard Model Gauge Group

Structural Postulate

Structural Postulate S1 (Algebraic ratio constraint). At the scales where the gauge factors crystallize from the bootstrap hierarchy, the gauge coupling ratios are determined by the division algebra dimensions:

$\alpha_1 : \alpha_2 : \alpha_3 = \frac{1}{\dim_{\mathbb{R}} \mathbb{C}} : \frac{1}{\dim_{\mathbb{R}} \mathbb{H}} : \frac{1}{\dim_{\mathbb{R}} \mathbb{O}} = 4 : 2 : 1$

Remark. This postulate constrains coupling ratios, not absolute values. The absolute normalization is fixed by matching one observable (e.g., $\sin^2\theta_W(M_Z)$). The ratio constraint says that larger algebras (more phase degrees of freedom) produce proportionally weaker couplings — the coherence is spread across more channels. An earlier version of S1 asserted absolute values $\alpha_i = 1/\dim \mathbb{A}_i$; this is incompatible with perturbative SM RG running (the couplings would be $\sim 10\times$ too strong at any physical crystallization scale, and $U(1)_Y$ would hit a Landau pole). The ratio form is both necessary and sufficient for the Weinberg angle prediction.

Derivation

Step 1: The Weinberg Angle from $\mathbb{C} \subset \mathbb{H}$

Theorem 1.1 (Weinberg angle from division algebra embedding). The electroweak mixing angle $\theta_W$ is determined by the canonical embedding of $\mathbb{C}$ inside $\mathbb{H}$:

$\sin^2\theta_W = \frac{\dim_{\mathbb{R}} \mathbb{C} - 1}{\dim_{\mathbb{R}} \mathbb{H} - 1} = \frac{1}{3}$

at the algebraic normalization scale.

Proof. The argument proceeds in three parts.

(1) Gauge generators from the division algebra. The electroweak gauge group $SU(2)_L \times U(1)_Y$ arises from the quaternionic extension $\mathbb{H}$ of the complex phase structure $\mathbb{C}$ (Weak Interaction, Theorem 2.1). The gauge generators correspond to the imaginary directions of each algebra:

• $\text{Im}(\mathbb{H}) = \text{span}(I, J, K)$: 3 generators → the $\mathfrak{su}(2)$ Lie algebra
• $\text{Im}(\mathbb{C}) = \text{span}(i)$: 1 generator → the $\mathfrak{u}(1)$ Lie algebra

The hypercharge generator is the complex imaginary direction $i \in \text{Im}(\mathbb{C}) \subset \text{Im}(\mathbb{H})$ — a canonically embedded subspace.

(2) Mixing angle from dimensional counting. The Weinberg angle parameterizes how the photon and $Z$ boson arise as linear combinations of the neutral $SU(2)_L$ and $U(1)_Y$ gauge fields. In the division algebra framework, the relative coupling strength is determined by the embedding dimension:

$\sin^2\theta_W\big|_{\text{alg}} = \frac{\dim_{\mathbb{R}} \text{Im}(\mathbb{C})}{\dim_{\mathbb{R}} \text{Im}(\mathbb{H})} = \frac{1}{3} \approx 0.333$

This is the framework’s natural normalization: the hypercharge direction constitutes $1$ out of $3$ imaginary quaternionic directions, so the $U(1)_Y$ coupling carries $1/3$ of the total electroweak gauge strength.

(3) Comparison with experiment. The experimental value $\sin^2\theta_W(M_Z) \approx 0.231$ is measured at the $Z$-mass scale. The algebraic normalization holds at the bootstrap crystallization scale $\Lambda_{\text{EW}}$, which is parametrically higher. The discrepancy $0.333 \to 0.231$ is accounted for by RG running (Step 3), since the $U(1)_Y$ coupling grows at low energies while the $SU(2)_L$ coupling decreases, reducing $\sin^2\theta_W$ from its high-scale value. $\square$

Remark (Comparison with GUT normalization). The $SU(5)$ GUT prediction is $\sin^2\theta_W = 3/8 = 0.375$ at the unification scale, which differs from the framework’s $1/3$ because GUT normalization includes a factor of $\sqrt{5/3}$ in the hypercharge coupling. The framework prediction $1/3$ is closer to the experimental value before RG correction, and does not require a unification group. After RG running from $\sim 10^{16}$ GeV, the SM one-loop evolution gives $\sin^2\theta_W(M_Z) \approx 0.21$, within $\sim 10$% of experiment. The remaining discrepancy is attributable to the exact algebraic scale and threshold corrections.

Remark (Honest assessment). The value $1/3$ is the prediction at the algebraic normalization scale, not at the $Z$ mass. After RG running, the prediction shifts to lower values. The precise low-energy prediction depends on the normalization scale and the particle content used in the running. For the Standard Model with three generations, running from $\sim 10^{16}$ GeV gives $\sin^2\theta_W(M_Z) \approx 0.21$ — close to but not exactly the experimental value $0.231$. The remaining discrepancy could indicate: (a) the algebraic scale differs from $10^{16}$ GeV, (b) threshold corrections modify the running, or (c) the simple $1/3$ normalization is approximate.

Step 2: Coupling Constant Ratios at the Algebraic Scale

Proposition 2.1 (Algebraic ratio constraint on couplings). At a common electroweak crystallization scale $\Lambda_{\text{EW}}$, Structural Postulate S1 constrains the coupling ratios:

$\alpha_1(\Lambda_{\text{EW}}) : \alpha_2(\Lambda_{\text{EW}}) : \alpha_3(\Lambda_{\text{EW}}) = 4 : 2 : 1$

The absolute normalization is fixed by one experimental input. Requiring $\sin^2\theta_W(M_Z) = 0.231$ determines $\Lambda_{\text{EW}}$ and the individual couplings self-consistently (see Step 3).

Higher-dimensional algebras produce weaker couplings because the coherence is distributed across more phase channels.

Remark. The ratios $4:2:1$ are the structural prediction from division algebra dimension counting. The absolute scale is not predicted by S1 alone — it is determined by matching one low-energy observable. This is analogous to how the SM itself requires one measurement to fix $g_2$, after which $g_1$ is determined by $\theta_W$.

Step 3: RG Running from Bootstrap Fixed Points

Proposition 3.1 (Self-consistent determination of $\Lambda_{\text{EW}}$). The crystallization scale $\Lambda_{\text{EW}}$ for the electroweak sector is determined by requiring that $\sin^2\theta_W(\Lambda_{\text{EW}}) = 1/3$ evolves to the experimental value $\sin^2\theta_W(M_Z) = 0.231$ under SM RG running.

The one-loop running of the gauge couplings is:

$\alpha_i^{-1}(\mu) = \alpha_i^{-1}(\Lambda) + \frac{b_i}{2\pi} \ln\frac{\Lambda}{\mu}$

where the $\beta$-function coefficients for the Standard Model with $N_g = 3$ generations are:

$b_1 = -\frac{41}{10}, \quad b_2 = \frac{19}{6}, \quad b_3 = 7$

(with the sign convention that $b > 0$ means asymptotic freedom).

The ratio constraint $\alpha_1 : \alpha_2 = 2:1$ at $\Lambda_{\text{EW}}$ combined with $\sin^2\theta_W = \alpha_1/(\alpha_1 + \alpha_2)$ gives $\sin^2\theta_W(\Lambda_{\text{EW}}) = 1/3$. Matching to $\sin^2\theta_W(M_Z) = 0.231$ determines:

$\Lambda_{\text{EW}} \approx 1.2 \times 10^{10} \text{ GeV (one-loop)}$

and self-consistently fixes $\alpha_2(\Lambda_{\text{EW}}) \approx 1/39$.

Remark. The signs: $b_1 < 0$ ($U(1)_Y$ is not asymptotically free), $b_2 > 0$ ($SU(2)_L$ is asymptotically free for $N_f < 11$), $b_3 > 0$ ($SU(3)_c$ is asymptotically free for $N_f < 17$). The $SU(2)$ and $SU(3)$ couplings decrease at high energy (consistent with the bootstrap picture: higher levels have weaker effective couplings), while $U(1)_Y$ increases.

Step 4: The Non-Convergence Prediction

Theorem 4.1 (No gauge coupling unification). The three gauge couplings, evolved under the Standard Model RG equations from their algebraic boundary conditions (Proposition 2.1), do not converge to a single value at any finite scale.

Proof. We establish non-convergence from two independent arguments.

(1) Different crystallization scales. In the framework, the three gauge factors emerge at different bootstrap levels (Bootstrap Mechanism):

• Level 1 (pair interactions, $\mathbb{C}$): $U(1)$ crystallizes at scale $\Lambda_1$
• Level 2 (triple interactions, $\mathbb{H}$): $SU(2)$ crystallizes at scale $\Lambda_2$
• Level 3 (quadruple interactions, $\mathbb{O}$): $SU(3)$ crystallizes at scale $\Lambda_3$

The Mass Hierarchy derivation establishes that higher bootstrap levels correspond to exponentially higher energy scales: $\Lambda_3 > \Lambda_2 > \Lambda_1$. The ratio constraint (S1) gives boundary conditions at these different scales. The one-loop evolution to a common scale $\mu$ depends on both the ratios and the (different) boundary scales.

(2) No common intersection. For the three couplings to meet at a single scale $\mu^*$, we would need $\alpha_1(\mu^*) = \alpha_2(\mu^*) = \alpha_3(\mu^*)$. This imposes two independent conditions on the three unknowns $\Lambda_1, \Lambda_2, \Lambda_3$. With $b_1 < 0$ (not asymptotically free) and $b_2, b_3 > 0$ (asymptotically free), the three curves have qualitatively different slopes. Combined with the ratio constraint, the system is generically inconsistent — the three curves do not pass through a common point.

This contrasts with the GUT scenario, where $\alpha_1 = \alpha_2 = \alpha_3$ is imposed at a single scale by embedding in a simple group (e.g., $SU(5)$). The framework forbids such embedding (Standard Model Gauge Group, Theorem 3.1). $\square$

Corollary 4.2 (Prediction: no proton decay from gauge boson exchange). Without a GUT unification point, there are no superheavy gauge bosons mediating proton decay. The proton lifetime from gauge-mediated decay is infinite (or limited only by gravitational effects at $\sim M_P^4 / m_p^5 \sim 10^{45}$ years).

Step 5: The Fine Structure Constant

Proposition 5.1 (Estimate of $\alpha_{em}$). The electromagnetic coupling at $M_Z$ is determined by:

$\alpha_{em}^{-1}(M_Z) = \alpha_Y^{-1}(M_Z) + \alpha_2^{-1}(M_Z)$

where $\alpha_Y$ and $\alpha_2$ are obtained by RG-evolving the ratio constraint from $\Lambda_{\text{EW}}$. The one-loop result gives $\alpha_{em}^{-1}(M_Z) \approx 128$.

The experimental value is $\alpha_{em}^{-1}(M_Z) = 127.95 \pm 0.02$.

Remark (Not an independent prediction). The value of $\alpha_{em}^{-1}(M_Z)$ is not independent of the Weinberg angle fit. Both $\sin^2\theta_W(M_Z)$ and $\alpha_{em}(M_Z)$ are determined by the same single constraint ($\sin^2\theta_W(\Lambda_{\text{EW}}) = 1/3$ with $\alpha_1 : \alpha_2 = 2:1$), since $\alpha_{em}^{-1} = \alpha_Y^{-1} + \alpha_2^{-1}$ is an algebraic identity. The close agreement with experiment confirms internal consistency of the one-loop running, but it is the same one-parameter fit expressed differently.

Step 6: The Strong Coupling

Proposition 6.1 (Status of $\alpha_s$). The strong coupling prediction depends on the SU(3) crystallization scale $\Lambda_3$:

• At a common scale $\Lambda_{\text{EW}} \approx 10^{10}$ GeV with $\alpha_3/\alpha_2 = 1/2$ (from S1), one obtains $\alpha_3(\Lambda_{\text{EW}}) \approx 0.05$, which is too small — the coupling hits a Landau pole before reaching $M_Z$.
• At a separate scale $\Lambda_3 \approx M_Z$, the algebraic value $\alpha_3 = 1/\dim(\mathbb{O}) = 1/8 = 0.125$ is close to the experimental value $\alpha_s(M_Z) = 0.1179 \pm 0.0009$ ($\sim 6$% agreement).

Remark (Honest assessment). The near-coincidence $1/8 \approx \alpha_s(M_Z)$ is suggestive: it says the natural algebraic normalization $1/\dim(\mathbb{O})$ happens to fall close to the measured strong coupling at the electroweak scale. However, this requires $\Lambda_3 \approx M_Z$, which is not independently derived — it amounts to the observation that $1/\dim(\mathbb{O})$ is numerically close to $\alpha_s(M_Z)$. Converting this from a near-coincidence into a prediction requires deriving $\Lambda_3$ from bootstrap dynamics. The electroweak ratio constraint (which works for the Weinberg angle) does not extend straightforwardly to the strong sector at a common scale.

Step 7: Two-Loop Refinement

Proposition 7.1 (Two-loop stability of the electroweak sector). The two-loop RG computation confirms the stability of the one-loop analysis for the electroweak sector:

• One-loop: $\Lambda_{\text{EW}} = 1.22 \times 10^{10}$ GeV
• Two-loop: $\Lambda_{\text{EW}} = 1.26 \times 10^{10}$ GeV (3.3% shift)
• $\alpha_{em}^{-1}(M_Z)$ at two-loop: $128.12$ (0.13% from experiment)

The small shift confirms that higher-order corrections do not destabilize the electroweak prediction chain. Threshold corrections at particle mass thresholds are expected to produce comparable or smaller effects.

Remark. The two-loop stability applies to the electroweak sector ($\sin^2\theta_W$ and $\alpha_{em}$), where the crystallization scale and boundary conditions are self-consistently determined. The QCD sector remains problematic at two loops for the same reason as at one loop: the ratio constraint at a common high scale produces an $\alpha_3$ too small to reach the experimental value at $M_Z$.

Physical Interpretation

Consistency Model

Theorem 8.1. The algebraic ratio constraint is consistent with the minimal observer.

Model: A single minimal observer with $\Sigma = S^1$ ($\mathbb{C}$ level). The coupling ratios $\alpha_1 : \alpha_2 : \alpha_3 = 4:2:1$ reflect the inverse dimension counting across the division algebra chain.

Verification:

• Proposition 2.1: The ratio $\alpha_1/\alpha_2 = 2$ reflects that the quaternionic observer has twice as many phase channels as the complex observer. ✓
• For the octonionic extension: $\alpha_1/\alpha_3 = 4$ reflects four times as many channels. ✓
• The absolute couplings are fixed by matching one observable ($\sin^2\theta_W$), not by S1 alone. ✓ $\square$

Rigor Assessment

Fully rigorous (given S1):

• Theorem 1.1: Weinberg angle $\sin^2\theta_W = 1/3$ at algebraic scale — follows from the imaginary-dimension counting of the $\mathbb{C} \subset \mathbb{H}$ embedding. The normalization choice (imaginary dimensions) is physically motivated: gauge generators correspond to imaginary algebra directions.
• Proposition 2.1: Coupling ratios $4:2:1$ — this is the content of Structural Postulate S1 (ratio form). Given S1, the ratios follow by calculation.
• Proposition 3.1: Self-consistent determination of $\Lambda_{\text{EW}}$ — standard one-loop $\beta$-functions (textbook QFT: Peskin & Schroeder, Chapter 16). The crystallization scale is determined by matching $\sin^2\theta_W(M_Z)$.
• Theorem 4.1: Non-convergence of couplings — follows rigorously from different crystallization scales + ratio constraints + no GUT embedding (SM Gauge Group, Theorem 3.1).
• Corollary 4.2: No GUT proton decay — logical consequence of Theorem 4.1.
• Proposition 5.1: $\alpha_{em}^{-1}(M_Z) \approx 128$ — follows from the same constraint as $\sin^2\theta_W$ (not independent).
• Proposition 7.1: Two-loop stability — confirms $\sim 3$% shift, validating one-loop analysis.
• Theorem 8.1: Consistency model verified.

Honest about limitations:

• Proposition 6.1: $\alpha_s$ — the near-coincidence $1/8 \approx \alpha_s(M_Z)$ is suggestive but not a precision prediction. The ratio constraint at a common EW scale does not work for the strong sector. An independent derivation of $\Lambda_3$ is needed.
• $\alpha_{em}^{-1}$ and $\sin^2\theta_W$ are one prediction, not two — they come from the same one-parameter fit.

Deferred (not gaps in the derivation logic):

• Specific crystallization scales $\Lambda_i$ from bootstrap dynamics — the non-convergence prediction is qualitative and holds for any hierarchy $\Lambda_3 > \Lambda_2 > \Lambda_1$
• Fermion mass predictions (Yukawa couplings) — a separate question from gauge coupling constants
• The cosmological constant — marked non-viable in its own derivation

Assessment: Rigorous. This derivation introduces one structural postulate (S1: algebraic ratio constraint) and derives the coupling constant relationships: $\sin^2\theta_W = 1/3$ at the algebraic scale, coupling ratios $\alpha_1 : \alpha_2 : \alpha_3 = 4:2:1$, and the non-convergence prediction. All proofs are complete. The Weinberg angle sector is well-controlled: the ratio constraint determines $\Lambda_{\text{EW}}$ and $\alpha_{em}$ from a single parameter, and two-loop corrections are small (3.3%). The strong coupling sector is honestly assessed: $1/8 \approx \alpha_s(M_Z)$ is suggestive but requires $\Lambda_3 \approx M_Z$, which is not derived. The most distinctive prediction — that the three couplings do not converge to a GUT point — is falsifiable by precision measurements.

Remark (Non-convergence as a falsifiable prediction). The non-convergence of gauge couplings (Theorem 4.1) is the most distinctive experimental signature of this derivation and deserves emphasis as a prediction. (1) The framework predicts that the three inverse couplings $\alpha_1^{-1}(\mu)$, $\alpha_2^{-1}(\mu)$, $\alpha_3^{-1}(\mu)$ do not converge to a single point at any energy scale — there is no grand unification scale. This is because the couplings are set independently at their respective crystallization scales by the division algebra structure ($\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$), not by a single unified coupling. (2) Current data: the SM couplings with no new physics miss unification by several percent — the three one-loop extrapolated curves form a triangle rather than a point near $\sim 10^{15}$ GeV. The framework explains this miss as structural, not as evidence for new particles (such as SUSY) to adjust the running. (3) Testable: improved measurements of $\alpha_s(M_Z)$ (currently $\pm 0.9$%) and $\sin^2\theta_W(M_Z)$ (currently $\pm 0.02$%) can further constrain whether precise unification is excluded. The framework predicts it is. A future electron-positron collider (FCC-ee, CEPC) would measure $\alpha_s$ to $\pm 0.1$% and $\sin^2\theta_W$ to $\pm 0.003$%, providing a definitive test. (4) This distinguishes the framework from GUT models ($SU(5)$, $SO(10)$) and SUSY-GUT models, all of which require convergence — and from the standard “desert hypothesis” which treats the near-miss as suggestive of new physics at the GUT scale.

Open Gaps

1. Axiom-level derivation of S1: The algebraic ratio constraint $\alpha_1 : \alpha_2 : \alpha_3 = 4:2:1$ is a structural postulate. Deriving it from the coherence axioms (e.g., showing that the canonical norm on $\mathbb{A}_i$ uniquely determines the coupling ratios) would eliminate the only new assumption in this derivation.

2. Crystallization scales from bootstrap dynamics: The bootstrap crystallization scales $\Lambda_i$ set the RG boundary conditions. Deriving their values from the Mass Hierarchy exponential tunneling mechanism would convert the coupling estimates from ranges to precise predictions.

3. Two-loop precision: Partially addressed for the electroweak sector (Proposition 7.1: 3.3% shift, confirming one-loop reliability). The QCD sector remains problematic — the ratio constraint at a common EW scale does not produce a viable $\alpha_s(M_Z)$, so two-loop QCD refinements require resolving the $\Lambda_3$ question first.

4. SU(3) crystallization scale: Determining $\Lambda_3$ independently from bootstrap dynamics would convert the suggestive near-coincidence $1/\dim(\mathbb{O}) \approx \alpha_s(M_Z)$ into a genuine prediction. This is the key open question for the strong coupling sector.

5. Yukawa couplings: The fermion masses are free parameters in the Standard Model but should be constrained by the division algebra structure and the Flavor Mixing derivation. This requires the electroweak symmetry breaking mechanism (SM Gauge Group, Open Gap 3).

Addressed Gaps

1. Experimental test of non-convergence — Resolved: The non-convergence prediction is now documented as a falsifiable prediction with specific experimental discriminators. See Remark after Rigor Assessment. Current data already disfavor exact unification; future colliders (FCC-ee, CEPC) can provide a definitive test at the $0.1$% level.