Gauge Couplings of the Standard Model from First Principles in the Octonionic Framework

1. Introduction

The octonionic programme aims at a pre-quantum, pre-spacetime description of elementary physics in which quantum field theory and classical spacetime both emerge from an underlying matrix-valued dynamics. In this approach, the fundamental variables are matrices living on an octonionic or split-bioctonionic spinor space and evolving in Connes time. The corresponding framework is a version of trace dynamics: one writes a trace Lagrangian for matrix degrees of freedom and does not impose canonical commutators at the outset. Instead, ordinary quantum theory is expected to emerge in an appropriate thermodynamic limit.

This framework has led to a sequence of concrete phenomenological claims. The 2022 Eur. Phys. J. Plus paper [1] proposed a derivation of the low-energy fine-structure constant together with fermion mass ratios from Jordan eigenvalues and the octonionic state structure. The 2022 bosonic-Lagrangian paper [2] derived the weak mixing angle from a spinorial rotation between the $(B,W_3)$ and $(\gamma ,Z^0)$ sectors. More recently, the visible gauge-sector relation ${\alpha }_s/{\alpha }_{\mathrm{em}}=16$ has emerged as the missing piece that connects the earlier Eur. Phys. J. Plus fine-structure seed to the strong coupling.

The purpose of the present paper is to make that gauge-sector chain readable in one place. The emphasis is not on presenting new ultraviolet dynamics, but on giving a clear and pedagogical account of how the various ingredients fit together. In particular, we want to answer three questions.

1.

Where does the factor 16 come from inside the visible bosonic sector?

2.

How does one go from the earlier Eur. Phys. J. Plus seed [1] to separate formulas for ${\alpha }_s\left(M_Z\right)$ and ${\alpha }_{\mathrm{em}}\left(0\right)$?

3.

Why is it legitimate to anchor the seed on the charge quantum $q_0=1/3$ even though the electron carries charge $1=3q_0$?

2. Trace-Dynamics Background and the Earlier Eur. Phys. J. Plus Seed

The gauge-sector story begins with the bosonic part of the trace-dynamics Lagrangian. In the notation of Ref. [2], one introduces bosonic and fermionic matrix variables $Q_B$ and $Q_F$, and then writes the bosonic velocity in the form

$${\dot{Q}}_B={\displaystyle \frac{1}{L}}\left(i\alpha q_B+L{\dot{q}}_B\right).$$

(1)

Here L is a characteristic length scale and $\alpha$ is a dimensionless parameter generated when scale invariance is broken. At the schematic level, the visible bosonic block takes the form

$${\mathcal{L}}_{\mathrm{vis}}\sim {\displaystyle \frac{L_P^2}{2L^2}}\mathrm{Tr}\left({\dot{q}}_B^{†}+{\displaystyle \frac{i\alpha }{L}}{q}_B^{†}\right)\left({\dot{q}}_B+{\displaystyle \frac{i\alpha }{L}}{q}_B\right).$$

(2)

When this is expanded, one finds three qualitatively different structures:

1.

a pure visible block $q_B^{†}{q}_B$, from which the photon and the gluons are identified;

2.

a mixed block $q_B^{†}{\dot{q}}_B$, from which the weak bosons are identified;

3.

fermionic terms and mixed boson-fermion terms, not needed for the present note.

This split is important. In the present gauge-sector package, color and electromagnetism are read from the same visible block, whereas the weak bosons are associated with the mixed sector.

The earlier Eur. Phys. J. Plus paper [1] then identifies the dimensionless coefficient of the charged visible terms as

$$C\equiv {\alpha }^2{\displaystyle \frac{L_P^4}{L^4}}.$$

(3)

To avoid confusion with the gauge couplings ${\alpha }_s$ and ${\alpha }_{\mathrm{em}}$, we denote the earlier exponential seed by A rather than by $\alpha$. In the Eur. Phys. J. Plus paper [1], the anti-down sector is used to define

$$lnA={\lambda }_{ad}{q}_{ad},q_{ad}={\displaystyle \frac{1}{3}},{\lambda }_{ad}={\displaystyle \frac{1}{3}}-\sqrt{{\displaystyle \frac{3}{8}}},$$

(4)

so that

$$A=exp\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{3}}-\sqrt{{\displaystyle \frac{3}{8}}}\right)\right].$$

(5)

The same Eur. Phys. J. Plus paper [1] then arrives at the low-energy fine-structure expression

$${\alpha }_{\mathrm{em}}^{\mathrm{EPJP}}\left(0\right)=A^2{\left({\displaystyle \frac{3}{32}}\right)}^2={\displaystyle \frac{9}{1024}}exp\left[{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{1}{3}}-\sqrt{{\displaystyle \frac{3}{8}}}\right)\right].$$

(6)

Numerically, this works remarkably well. Conceptually, however, the origin of the extra factor $1/16$ inside ${(3/32)}^2$ is not as transparent as it could be, because in the original Eur. Phys. J. Plus presentation [1] it is tied to a length-identification step. The aim of the present paper is to separate that bookkeeping into two cleaner pieces:

$${\left({\displaystyle \frac{3}{32}}\right)}^2={\left({\displaystyle \frac{3}{8}}\right)}^2\times {\displaystyle \frac{1}{16}}.$$

(7)

We will interpret $3/8$ as the primary charged-sector datum and $1/16$ as the visible broken-phase gauge normalization.

3. Broken-Phase Derivation of ${\alpha }_s/{\alpha }_{\mathrm{em}}=16$

3.1. The Standard Visible Charge-Trace Factor $8/3$

Before introducing any octonionic support hypothesis, one first recalls a standard one-generation normalization. For visible matter with quark charges $2/3$ and $-1/3$ and charged-lepton charge $-1$, the electromagnetic charge trace is

$$k_{\mathrm{em}}=3\left[{\left({\displaystyle \frac{2}{3}}\right)}^2+{\left({\displaystyle \frac{1}{3}}\right)}^2\right]+1={\displaystyle \frac{8}{3}}.$$

(8)

The factor 3 counts color, and the final $+1$ is precisely the charged-lepton contribution. If the visible bosonic sector carries a single Yang–Mills coupling g before symmetry breaking, then the canonically normalized abelian coupling at equal support is

$$e_0={\displaystyle \frac{g}{\sqrt{k_{\mathrm{em}}}}}=\sqrt{{\displaystyle \frac{3}{8}}}g,$$

(9)

so that

$${\displaystyle \frac{{\alpha }_s}{{\alpha }_{\mathrm{em}}^{\left(0\right)}}}={\displaystyle \frac{g^2}{e_0^2}}={\displaystyle \frac{8}{3}}.$$

(10)

This $8/3$ factor is conventional. The new step is the broken-phase support factor.

3.2. The Six Real Ladder Directions and the Support Factor 6

The visible ladder operators employed in the octonionic construction are [2,3]

$${\alpha }_1={\displaystyle \frac{-e_5+i{e}_4}{2}},{\alpha }_2={\displaystyle \frac{-e_3+i{e}_1}{2}},{\alpha }_3={\displaystyle \frac{-e_6+i{e}_2}{2}}.$$

(11)

They single out the real six-dimensional space

$$H_6:={\mathrm{span}}_{\mathbb{R}}\{e_1,e_2,e_3,e_4,e_5,e_6\}.$$

(12)

In the visible bosonic block, the gluon representatives are built from antisymmetric bilinears involving exactly these six directions. This suggests treating $H_6$ as the relevant broken-phase support space for the visible gauge sector.

We now make the effective broken-phase hypothesis used throughout this paper.

Choose an orthonormal support basis ${\chi }_1,\dots ,{\chi }_6$ for $H_6$. The normalized abelian trace mode is then

$${\psi }_{\mathrm{em}}={\displaystyle \frac{1}{\sqrt{6}}}\sum _{A=1}^6{\chi }_A,$$

(13)

whereas a minimal localized color mode and the relevant matter zero mode may be taken as

$${\psi }_c={\chi }_1,\eta ={\chi }_1.$$

(14)

Modeling the effective couplings by support overlaps gives

$$g_s\propto g\langle \eta ,{\psi }_c\rangle ,e\propto e_0\langle \eta ,{\psi }_{\mathrm{em}}\rangle .$$

(15)

Using Equations (13) and (14), one finds

$$\langle \eta ,{\psi }_c\rangle =1,\langle \eta ,{\psi }_{\mathrm{em}}\rangle ={\displaystyle \frac{1}{\sqrt{6}}},$$

(16)

so that

$$g_s=g,e={\displaystyle \frac{e_0}{\sqrt{6}}}.$$

(17)

This is the origin of the support factor 6.

3.3. The Ratio 16

Combining Equations (9) and (17) gives

$$e=\sqrt{{\displaystyle \frac{3}{8}}}{\displaystyle \frac{g}{\sqrt{6}}}={\displaystyle \frac{g}{4}},$$

(18)

so that

$$\begin{array}{|c|}\hline {\displaystyle \frac{{\alpha }_s}{{\alpha }_{\mathrm{em}}}}=16.\\ \hline\end{array}$$

(19)

Equivalently,

$${\displaystyle \frac{e^2}{g^2}}={\displaystyle \frac{3}{8}}\times {\displaystyle \frac{1}{6}}={\displaystyle \frac{1}{16}}.$$

(20)

The factor $3/8$ is the standard visible charge-trace normalization, and the factor $1/6$ is the broken-phase support dilution of the democratic electromagnetic mode.

It is important to be honest about the status of this result. The derivation of Equation (19) is conditional: the support factor 6 is not yet derived from a microscopic localization operator. What has been achieved is a precise and geometrically motivated broken-phase mechanism under which the factor 16 follows.

3.3.0.1. Comparison with experiment.

Using the current PDG/CODATA reference values [4,5,6],

$${\alpha }_s\left(M_Z\right)=0.1180\pm 0.0009,{\left[{\widehat{\alpha }}^{\left(5\right)}\left(M_Z^2\right)\right]}^{-1}=127.930\pm 0.008,{\alpha }^{-1}\left(0\right)=137.035999177\left(21\right),$$

(21)

one obtains

$${\displaystyle \frac{{\alpha }_s\left(M_Z\right)}{{\widehat{\alpha }}^{\left(5\right)}\left(M_Z^2\right)}}=15.096\pm 0.115,{\displaystyle \frac{{\alpha }_s\left(M_Z\right)}{\alpha \left(0\right)}}=16.170\pm 0.123.$$

(22)

The first comparison is the conservative same-scale benchmark; the second is the mixed-scale comparison that turns out to be especially relevant in the present framework.