13.4.  $\mathrm{SU}(2)_{L}$ from threshold topology: the electroweak doublet

The single complex field $\psi$ is the leading-order description of the modal order parameter. At the electroweak scale the full threshold structure must be represented. The threshold $\tau$ is a hypersurface in $\mathcal{C}$; near any crossing point the configuration explores the level set $T(C)=\tau$. The minimal order parameter consistent with both $\mathrm{U}(1)_{Y}$ and $\mathrm{SU}(2)_{L}$ is a complex doublet:

$$H\;=\;\begin{pmatrix}H^{+}\\ H^{0}\end{pmatrix}\;\in\;\mathbb{C}^{2},\qquad|H|^{2}\;=\;|H^{+}|^{2}+|H^{0}|^{2}.$$ (158)

The scalar $\psi$ of Chapter 7 is the neutral component $H^{0}$ in the broken phase; $H$ is the Standard Model Higgs doublet [75, 76].

For $H\in\mathbb{C}^{2}\cong\mathbb{R}^{4}$, the vacuum manifold after crossing is:

$$\mathcal{V}_{\mathrm{EW}}\;=\;\bigl{\{}\,H\in\mathbb{C}^{2}\;:\;|H|^{2}=v^{2}/2\,\bigr{\}}\;\cong\;S^{3},$$ (159)

where $v=246\;\mathrm{GeV}$ is the electroweak vev [93] ($r_{0}\leftrightarrow v/\sqrt{2}$ in the doublet convention of (201); $r_{0}=v$ in the single-component normalisation of Chapter 7, the two reconciled in Chapter 15). The three-sphere $S^{3}$ is diffeomorphic to the group manifold of $\mathrm{SU}(2)$:

$$S^{3}\;\cong\;\mathrm{SU}(2),$$ (160)

and its isometry group is $\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$. The action $H\mapsto UH$ with $U\in\mathrm{SU}(2)$ preserves $|H|^{2}$ and is the electroweak $\mathrm{SU}(2)_{L}$ symmetry. Three of the four real degrees of freedom in $H$ are Goldstone bosons of the breaking $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}\to\mathrm{U}(1)_{\mathrm{em}}$; they are absorbed by the $W^{\pm}$ and $Z$ bosons, giving them mass [77, 78, 79]. The remaining degree of freedom is the physical Higgs boson of mass $m_{h}^{2}=4\varepsilon$, identified in equation (122).

The $\mathrm{SU}(2)_{R}$ isometry is broken by the directional character of the threshold crossing: $\varepsilon=|T(C)|-\tau>0$ is an oriented statement in which tension exceeds the threshold from below, selecting a preferred chiral handedness. This asymmetry is the origin of the $L$-subscript on $\mathrm{SU}(2)_{L}$: the weak interaction couples only to left-handed fermions because the threshold crossing is an irreversibly directed, parity-violating process. Parity violation in the weak sector is therefore not an external assumption but a theorem of the asymmetric modal threshold [83, 84].

13.4.1.  Two descriptions of one would-be Goldstone phase

The single Landau–Ginzburg (LG) phase mode $\theta$ on the substrate side appears in two formally different but physically identical guises, describing the same field at the same scale in different language. The first edition listed three and the first two were the same object counted twice, with the phase of $H^{0}$ appearing once among the eaten Goldstones and once again as a separate $\mathrm{U}(1)_{\mathrm{em}}$ gauge redundancy; the correction is Computation 172 and the arithmetic is that a four-component doublet cannot carry five real degrees of freedom.

1. Substrate view (§13.3, §9.2): $\theta$ is the phase of the complexified order parameter $\psi$, which P2’s determination of the $V_{7}$ direction does not fix, $\hat{\tau}$ being a real direction carrying no phase (Computation 171). It is the mode whose wave equation §4.12 argues from, described before it is eaten.

2. Electroweak doublet view (this section): with $\psi=H^{0}$ enriched to the Standard Model (SM) Higgs doublet $H=(H^{+},H^{0})^{\mathrm{T}}$, $\theta$ is the phase of the neutral component, and it is one of the three angular Goldstones absorbed into the longitudinal polarisations of $W^{\pm}$ and $Z$ under $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}\to\mathrm{U}(1)_{\mathrm{em}}$. The broken diagonal combination, the $Z$, is the one that takes it.

   It is not eliminable by $\mathrm{U}(1)_{\mathrm{em}}$, and the reason is the same fact that makes that symmetry unbroken: $Q=T_{3}+Y/2$ annihilates the vacuum, and it annihilates $H^{0}$ as well, $H^{0}$ carrying charge zero, so a $\mathrm{U}(1)_{\mathrm{em}}$ transformation does not move $\theta$ at all and has nothing to gauge away (Computation 172). Four real components, three eaten, one physical and massive; no massless scalar survives.

A third statement, which is about the zero mode rather than a third view of the same one. A spatially uniform rotation $\psi=r_{0}\mathrm{e}^{i\omega t}$ would carry a conjugate Noether charge, the global $\mathrm{U}(1)_{\mathrm{em}}$ charge of the universe, and the observed cosmic charge-neutrality pins $\omega=0$ (§9.1, where the charge is derived as $J^{0}=2r_{0}^{2}\omega$). That is consistent with $\theta$ being eaten: a mode absorbed into the longitudinal $Z$ carries no independent zero mode either. The first edition gave the charge here as $J^{0}=4\varepsilon\omega$ and cited §16.5, which derives neither; the form and the reference now follow the section that does.

These are not different fields nor different mechanisms; the first two are one Goldstone phase described in two vocabularies appropriate to two observational regimes (the infrared (IR) and the electroweak), and the third is a statement about the zero mode rather than a third view of the same one. The unifying statement is: $\theta$ does not propagate as an independent physical scalar in emergent spacetime. The only propagating scalar dof of the LG–Higgs sector is the radial mode, the physical Higgs $h$ of mass $m_{h}\simeq 125$ GeV.

13.4.2.  Relation to existing electroweak approaches

PST’s substrate-Higgs mechanism is this section together with the identification of §15.2 and the gauging of §13.3. How it compares against the Connes–Chamseddine–Marcolli spectral action, Technicolor and composite-Higgs models, Coleman–Weinberg radiative spontaneous symmetry breaking (SSB) [91], Little Higgs models, and supersymmetric extended Higgs sectors is treated in §17.5.

13.4.3.  The doublet does not fix chirality

The derivation above gives the $\mathrm{SU}(2)_{L}$ doublet but not its chirality (its action on left-handed components only). Chirality is a separate structural consequence of the directed modal threshold, treated in §14.4. The same threshold-direction parity argument that drives parity violation selects which of the two $\mathrm{su}(2)$ factors of $\mathrm{so}(4)=\mathrm{su}(2)_{L}\oplus\mathrm{su}(2)_{R}$ on the spatial $S^{3}$ is identified with the physical $\mathrm{SU}(2)_{L}$.