13.9.  $\mathrm{U}(1)_{Y}$ hypercharges compatible with PST structure

Four structural facts fix all six Standard Model hypercharges with no free hypercharge parameter beyond the $\mathrm{U}(1)_{Y}$ normalisation. (The Gell-Mann–Nishijima relation and the empirical nucleon charges enter as inputs to the determination, as the closing paragraph makes explicit.)

(i) Identification of $\mathrm{U}(1)_{Y}$. The PST order parameter $H\in\mathbb{C}^{2}$ is a complex doublet. Under $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$, the $\mathrm{U}(1)_{Y}$ generator acts as an overall phase rotation $H\to e^{i\alpha}H$ on both isospin components. The weak hypercharge relation is

$$Q\;=\;T_{3}+\tfrac{Y}{2},$$ (186)

where $Q$ is electric charge, $T_{3}=\pm\tfrac{1}{2}$ for an $\mathrm{SU}(2)_{L}$ doublet, and $Y$ is the $\mathrm{U}(1)_{Y}$ hypercharge.

(ii) Higgs hypercharge from photon masslessness. After the $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}\to\mathrm{U}(1)_{\mathrm{em}}$ breaking at $\langle H\rangle=(0,\,v/\sqrt{2})^{T}$, the photon is the unique massless gauge boson. This requires the neutral Higgs component $H^{0}$ to be electrically neutral: $Q(H^{0})=0$. With $T_{3}(H^{0})=-\tfrac{1}{2}$ (lower component of the doublet), equation (186) gives

$$Y(H)\;=\;2\bigl{[}Q(H^{0})-T_{3}(H^{0})\bigr{]}\;=\;2\!\left[0+\tfrac{1}{2}\right]\;=\;+1.$$ (187)

(iii) Quark charges from $\mathrm{SU}(3)_{c}$ confinement. Colour confinement (§13.5) requires all asymptotic states to be colour singlets. For the lightest colour-singlet baryons, the proton ($|uud\rangle$, $Q=+1$) and neutron ($|udd\rangle$, $Q=0$) give

$$2Q_{u}+Q_{d}=1,\quad Q_{u}+2Q_{d}=0\;\;\Longrightarrow\;\;Q_{u}=\tfrac{2}{3},\quad Q_{d}=-\tfrac{1}{3}.$$ (188)

Applied to the left-handed quark doublet $Q_{L}=(u_{L},d_{L})$ via equation (186):

$$Y(Q_{L})\;=\;2\bigl{[}Q(u_{L})-T_{3}(u_{L})\bigr{]}\;=\;2\!\left[\tfrac{2}{3}-\tfrac{1}{2}\right]\;=\;+\tfrac{1}{3}.$$ (189)

(iv) Remaining assignments from Yukawa invariance and anomaly cancellation. The Yukawa couplings $\bar{Q}_{L}Hd_{R}$, $\bar{Q}_{L}\tilde{H}u_{R}$ (where $\tilde{H}=i\sigma_{2}H^{*}$, $Y(\tilde{H})=-1$), and $\bar{L}_{L}He_{R}$ must each be $\mathrm{U}(1)_{Y}$ singlets (total charge $Y=0$):

$$-Y(Q_{L})+Y(H)+Y(d_{R}) =0 \Rightarrow\;Y(d_{R})=-\tfrac{2}{3},$$ (190)

$$-Y(Q_{L})-Y(H)+Y(u_{R}) =0 \Rightarrow\;Y(u_{R})=+\tfrac{4}{3},$$ (191)

$$-Y(L_{L})+Y(H)+Y(e_{R}) =0 \Rightarrow\;Y(e_{R})=Y(L_{L})-1.$$ (192)

The remaining unknown $Y(L_{L})$ is fixed by the $[\mathrm{SU}(2)_{L}]^{2}[\mathrm{U}(1)_{Y}]$ anomaly condition (equation (185)), which requires $N_{c}Y(Q_{L})+Y(L_{L})=0$. With $N_{c}=3$:

$$Y(L_{L})\;=\;-N_{c}\,Y(Q_{L})\;=\;-3\cdot\tfrac{1}{3}\;=\;-1,\qquad Y(e_{R})\;=\;-2.$$ (193)

The complete Standard Model hypercharge assignment is therefore

$$Y(Q_{L})=+\tfrac{1}{3},\quad Y(u_{R})=+\tfrac{4}{3},\quad Y(d_{R})=-\tfrac{2}{3},$$ (194)

$$Y(L_{L})=-1,\quad Y(e_{R})=-2,\quad Y(H)=+1.$$

Substituting into equation (184) gives $-2N_{c}+6=0$ at $N_{c}=3$, a non-trivial cross-check that no value was inserted by hand; Computation 136 evaluates all six anomaly coefficients as functions of $N_{c}$ and confirms anomaly freedom forces $N_{c}=3$.

Given PST’s representation content and $N_{c}=3$, the six SM hypercharges follow from the standard determination: the Gell-Mann–Nishijima relation $Q=T_{3}+Y/2$, photon masslessness (fixing $Y(H)=+1$), the observed proton and neutron charges (fixing the $\mathrm{U}(1)_{Y}$ normalisation via $Q_{u}=+\tfrac{2}{3}$, $Q_{d}=-\tfrac{1}{3}$), and gauge-anomaly cancellation together with Yukawa gauge-invariance (fixing the remainder). The non-trivial PST content is that its representation content is anomaly-consistent and reproduces the SM hypercharge pattern with no free hypercharge parameter beyond this normalisation; the Gell-Mann–Nishijima relation and the empirical nucleon charges are inputs to the determination, not PST outputs. Standard Model hypercharge quantisation is thus shown compatible with, and reproduced by, PST structure, rather than postulated independently. This closes the gauge group derivation for the bosonic sector. The one-generation fermion-sector representation content is determined explicitly in §14.1, matching the hypercharge assignments derived above; the generation count $N_{\mathrm{gen}}=3$ is read structurally in §14.2; and the Yukawa hierarchy and Cabibbo–Kobayashi–Maskawa (CKM) matrix are addressed by the structural-scope theorem of §14.3.