13.8.  Gauge anomaly cancellation as a structural check

A gauge theory is quantum-mechanically consistent only if its triangle anomalies vanish [85, 86]: the Ward identities of the gauge currents must hold at the quantum level, otherwise the gauge symmetry is destroyed by radiative corrections and the theory is ill-defined. In the Standard Model, anomaly cancellation holds by a precise numerical relationship between the number of colours and the hypercharge assignments of quarks and leptons.

Important caveat on hypercharges. The $\mathrm{U}(1)_{Y}$ hypercharge assignments used below, $Y=+1/3$ for the quark doublet, $Y=-4/3$ for the right-handed up quark, etc., are taken directly from the Standard Model. The structural derivation of these values is inherited from the Chamseddine–Connes–Marcolli framework [50] applied to the substrate-derived spectral triple. Given $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ (Computation 6), two ingredients fix the action of the $\mathbb{C}$-factor’s $\mathrm{U}(1)$ on each chirality block: the chiral bimodule support delivered by the directed modal threshold (Computations 7, 8, 11), and the order-one condition. The unimodularity constraint of Computation 6 then removes the overall trace. The remaining freedom is a single integer (the unimodularity assignment selecting which $\mathrm{U}(1)\subset\mathrm{U}(A_{F})$ is gauged), and the SM hypercharge pattern is the unique choice consistent with anomaly cancellation under $N_{c}=3$. The structural derivation walked through in Chamseddine–Connes–Marcolli (CCM) 2007 inherits into PST through PST’s structural delivery of $A_{F}$, the chiral bimodule, and the order-one condition. All six hypercharges are determined explicitly below in §13.9, from Gell-Mann–Nishijima, photon masslessness, colour confinement, Yukawa gauge-invariance, and anomaly cancellation. There the Gell-Mann–Nishijima relation and the empirical nucleon charges enter as inputs rather than PST outputs. The computation below is therefore not a derivation of anomaly cancellation from PST but a demonstration that given the SM hypercharges, the PST result $N_{c}=3$ is the unique value that makes the theory consistent. The result is: if the hypercharges are as in the Standard Model, then $N_{c}=3$ is forced by internal consistency, the value PST fixes independently.

Express the Standard Model fermion content of one generation as left-handed Weyl spinors [162], replacing each right-handed field $\chi_{R}$ by its left-handed conjugate $\chi_{R}^{\;c}$ with opposite quantum numbers. With $N_{c}$ colours, the fields and their hypercharges are: the quark doublet $Q_{L}$ ($N_{c}$ colours, $Y=+1/3$), the right-handed up quark $(u_{R})^{c}$ ($N_{c}$ colours, $Y=-4/3$), the right-handed down quark $(d_{R})^{c}$ ($N_{c}$ colours, $Y=+2/3$), the lepton doublet $L$ ($Y=-1$), and the right-handed charged lepton $(e_{R})^{c}$ ($Y=+2$). The two independent anomaly conditions that are not automatically satisfied are:

$$\bigl{[}\mathrm{U}(1)_{Y}\bigr{]}^{3}=0:\quad N_{c}\!\left[2\!\left(\tfrac{1}{3}\right)^{3}+\!\left(-\tfrac{4}{3}\right)^{3}+\!\left(\tfrac{2}{3}\right)^{3}\right]+2(-1)^{3}+(2)^{3}$$

$$\hphantom{N_{c}\!\left[\right.}\;=\;-2N_{c}+6,$$ (184)

$$\bigl{[}\mathrm{SU}(2)_{L}\bigr{]}^{2}\!\bigl{[}\mathrm{U}(1)_{Y}\bigr{]}=0:\quad N_{c}\cdot\tfrac{1}{3}+1\cdot(-1)\;=\;\tfrac{N_{c}}{3}-1.$$ (185)

Both vanish if and only if $N_{c}=3$, so the following holds as a theorem (the colour factor is §13.5’s):

A projected theory with $N_{c}$ colours is anomaly-free (given the Standard Model hypercharge assignments) if and only if $N_{c}=3$. PST fixes $N_{c}=3$ from the colour factor $M_{3}(\mathbb{C})$ of the internal algebra, so anomaly cancellation is met at the value the algebra already requires. The Standard Model hypercharge assignments themselves are reproduced by PST structure in §13.9, given the Gell-Mann–Nishijima relation and the empirical nucleon charges as inputs.

Two further conditions are automatically satisfied for any $N_{c}$: $[\mathrm{SU}(3)_{c}]^{3}=0$ because the quark sector contains equal numbers of fundamental and anti-fundamental representations, and $[\mathrm{SU}(2)_{L}]^{3}=0$ because $\mathrm{SU}(2)$ has a vanishing cubic Casimir. The mixed gravitational anomaly $[\mathrm{grav}]^{2}[\mathrm{U}(1)_{Y}]\propto\sum_{i}Y_{i}=0$ also vanishes per generation, independently of $N_{c}$.