Gauge-Anomaly Cancellation and Quantum Consistency

One Standard Model generation is a chiral gauge theory whose perturbative gauge anomalies cancel exactly: color is vectorlike, the mixed non-Abelian–hypercharge sums vanish, and the cubic and gravitational hypercharge sums vanish. The cancellation must be performed with left-handed Weyl fields and does not, by itself, settle global anomalies or the global form of the gauge group.

Required background. Use the one-generation field table in the Standard Model Lagrangian and the triangle-anomaly conventions from perturbative chiral gauge anomalies.

Helpful background. Global and torsion anomalies explain why vanishing anomaly polynomials are necessary but not sufficient for quantum consistency.

Put every fermion in one chirality convention

Write the fermion content entirely as left-handed Weyl fields:

$$Q:(\mathbf3,\mathbf2)_{1/6},\quad u^c:(\bar{\mathbf3},\mathbf1)_{-2/3},\quad d^c:(\bar{\mathbf3},\mathbf1)_{1/3},\quad L:(\mathbf1,\mathbf2)_{-1/2},\quad e^c:(\mathbf1,\mathbf1)_1.$$

Conjugating a right-handed field reverses its Abelian charge and replaces a complex representation by its conjugate. For a non-Abelian representation $R$, use

$$\operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab}, \qquad \operatorname{tr}_R\!\left(T^a\{T^b,T^c\}\right)=A(R)d^{abc},$$

with $T(\mathbf N)=T(\bar{\mathbf N})=1/2$ and $A(\bar R)=-A(R)$. Multiplicities from the spectator gauge factors must be included. This convention makes the color check especially transparent.

Exact local anomaly sums

Pure non-Abelian anomalies

For $SU(3)_C^3$, the weak doublet contains two color triplets, whereas $u^c$ and $d^c$ are antitriplets:

$$\mathcal A_{SU(3)^3} =2A(\mathbf3)+A(\bar{\mathbf3})+A(\bar{\mathbf3}) =2-1-1=0.$$

This is the exact statement that color is vectorlike. A proposed generation with only one of the two conjugate singlets would fail this check. The perturbative $SU(2)_L^3$ anomaly vanishes because every finite-dimensional $SU(2)$ representation is real or pseudoreal, so its symmetric cubic invariant is zero.

Mixed non-Abelian–hypercharge anomalies

With $T_F=1/2$,

$$\begin{aligned} \mathcal A_{SU(3)^2U(1)_Y} &=2\left(\frac16\right)T_F +\left(-\frac23\right)T_F +\left(\frac13\right)T_F=0,\\ \mathcal A_{SU(2)^2U(1)_Y} &=3\left(\frac16\right)T_F +\left(-\frac12\right)T_F=0. \end{aligned}$$

The factors two and three are, respectively, weak and color multiplicities. Diagrams with one non-Abelian current and two $U(1)$ currents vanish representation by representation because the single non-Abelian generator is traceless.

Cubic and gravitational hypercharge anomalies

The complete cubic sum is

$$\begin{aligned} \mathcal A_{Y^3} ={}&6\left(\frac16\right)^3 +3\left(-\frac23\right)^3 +3\left(\frac13\right)^3 +2\left(-\frac12\right)^3+1^3\\ ={}&\frac1{36}-\frac89+\frac19-\frac14+1=0. \end{aligned}$$

The mixed gravitational–hypercharge coefficient is the corresponding linear trace,

$$\mathcal A_{\mathrm{grav}^2Y} =6\left(\frac16\right)+3\left(-\frac23\right) +3\left(\frac13\right)+2\left(-\frac12\right)+1=0.$$

These formulas, including the spectator multiplicities, are the standard one-generation cancellation shown in Schwartz 2014, §30.4, pp. 631–634. Since the cancellation occurs within each generation, merely repeating the same representations three times neither helps nor harms the perturbative sums.

The global SU(2) parity test

Pseudoreality removes the local $SU(2)^3$ anomaly but permits Witten’s mod-two global anomaly. For doublets in the Standard Model generation, color supplies three copies of $Q$ and leptons supply one copy of $L$:

$$N_{\mathbf2}=3+1=4=0\pmod2.$$

The fermion determinant therefore has no sign obstruction of the original doublet type under the nontrivial large $SU(2)$ gauge transformation. The hypothesis and mod-two result are those of Witten 1982, pp. 324–328. A new chiral extension must repeat the parity test with all half-integer isospin representations; counting only visually obvious doublets can miss the more general index criterion.

What cancellation proves—and what it does not

The anomaly calculation establishes that the local gauge symmetry can survive quantization for the stated perturbative spectrum. It does not imply any of the following:

• that an arbitrary global quotient of the gauge group is consistent on every spacetime or bundle;
• that a global symmetry such as baryon or lepton number is exact;
• that heavy anomalous matter may be deleted without its Wess–Zumino or inflow remnant;
• that gauge-invariant higher-dimensional operators automatically respect accidental selection rules.

The distinction is structural. Perturbative anomalies are encoded in the anomaly polynomial and descent Alvarez-Gaumé and Ginsparg 1985, §§3–5. Torsion or global anomalies require additional global data; Witten’s mod-two obstruction is the relevant example here Witten 1982, pp. 324–328.

A diagnostic for proposed spectra

Given a set of new left-handed Weyl fields, build the following table before attempting phenomenology:

Check	Sum to evaluate	Failure means
$G^3$	$\sum_i A(R_i)$ with spectator multiplicities	local non-Abelian gauge anomaly
$G^2U(1)$	$\sum_i Y_iT(R_i)$	mixed gauge anomaly
$U(1)^3$	$\sum_i d(R_i)Y_i^3$	Abelian gauge anomaly
$\mathrm{grav}^2U(1)$	$\sum_i d(R_i)Y_i$	mixed gravitational anomaly
global $SU(2)$	appropriate mod-two index	large-gauge-transformation obstruction
chosen global form	descent of representations and global anomaly test	locally valid data fail globally

Vectorlike pairs cancel all local gauge anomalies because $R_Y$ and $\bar R_{-Y}$ contribute with opposite chirality. This provides a strong limiting check on any implementation. It is not a license to ignore their masses, threshold matching, or possible global quantum numbers.

The anomaly-free spectrum should next be combined with the chosen global quotient and the classification of accidental symmetries. Only that combined object is a meaningful quantum-consistency input for an extension.

References

• Alvarez-Gaumé, Luis, and Paul Ginsparg. “The Structure of Gauge and Gravitational Anomalies.” Annals of Physics 161, no. 2 (1985): 423–490, §§3–5. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §30.4, pp. 631–634. DOI.
• Witten, Edward. “An SU(2) Anomaly.” Physics Letters B 117, nos. 5–6 (1982): 324–328. DOI.