Why does the weak force act only on left-handed particles?

Object. In the Standard Model, SU(2) acts on left-handed doublets and not on right-handed singlets. This is an input to the unbroken Lagrangian; the Higgs mechanism hides it (masses couple the two hands) rather than creating it. Electromagnetism is vector-like because the photon is the unbroken diagonal combination.

Presenting problem. Three candidate origins, all consistent, none derived. - Placement facts (not origins). Chirality is possible only in even dimensions because so(2n)'s Dynkin diagram has an automorphism swapping the two spinor nodes; a chiral theory is one whose matter is a Spin representation not extending to Pin. Anomaly cancellation constrains chiral assignments hard and vector-like ones not at all; given the field content, the hypercharges are essentially forced. Chirality is what allows light fermions (mass terms are excluded from the symmetry-closed operator space). - Origin A, symmetry breaking. Left–right symmetric models: SU(2)_L × SU(2)_R × U(1)_{B−L} with parity exact; the vacuum breaks SU(2)_R at a high scale, W_R is heavy, and low-energy physics looks purely left-handed. Explicit breaking becomes spontaneous; "why left" becomes "why this vacuum." Costs domain walls; buys charge quantization Y = T_{3R} + (B−L)/2 and the seesaw. - Origin B, algebra. In the division-algebra construction the quaternionic part supplies Cl(4) and, per Furey, "the electroweak theory can violate parity automatically" for leptons: chirality from the closure of the algebra rather than from dynamics. - Origin C, none. Chirality is a representation-content input with no deeper rung.

Differential. 1. A: testable — a W_R at colliders, heavy right-handed neutrinos, neutrinoless double beta decay. Current limits (LHC √s = 13 TeV, 139 fb⁻¹, as of 2026-09) exclude m(W_R) to 5.0–5.2 TeV in the e/μ channels, with dilepton bounds pushing Z_R past 5.4–6.1 TeV — roughly twice the "several TeV" recorded below, and past comfortable near-term reach. This raises branch A's cost; it does not kill it. 2. B: testable only through the rest of the framework's consequences (global form ℤ₆, three generations via triality, S₃ flavor structure), one of which (S₃ in the mixings) already fails in its naive form. 3. C: the honest default.

Discriminating tests. Discovery or exclusion of W_R; whether any algebraic construction produces parity violation for quarks as well as leptons without an added assumption.

Stakes. Whether the most conspicuous asymmetry in particle physics is a vacuum choice, an algebraic necessity, or a brute fact.

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Update (2026-09-12): the case is two questions, and separating them dissolves half the disagreement.

There is a computable invariant sitting under this case that the body above gestures at without naming. The Frobenius–Schur indicator of a representation is +1, 0 or −1 according as it is real, complex or quaternionic — the same trichotomy that labels GOE/GUE/GSE spectral statistics, and the same one behind Arf invariants and quadratic refinements. Spinor representations have a reality type that depends on dimension mod 8 (Bott periodicity, read as reality).

Chirality lives there. The case's own placement facts already point at it: chirality is possible only in even dimensions because so(2n)'s diagram automorphism swaps the two spinor nodes, and a chiral theory is one whose matter is a Spin representation not extending to Pin. Both are statements about whether a structure map exchanges the two half-spinor representations, which is exactly what the indicator measures. In 4d the Weyl representation has indicator 0 — complex, the two chiralities genuinely distinct representations — which is what makes a chiral assignment available at all. Where the spinor representation is real or quaternionic, conjugation relates the hands and no chiral assignment exists.

The split.

• Existence of chirality is a reality-type question: does the spinor representation have indicator 0 in this dimension and signature? Computable, answered, not a mystery.
• Which hand is a selection question: what picks left out of a symmetric pair? This is where Origin A (left–right symmetric models, the vacuum picks) and Origin C (brute fact) actually compete. Origin B (algebraic closure) is the claim that the room forecloses the choice rather than that something selects within it, which is a third kind of answer.

The body of this case runs both questions at once, which is why the three origins read as incommensurable rather than as competing answers. They are competing answers to the second.

Additional discriminating test. Compute the Frobenius–Schur indicator of the relevant spinor representation in each construction, and check whether Origin B's claim — parity violation arising automatically, for leptons — is an indicator-0 statement in disguise. If it is, Origin B answers the existence question and is silent on selection, and the central disagreement of this case was never a disagreement. If it is not, Origin B is claiming something strictly stronger than reality type supports, and that stronger claim is the thing to attack.

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Update (2026-09-14): the W_R bound was stale; corrected and dated. The "several TeV" figure was current when this case was written and is now about half the real limit. ATLAS (139 fb⁻¹, √s = 13 TeV) excludes m(W_R) to 5.0–5.2 TeV in the electron/muon channels, and dilepton searches push Z_R past 5.4–6.1 TeV. The left–right symmetric account (Origin A) now needs W_R heavier than ~5 TeV — roughly twice as constrained as recorded, and moving away from near-term reach. Its cost rises, which is what the case asked to track; it is not excluded. Case stays open; moved by a literature pass, no run. Sources: ATLAS 139 fb⁻¹ LRSM search (arXiv:2304.09553); dilepton Z_R bounds (arXiv:2507.10684); figures current as of 2026-09.