The selection rule for the charge of the breaking scalar is a half-line

A vacuum expectation value of charge q > 0 breaks a U(1) to a discrete subgroup, and an operator of charge A can be dressed with insertions of it iff A / q ∈ ℤ: protection fails at q. For A ≠ 0 this happens iff q = |A| / n for some integer n ≥ 1, so over finitely many nonzero charges the failure set has a maximum, max_j |A_j|, and protection holds on the half-line above it.

The hypothesis A ≠ 0 is implicit in the paper: for A = 0 protection fails at every q.

Main results

• GaugeHiggsUnification.SU7QPhi.fails_iff: protection fails at q iff q = |A| / n, n ≥ 1.
• GaugeHiggsUnification.SU7QPhi.isGreatest_failSet: the failure set has maximum max_j |A_j|.

References

• C. Marín, Proton Decay in SU(7) Grand Gauge-Higgs Unification: An Obstruction, Its Minimal Escapes, and the One Row of Their Table 1 That Can Pay for Them (Part VI), Zenodo (2026), doi:10.5281/zenodo.22033303, Proposition prop:closed.

def GaugeHiggsUnification.SU7QPhi.Fails (A q : ℚ) : Prop

Protection fails at q for the charge A: the operator can be dressed, A / q is an integer.

Equations

• GaugeHiggsUnification.SU7QPhi.Fails A q = ∃ (k : ℤ), A / q = ↑k

theorem GaugeHiggsUnification.SU7QPhi.fails_iff {A q : ℚ} (hA : A ≠ 0) (hq : 0 < q) : Fails A q ↔ ∃ (n : ℕ), 1 ≤ n ∧ q = |A| / ↑n

theorem GaugeHiggsUnification.SU7QPhi.isGreatest_failSet {ι : Type u_1} [Fintype ι] [Nonempty ι] (A : ι → ℚ) (hA : ∀ (j : ι), A j ≠ 0) : IsGreatest {q : ℚ | 0 < q ∧ ∃ (j : ι), Fails (A j) q} (Finset.univ.sup' ⋯ fun (j : ι) => |A j|)

The failure set of finitely many nonzero charges has a maximum, attained at max_j |A_j|.