Parity Engine · Bet A · snapshot 25 Oct 2026

Which Nelson-Barr models exist with small discrete symmetries?

We made a list of all the Nelson-Barr models in one class. Each model uses the Standard Model, one or two heavy quarks, one or two scalars and a ZN symmetry. N is 2, 3, 4, 5 or 6. Lean checked the full list. A physicist must examine two of our definitions. One of these definitions changes the number of models.

Class: conventional CP × ZN, renormalizable, tree level · lab commit 078b69b

Written in ASD-STE100 Simplified Technical English
Proved The full list has a machine proof

We examined 2.6 million charge assignments. Each one has a result. The Lean kernel checked each statement. The proofs use only the three standard axioms.

Possibly new The full list is new. Most of the physics is known.

We did not find a full list of this class in the literature. Other papers contain most of the separate results. We cite these papers.

Needs a physicist Two questions, approximately 30 minutes

Question 1 is about models with a massless Goldstone boson. Question 2 is about cubic terms in the scalar potential.

The mechanism

How a Nelson-Barr model removes the strong CP problem

The strong CP problem is this: the angle θ̄ is less than 10−10, but the Standard Model does not explain this small value. A Nelson-Barr model makes CP an exact symmetry of the Lagrangian. Thus, all couplings are real. The scalars then break CP when they get a vacuum value with a complex phase. The model must put this phase into the quark mixing (CKM). It must keep the phase out of θ̄.

Flow: exact CP gives real couplings; scalar vacuum gives a phase; the phase goes into F only; the determinant stays real so theta-bar is zero; the light-quark mixing gets the phase so the CKM phase is not zero. CP is exact All couplings are real Scalars get a vacuum The vacuum has a phase eiφ The phase goes into F Only the heavy-light block det M is real θ̄ = 0 at tree level CKM phase ≠ 0 J ≠ 0, as measured
Figure 1. The order of the mechanism. The symmetry ZN controls which couplings can exist. Thus, ZN controls where the phase can go.
The down-type quark mass matrix in four blocks: m_d real, zero, F complex, M real. right-handed: dR (×3) · DR left-handed: q (×3) · D̄L md real · from the Higgs 0 ZN forbids F complex · from the scalars M
Figure 2. The Barr texture. The zero block makes the determinant equal to det md × M. Both factors are real. Thus, θ̄ is zero at tree level. The phase in F still gets into the light-quark mixing.
realhas the CP phasezero, the symmetry forbids it
Funnel from 2.6 million charge assignments to 85 classes, 75 distinct models and 1 minimal model. 2,600,000 charge assignments, all N 85 classes that pass, counted per N 75 different models 1 minimal (BBP) texture, CP test, vacuum
Figure 3. The search. Each assignment must have the Barr texture. It must also give CP violation at a real vacuum. Some classes for large N are copies of classes for small N. Thus, 85 classes give 75 different models.
Confidence

The checks that the result passed

Read from top to bottom. A full dot shows a check that is complete. The dashed dot shows the step that a person must do.

  1. Exact enumerationTwo different programs examined each assignment. They use exact arithmetic. Their results agree for each class.
  2. Kernel proofLean 4 on PhysLib proved each statement. The proofs do not use native_decide.
  3. Second, blind proofA second prover did not see the first proof. It proved the statements again. The same gates accepted it.
  4. Critic reviewThe critic compares the Lean statement with the English text. It found 0 major problems and 2 minor problems.
  5. Red teamThe red team tries to find the result in the literature. It also tries to show that the result is trivial. The result passed.
  6. Literature searchWe did 54 searches of INSPIRE, arXiv and OpenAlex. We read 470 records. No record has a full list of this class.
  7. Stricter vacuumOn 25 October we used a stricter definition of the vacuum. No list changed. Lean proved this.
  8. A physicist agrees with the definitionsThis step is not complete. The two questions at the end of this page are for this step.
The result

The number of models for each symmetry

Each cell gives the number of classes. Two assignments are in the same class if a relabel, a conjugate scalar or a ZN symmetry changes one into the other. Use the two switches. They show the result for the other answers to the two questions.

Models with a massless Goldstone (question 1)
Cubic scalar terms (question 2)
Number of classes for each Z_N, all slices together. Dashed parts are classes that the current switches remove.
Heavy quarks + scalarsZ₂Z₃Z₄Z₅Z₆Total

The chart adds the four rows of the table. The dashed part of a bar shows the classes that the switches remove.

One statement in detail

Z₄ with one heavy quark and two scalars: five models

A Z4 charge is a position on a clock with 4 hours. Each clock below is one model. The dots show the charge of the heavy quark and the charges of the two scalars.

Five Z4 models shown as clocks with four positions.
heavy quark Dscalar SQuark charges by generation are under each clock.
Figure 4. Models 4 and 5 give different charges to the generations. Some of the five models are Z2 models inside Z4. Lean proves that each Z4 model with this field content is one of these five.
theorem nbCompatible_iff_classes_N4 (z : ChargeAssignment 4 1 2) :
  NBCompatible z ↔ ∃ r ∈ [ model 1, model 2, model 3, model 4, model 5 ], ChargeEquiv z r
New and known

Each claim and the result of the literature search

Not found

A full list of the CP × ZN Nelson-Barr charge assignments with heavy quarks, for this class.

The nearest method: Hall, Manzari and Noether (arXiv:2407.14585). They examined multi-Higgs models. They did not examine Nelson-Barr models.

Partly known

BBP is the only minimal model. This is also true when the quark charges or the Higgs charge are different.

Physicists know this result, but no paper proves it. Haber and Surujon (arXiv:1201.1730) give the scalar part.

Partly known

Models with different charges for each generation exist only with two scalars. An example is Z₃ with the charges (0, 0, 1).

Other papers use such charges to get θ̄ = 0. They do not use them in Nelson-Barr models.

Partly known

Three Z₆ "geometric" models. Their vacuum phase is fixed at multiples of 2π/9.

Björkeroth et al. (arXiv:1503.03306) have a similar model. We must compare the full texts before we make a claim.

Known

The Z₄ model with only S⁴ has a CKM phase, but no CP violation at a vacuum.

Branco, Gerard and Grimus (1984); Haber and Surujon (2012). We use this result as a check.

Partly known

The first machine proofs about the Nelson-Barr mechanism and θ̄ at tree level.

Krippendorf and Tooby-Smith (arXiv:2603.28406) did the first Lean proof of a charge classification in physics. PhysLib proves that the Jarlskog invariant does not change under a rephasing.

One slice alone has little new physics. The papers above contain that physics. The literature does not have the full list for the class. It also does not have a proof that the list is complete.

Limits

What the result does not include

For a physicist

Two questions that change the result

We compared all other definitions with the literature. Our definitions are the same or stricter. When you select an answer, the table above changes. A short answer is sufficient, for example "1b, 2 yes".

Question 1 · accidental U(1)

18 of the 75 models have an accidental U(1) symmetry. All Z₅ models are in this group. At the renormalizable level, this symmetry gives a Goldstone boson with zero mass. What must a classification do with these models?

The U(1) has no anomaly. It is vector-like on the heavy quarks. It couples to the light quarks. BBP, Murai-Nakayama and Alves-Nishi add a term that gives the phase a mass. No paper that we read accepts this Goldstone boson without such a term.

Question 2 · cubic terms

Must the potential include all renormalizable terms that ZN × CP permits, cubic singlet terms included?

BBP potentials do not have cubic terms, because Z₂ and Z₄ forbid them. The general potential of Alves-Nishi has cubic terms.

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