Solving the strong CP problem

Alessandro Strumia

REVIEW 3 major objections 4 minor 1 cited by

Solving the strong CP problem

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a modular flavour symmetry, broken by the modulus tau and protected by supersymmetry, forces the quark mass matrix determinant to be real and thereby sets the QCD theta angle to zero, while leaving the CKM phase large.

desk verdict A clear and honest proceedings review that presents the modular-CP determinant mechanism well, but the core idea is not new and the supergravity step is left as an assumption; still worth refereeing. read the letter →

arxiv 2501.16427 v1 pith:DNIX77MJ submitted 2025-01-27 hep-ph

Alessandro Strumia This is my paper · ORCID

classification hep-ph

keywords strongCPproblemQCDthetaanglemodularinvarianceformssupersymmetryflavoursymmetryCKMphaseaxion

topics The Strong CP Problem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a mechanism to solve the strong CP problem without an axion: make the QCD theta angle vanish because the quark mass matrix determinant is forced to be real by a modular flavour symmetry. Under supersymmetry and the condition that the two Higgs doublets carry opposite modular weights, Yukawa couplings are holomorphic modular forms with real coefficients, and the determinant of the quark mass matrix is a modular form of weight zero, hence a constant, and with CP imposed, a real one. This gives arg det M_u M_d = 0 and theta_QCD = 0, while the CKM phase naturally comes out of order one because different modular forms carry different phases. The paper also reviews the standard axion and parity solutions and argues that the modular-invariance route is a plausible alternative that avoids the axion's quality problem.

What carries the argument

The mechanism is carried by modular invariance as a flavour symmetry, together with the holomorphicity enforced by supersymmetry. The complex modulus tau parameterises the shape of a two-torus compactification and transforms under SL(2,Z); matter fields carry modular weights k, and Yukawa couplings must be modular forms, holomorphic functions of tau of fixed weight, constructed from the Eisenstein series E_4 and E_6. The determinant of the quark mass matrix is itself a modular form of weight A, and when A = 0 the only modular form of weight zero is a constant; with CP imposed, that constant is real, so arg det M = 0. The CKM phase arises because $E_4^{3}$ and $E_6^{2}$ have different phases, yielding an order-one CP-violating phase in the mixing matrix.

What would settle it

A decisive check is to compute the one-loop effective quark mass matrix in the model of eq. (30), including the Kaehler potential of eq. (23) and its supergravity corrections; if the determinant of the canonically normalised mass matrix acquires a non-zero phase for any real coefficients c_ij, the claim that theta_QCD = 0 is false.

Watch

Extended reading notes

Core claim

The central claim is that the strong CP problem disappears if CP is identified with a modular symmetry that is spontaneously broken only by the real part of the modulus tau. The quark mass matrix M_q then has entries built from real coefficients times modular forms of positive weight; supersymmetry guarantees that no conjugated fields or negative powers appear. The determinant det M_q transforms as a modular form whose weight is the QCD modular anomaly coefficient A = sum_i (2k_Q_i + k_U_i + k_D_i) + k_Hu + k_Hd. Requiring the Higgs doublets to have k_Hu + k_Hd = 0 makes this weight zero, so det M_q is a modular form of weight zero and therefore a constant; CP then forces it real, giving theta_QCD = 0. The CKM phase, by contrast, is controlled by the relative phase of the Eisenstein series E_4 and E_6 and is generically of order unity. The paper presents a concrete three-generation model with modular weights {-6, 0, +6} that reproduces the observed quark mass hierarchies and mixing angles.

Load-bearing premise

Everything hinges on the holomorphicity assumption: Yukawa couplings must be sums of real coefficients times modular forms with non-negative powers, never involving the conjugate modulus (tau-dagger) or conjugated flavons, and the paper itself admits this assumption is so far unjustified; if it fails, the determinant acquires a phase and theta_QCD reappears.

Editorial extensions

If this is right

theta_QCD = 0 is imposed by symmetry, so no axion or new light particle is needed to explain the smallness of the strong CP phase.
The CKM phase is naturally of order unity, so the scheme is compatible with observed CP violation in the weak sector.
Quark mass hierarchies emerge from modular weights, for example (2 Im tau)^6 = 64 for tau ~ i, reproducing the observed pattern with order-one coefficients.
The mechanism extends to charged leptons and neutrinos with the same modular weights, connecting the strong CP solution to models of large neutrino mixing angles.
If the modulus tau is light enough it produces flavour-violating effects that are experimentally testable; otherwise it sits near the Planck scale and the main signals are cosmological.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The holomorphicity assumption is the fragile link: in any supergravity or string completion, Kaehler corrections generically depend on tau-dagger, so the solution is most secure in a limit where those corrections are negligible or protected by a non-renormalisation theorem.
The modular mechanism can be viewed as a dynamical derivation of the Nelson-Barr structure: the zero entries and real determinant follow from modular weights rather than by fiat, which suggests a family of UV completions.
The condition k_Hu + k_Hd = 0 is restrictive; if realistic models require Higgs doublets with non-zero modular weight, the mechanism may need to be relaxed to higher-level modular groups or include heavy quarks to restore a real full determinant.
An experimental discovery of a QCD axion would not falsify this scheme, but would make it less compelling; conversely, continued null axion searches strengthen the case for exploring modular solutions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This is a conference proceedings paper, not a full research article. Strumia reviews the standard axion, parity, and CP-based fixes for the strong CP problem, then spends the second half presenting the modular-invariance mechanism from his earlier work with Feruglio and Titov. The thing done well: the determinant identity is stated cleanly. If Yukawa couplings are holomorphic in the modulus with real coefficients, the quark determinant is a modular form of weight zero and therefore a real constant. That gives arg det M_u M_d = 0 while allowing the CKM phase to be order one. The U(1) toy examples make the logic transparent, and the paper flags, in footnote 1 and section 4.2, exactly which steps are assumptions rather than consequences.

The soft spots are real and acknowledged. The 'no conjugated fields' assumption is justified only by an appeal to global supersymmetry, and the extension to N=1 supergravity is postponed. The stress-test concern is on target: going from the holomorphic superpotential to the physical mass matrix in supergravity involves e^{K/2} and Kähler-connection terms that are not shown to preserve a real determinant. So the mechanism, as presented here, solves the strong CP problem in a global-SUSY toy model, not yet in a realistic string embedding. The paper does not pretend otherwise, but it does frame the idea as a solution, and that framing outruns the proof.

On novelty: the core result is already in refs. [6] and [7], both with the present author. This paper adds a compact review and a concrete toy model, which is fair for a proceedings, but it is not a new result. A referee should not be asked to verify a new claim; they should assess the pedagogical clarity and the honesty of the limitations.

Overall: this is a coherent, honest, useful presentation of an ongoing research program. I would send it to peer review—not because it settles anything, but because the determinant argument is correct and the limitations are stated in the text. The referee should push for a clear statement that the global-SUSY result does not automatically extend to supergravity, or for a reference that fills that gap.

For a reading group it is a good half-hour read on the state of non-axion strong-CP model building. I would probably cite [6,7] rather than this paper if I needed the mechanism, but the review sections are a handy entry point.

My recommendation: accept after the author either proves the SUGRA step or tones down the claim.

Referee Report

3 major / 4 minor

Summary. The paper reviews the main approaches to the strong CP problem (axion, parity, and CP-based Nelson-Barr constructions) and then presents a new mechanism in which CP is a spontaneously broken flavour symmetry. The central idea, developed in Secs. 4.1 and 4.2, is that if the quark Yukawa couplings are sums of real coefficients times holomorphic modular forms (or U(1) monomials with non-negative powers of complex scalars), and if the total modular weight or U(1) charge of the determinant vanishes (anomaly cancellation condition), then det M_u M_d is a modular form of weight zero, hence a real constant, so that \bar theta = arg det M_u M_d + theta_QCD = 0, while the CKM phase can be large. The paper gives a toy U(1) example and a modular-invariant example with modular weights {-6,0,+6} that reproduces the observed quark mass hierarchies. The manuscript explicitly states that the construction relies on assumptions: the absence of conjugates and negative powers (footnote 1), the existence of supersymmetry, and a minimal Kähler potential; it also postpones the supergravity embedding at the end of Sec. 4.2.

Significance. If the mechanism is correct, it provides a genuinely new, axion-free solution to the strong CP problem with a natural connection to string compactifications and modular flavour symmetries. The mathematical core of the modular argument is sound under its stated assumptions: a holomorphic weight-zero modular form is a constant, and with real coefficients the determinant is real. The paper also gives a concise and useful review of axion, parity, and Nelson-Barr solutions. A particular strength is that the manuscript is unusually transparent about its own limitations, explicitly flagging the unjustified assumptions and the deferred supergravity step. However, those limitations are load-bearing: the physical implementation is not yet complete, and the U(1) generalization as stated actually contains a mathematical overgeneralization. The result, as presented, is therefore a plausible but not yet fully established solution.

major comments (3)

[Sec. 4.2, Eqs. (23)-(29) and final paragraph of Sec. 4.2] The proof that arg det M_q = 0 is carried out in global supersymmetry with the minimal Kähler potential of Eq. (23). In N=1 supergravity, the physical fermion mass matrix receives e^{K/2} and Kähler-connection contributions, so its determinant phase need not equal arg det W_ij; only the minimal-K canonicalization leading to Eq. (29) gives real positive rescaling factors. The manuscript defers this step, stating "The mechanism can be extended to super-gravity [6], but I would not be able to present it in a simple way." Since the string-motivated value h ~ M_Pl makes the supergravity regime the one of primary interest, this is a load-bearing gap: if the supergravity corrections introduce a phase, \bar theta reappears exactly in the parameter region the mechanism is supposed to cover.
[Sec. 4.1, Eq. (7) and footnote 1] The assumption that Yukawa couplings contain only non-negative powers of the scalars z_a and never the conjugates z_a^\dagger is called "our (so far unjustified) assumption" in footnote 1, and the text later repeats "The proposed idea is so far based on assumptions." The only justification offered is an appeal to supersymmetry and holomorphy of the superpotential. However, Kähler corrections and higher-dimensional operators can reintroduce conjugated fields or negative powers even in supersymmetric theories, and the paper does not provide a concrete symmetry or mechanism that forbids such terms beyond the minimal Kähler/superpotential structure. Without this, the reality of det M_q is not protected.
[Sec. 4.1, Eqs. (11)-(12)] The claim that det M_q is real whenever the total charge vanishes is not correct for a general U(1) model with multiple scalars. The scaling identity det M_q ∝ λ^k only shows that the determinant is invariant under the simultaneous rescalings; it does not imply the determinant is a constant. For example, with two scalars z_1 and z_2 of charges +1 and -1, the determinant can contain a term proportional to z_1 z_2, which has zero total charge but a non-trivial phase for generic complex vevs. The condition (12) alone is insufficient; one also needs that no non-constant invariant monomials with non-negative powers exist, which is not automatic and is not stated as an assumption. The modular-invariance version avoids this problem because weight-zero modular forms are constant, but the U(1) presentation as written overstates the generality of the result.

minor comments (4)

[Throughout] The spelling "Kahler" should be consistently rendered as "Kähler".
[Sec. 4.1, after Eq. (13)] The phrase "let's use consider" is a typo and should read "let us consider".
[Sec. 4.2, Eq. (26)] The statement that the Eisenstein series summation is divergent for k ≤ 2 is slightly imprecise: E_2 is a quasi-modular form, and the ordinary Eisenstein series for k=2 is not modular but has well-known transformation properties. The discussion would benefit from a brief clarification.
[Sec. 4.2, after Eq. (28)] The claim that "modular invariance cannot have anomalies, since it's just a remnant of general covariance in higher dimensions" is a non-trivial statement; discrete anomalies are subtle, and a reference or a short argument would be helpful.

Circularity Check

1 steps flagged · score 4.0 of 10

Central theta=0 proof is self-contained in global SUSY; the SUGRA completion is deferred to the author's own prior work, a load-bearing self-citation.

self citation load bearing [Section 4.2, final paragraph (after eq. (30))]
"The mechanism can be extended to super-gravity [6], but I would not be able to present it in a simple way."

The paper proves theta=0 only for global supersymmetry with h << M_Pl, explicitly assuming 'We here assumed h << M_Pl to avoid writing more complicated super-gravity actions.' A realistic string compactification gives h ~ M_Pl, so the physical solution requires the supergravity extension. That extension is not derived; it is referred to [6], whose authors include the present author. The load-bearing step from the global-SUSY proof to the string-theory regime is thus justified solely by a self-citation, and the assumptions underlying that cited work (holomorphic Yukawas, no negative powers) are the same 'so far unjustified' assumptions flagged in this paper.

full rationale

The central derivation in Sec. 4.2 is not circular: assuming CP, global SUSY, modular invariance with total anomaly A=0, and holomorphic Yukawa couplings built from real coefficients times modular forms, the determinant of the quark mass matrix is a modular form of weight zero, hence a holomorphic bounded function of tau and therefore a real constant. This is a direct mathematical consequence of eqs. (25)-(29), not a fit or a redefinition. The paper explicitly labels the holomorphy/no-negative-powers input as a 'so far unjustified' assumption before motivating it with supersymmetry; that is a stated assumption rather than a circularity. The only notable load-bearing self-citation is the supergravity extension, deferred to the author's own [6] with the admission 'I would not be able to present it in a simple way'. Because string compactifications give h ~ M_Pl, the realistic solution depends on that cited extension; however, the global-SUSY proof is self-contained and gives the central idea independent content. The flavour fit and large-CKM statements are supporting and not needed for theta=0. Hence score 4.

Assumptions & free parameters 3 free parameters · 7 assumptions · 2 invented entities

The central claim rests on a small set of symmetry assumptions: exact CP, holomorphic Yukawa couplings, non-anomalous modular symmetry, zero Kaehler contribution to theta, and neutral Higgs modular weights. No new particles beyond the modulus and toy flavons are required, but the modulus has no independent experimental evidence. The free parameters are the real Yukawa coefficients and the chosen modular weights; they are not fitted in this paper but are input model choices.

free parameters (3)

Real Yukawa coefficients c_ij and c'_33 = not fitted in this paper; comparable values claimed in ref. [6]
These real constants set the size of each Yukawa entry in eqs. (13)-(15) and (30). They do not affect the theta = 0 result, but they are free parameters used to reproduce quark masses and CKM mixing.
Modular weights for SM quarks and leptons = k_Q = k_U = k_D = {-6, 0, +6}; k_L = k_E = k_Q
Chosen by hand in eq. (30) to generate the zero structure and mass hierarchy. The real-determinant mechanism only needs the total anomaly to vanish, so the particular weight assignment is a model choice, not a prediction.
Modulus vacuum expectation value tau = unspecified, with Im tau giving O(1) factors such as (2 Im tau)^6 = 64 at tau ~ i
Re tau breaks CP and Im tau sets the scale of Yukawa suppression. No mechanism or numerical value for tau is derived in the paper.

assumptions (7)

domain assumption CP is an exact symmetry of the full theory with theta_QCD = 0 at tree level, broken only by the modulus VEV Re tau.
Stated in the bullet list in section 4.2. If CP is not a high-energy symmetry, the determinant-real argument has no starting point.
domain assumption Yukawa couplings depend on the modulus or flavons only through positive powers, never through complex conjugates or negative powers.
Introduced in section 4.1 after eq. (7) and justified only by an appeal to global supersymmetry. This is the load-bearing premise that keeps phases out of the determinant.
domain assumption The modular symmetry is non-anomalous because it descends from general covariance, so the QCD modular anomaly coefficient A in eq. (28) vanishes.
Used to identify the total modular weight with zero, making det M_q a weight-zero modular form. If A were nonzero, theta would not be protected.
domain assumption The Kaehler function does not generate a contribution to theta_bar, as asserted citing ref. [11].
Invoked in section 4.1 after eq. (18). If canonical normalization of quark kinetic terms introduced phases, the effective Yukawa matrices would not preserve the real-determinant property.
domain assumption The MSSM Higgs doublets have modular weights satisfying k_Hu + k_Hd = 0, so they do not break the modular flavour symmetry.
Required in eqs. (12) and (28) so that anomaly cancellation is equivalent to the total charge condition. Stated as a bullet in section 4.2.
domain assumption The scalar potential has a vacuum with Re tau different from zero, breaking CP, while other VEVs do not introduce extra CP violation.
Needed for a large CKM phase. No explicit potential is given in this paper, and the author notes for the U(1) version that realizing such conditions 'seems possible but not nice.'
domain assumption Supersymmetry is broken at high scale by a mechanism that does not introduce new CP sources.
Stated as a bullet in section 4.2. Needed to preserve the holomorphy argument and the tree-level control of theta_bar.

invented entities (2)

Complex structure modulus field tau
purpose: Order parameter that spontaneously breaks CP via Re tau, and whose modular weight assignments generate the Yukawa textures and mass hierarchies.
No direct experimental evidence is given. The paper notes that if tau is light it could give flavour effects, but more plausibly its mass is near the Planck scale, leaving no near-term experimental handle.
Flavon scalars z_a in the U(1) toy model
purpose: Pedagogical stand-ins whose VEV phases break CP and whose positive-power monomials build the mass matrix in eq. (7).
The author presents the U(1) version only as an introduction and says realizing it in supersymmetric theories 'seems possible but not nice.' The modular version replaces these scalars with the modulus tau.

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Cite this review

Pith. "Pith review of Solving the strong CP problem." pith.science (2026). https://pith.science/paper/DNIX77MJ

@misc{pith2026250116427,
  author       = {Pith},
  title        = {Pith review of: Solving the strong CP problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNIX77MJ}},
  note         = {Machine review of arXiv:2501.16427}
}

read the original abstract

I briefly review solutions to the strong CP problem based on axions, parity invariance, CP-invariance, and present a new idea based on CP as part of a spontaneously broken flavour symmetry such as a U(1) or modular invariance.

Figures

Figures reproduced from arXiv: 2501.16427 by the authors.

**Figure 1.** Summary of axion searches, from [2]. The green band in the plane (axion mass, axion coupling to photons) is favoured by axion models. Axions produced from the misalignment mechanism match the DM abundance along the blue dashed lines, for different vales of the initial axion vacuum expectation value 𝑎∗. All the other shaded regions are exclusion bounds. 2. The axion solution The axion is a well known topic, summarise… view at source ↗

**Figure 2.** Basic modular transformations that leave invariant a 2-dimensional flat torus. Ricci-flat spaces that preserve 𝑁 = 1 supersymmetry. The simplest example is orbi-folded flat tori in 6 dimensions. For our purposes, it’s enough to consider a 2 dimensional torus, such that the needed mathematics gets intuitive. A 2-dimensional flat torus is simply built by folding a 2 dimensional plane 𝑥, 𝑦, merging opposite edges to fo… view at source ↗

**Figure 3.** A 2-dimensional flat space 𝑥, 𝑦 is decomposed in a lattice with periodicities given by two vectors 𝜔1 and 𝜔2. on a flat torus, is described by a 3+1 dimensional effective field theory containing a field 𝜏 = 𝜔1/𝜔2 that parameterizes the higher dimensional shape, with modular-invariant action. We are interested in theories that also contain matter super-fields Φ. They transform under modular invariance as Φ → (𝑐𝜏 + 𝑑)… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the Origins of the Strong CP Problem
hep-ph 2026-07 accept novelty 6.0 of 10

The conventional strong CP problem is contingent on extra global topological assumptions about gauge fields that no established QCD observable is shown to require.

Reference graph

Works this paper leans on

13 extracted references · 2 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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F.Feruglio,A.Strumia,A.Titov, ‘ModularinvarianceandtheQCDangle’ ,JHEP07(2023) 027 [arXiv:2305.08908]

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Feruglio, M

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