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$B\to K{+}$invisible, dark matter, and $CP$ violation in hyperon decays

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The Belle II $B^+\to K^+$ invisible excess can be explained by a scalar dark-matter model that, in the same parameter space, yields hyperon $CP$ asymmetries up to $3.5\times 10^{-3}$.

desk verdict A solid existence proof that the THDM+D model can produce hyperon CP asymmetries near 1e-3 while evading kaon, DM, and collider constraints, with the main caveat being the leading-order chiral LEC estimates. read the letter →

arxiv 2502.09603 v2 pith:FVGLNG5Y submitted 2025-02-13 hep-ph hep-ex

Xiao-Gang He , Xiao-Dong Ma , Jusak Tandean , German Valencia This is my paper · ORCID
classification hep-phhep-ex
keywords B+toK+invisibledecaydarkmattertwo-Higgs-doubletmodeldarkonCPviolationhyperondecayskaonmixingBelleIIanomaly
topics Dark Matter
open problems Dark Matter
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 tries to establish that the Belle II excess in $B^+\to K^+\nu\bar\nu$ can be explained by invisible dark-matter pairs in a specific two-Higgs-doublet model, and that the same model produces $CP$ violation in hyperon decays large enough for coming experiments. The model, THDM+D, adds a real scalar 'darkon' to a type-III two-Higgs-doublet model; its tree-level flavor-changing couplings generate both $b\to sDD$ and new strangeness-changing four-quark operators. After imposing dark-matter relic density, direct-detection, kaon, and collider constraints, the paper's central numerical result is that the new $CP$ asymmetries in $\Lambda$ and $\Xi$ nonleptonic decays can reach the $10^{-3}$ level, compared with standard-model values near $10^{-5}$. If true, this converts the Belle II anomaly into a concrete target for hyperon $CP$ searches at BESIII, Belle II, LHCb, PANDA, and the Super Tau Charm Facility.

What carries the argument

The central object is THDM+D: a type-III two-Higgs-doublet model augmented by a real $Z_2$-odd scalar singlet $D$ (the darkon) that plays the role of dark matter. After integrating out the heavy neutral Higgses, the model produces a dimension-six operator $D^2\bar q q'$ responsible for $b\to sDD$, and tree-level four-quark operators $Q_u$, $Q_\pm$ responsible for strangeness-changing transitions. The argument is carried by the leading-order chiral realization of these four-quark operators in eqs. (31)-(32), with low-energy constants estimated from factorization and bag-model matrix elements. That realization yields new weak-phase differences $\xi_{1B}^{\Lambda,\mathrm{new}}-\xi_{1A}^{\Lambda,\mathrm{new}}$ and the corresponding $\Xi$ version, which enter the $CP$ asymmetry through $A_{\mathrm{CP}}\simeq -\tan(\delta_{1B}-\delta_{1A})\tan(\xi_{1B}-\xi_{1A})$. The resulting correlation $A_{\mathrm{CP}}^{\Xi,\mathrm{new}}\simeq -0.11\,A_{\mathrm{CP}}^{\Lambda,\mathrm{new}}$ is what makes the hyperon observables a distinctive test of the scenario.

What would settle it

Run a high-statistics measurement of $A_{\mathrm{CP}}$ in $\Lambda\to p\pi^-$ and $\Xi^-\to\Lambda\pi^-$ with sensitivity below $10^{-4}$: finding no asymmetry and no weak-phase difference at the predicted $10^{-2}$ level would directly contradict the paper's claim. Alternatively, a lattice-QCD evaluation of the hyperon nonleptonic matrix elements of $Q_u$ and $Q_\pm$ could show that the chiral estimates are off by more than the claimed factor of two.

Watch

Extended reading notes

Core claim

The authors claim that one ultraviolet-complete model can simultaneously explain the Belle II $B^+\to K^+\nu\bar\nu$ excess (with the invisibles being $D\bar D$ pairs), reproduce the observed dark-matter relic density, evade direct-detection bounds, and generate $CP$ violation in hyperon decays well above standard-model expectations. With the parameter space constrained by $\Delta M_K$, $\varepsilon$, $\varepsilon'/\varepsilon$, kaon decay amplitudes, dark-matter relic density, Migdal-effect direct searches, and global SMEFT limits, they find new contributions to the $\Lambda$ and $\Xi$ decay asymmetries up to $A_{\mathrm{CP}}^{\Lambda,\mathrm{new}}\simeq 3.5\times 10^{-3}$, and $A_{\mathrm{CP}}^{\Xi,\mathrm{new}}\simeq -0.11\,A_{\mathrm{CP}}^{\Lambda,\mathrm{new}}$. Taking the current $2\sigma$ bound on $A_{\mathrm{CP}}^{\Lambda}+A_{\mathrm{CP}}^{\Xi}$ at face value reduces the reachable $\Lambda$ asymmetry to about $1.6\times 10^{-3}$. Both ranges sit orders of magnitude above the standard-model central expectations and are potentially testable with upcoming data.

Load-bearing premise

The load-bearing assumption is that the leading-order chiral realization of the new four-quark operators, with its factorization and bag-model low-energy constants, gives the new weak-phase differences to within about a factor of two; if those constants or the chiral truncation are wrong, the predicted hyperon $CP$ asymmetries could be much smaller and the observability claim would not survive.

Editorial extensions

If this is right

  • If the central claim is right, hyperon $CP$ asymmetries near $10^{-3}$ become discoverable with the sensitivities projected for BESIII, Belle II, LHCb, PANDA, and the Super Tau Charm Facility, instead of being buried below the $10^{-5}$ standard-model level.
  • The same couplings that fit the Belle II excess and the dark-matter relic density can be probed through kaon observables ($\Delta M_K$, $\varepsilon$, $\varepsilon'/\varepsilon$), so kaon and hyperon measurements provide complementary handles on the model.
  • The predicted sign and size correlation $A_{\mathrm{CP}}^{\Xi,\mathrm{new}}\simeq -0.11\,A_{\mathrm{CP}}^{\Lambda,\mathrm{new}}$ gives a specific two-asymmetry pattern that future data can confirm or refute.
  • Current data already place nontrivial pressure on the model: taking the $2\sigma$ range of $A_{\mathrm{CP}}^{\Lambda}+A_{\mathrm{CP}}^{\Xi}$ seriously caps the new $\Lambda$ asymmetry near $1.6\times 10^{-3}$, so near-term measurements can test the upper part of the allowed range.

Reading between the lines

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

  • If the Belle II excess survives with more data, hyperon $CP$ searches effectively become a dark-matter probe, since a positive asymmetry near the predicted size would be hard to produce without the same flavor-changing couplings.
  • The narrow phase requirement ($|\arg Y_{sd}|\simeq 84^\circ$) implies the observable prediction is fine-tuned; a null result would therefore not rule out the model but would restrict it to a small region, which future global fits could test.
  • A lattice-QCD determination of the nonleptonic hyperon matrix elements, which the authors call for, could reduce the factor-of-two uncertainty and turn the predicted range into a sharp benchmark for the planned experiments.
  • The mechanism suggests that searches for $CP$ violation in other hyperon modes, such as $\Sigma$ decays, may also carry new-physics sensitivity in this model, though the paper does not compute those rates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. Motivated by the Belle II measurement of B^+ -> K^+ + invisible, this paper studies a type-III two-Higgs-doublet model with a real singlet scalar darkon (THDM+D). The heavy neutral Higgses mediate b -> s D D transitions and also generate, through complex FCNC Yukawa couplings, the four-quark operators Q_u and Q_± at tree level. The authors match these operators onto a chiral Lagrangian for hyperon and kaon decays, derive explicit expressions for the new weak-phase differences and the CP asymmetries A_Λ_CP and A_Ξ_CP in eqs. (40)-(41), and then subject the parameter space to constraints from the B→KDD rate, DM relic density via darkon annihilation into light mesons, DM direct detection including the Migdal effect, kaon mixing and ε/ε′/ε, K→ππ amplitudes, and global SMEFT bounds. A random scan with m_H = 1 TeV and a representative relic-density-compatible point yields |A_Λ_CP^new| values up to about 3.5×10^{-3} (and up to about 1.6×10^{-3} after taking the 2σ A_Λ + A_Ξ data at face value), with A_Ξ_CP^new ≈ -0.11 A_Λ_CP^new. The authors conclude that hyperon CP asymmetries can substantially exceed the SM expectations and be testable at BESIII, Belle II, LHCb, PANDA, and a future Super Tau Charm Facility.

Significance. If correct, the paper provides a coherent, UV-complete dark-sector scenario in which one set of complex FCNC couplings simultaneously addresses the B→K+invisible excess, the DM relic density, and kaon/hyperon CP observables. The strengths of the manuscript are that the model is explicitly defined, the matching and the scan are reported in enough detail to reproduce the main steps, the predicted hyperon asymmetries are outputs of a random scan rather than fits to CP-violating data, and the paper is candid about the uncertainties in the hyperon amplitude estimates. The claimed complementarity between kaon and hyperon probes is substantiated by the different ways C_- and C_±,u enter eqs. (45), (46), (52), and (41). The main weakness is that the numerical reach of the central prediction is controlled by low-energy constants estimated at leading order from factorization plus the MIT bag model; without a robustness study, the statement that values around 10^{-3} are 'potentially discoverable' remains fragile.

major comments (3)
  1. [Section V.C and Appendix B, eqs. (B2)-(B3), (38), (40)-(41)] The central quantitative claim—that |A_Λ_CP| can reach ~10^{-3}—is linearly controlled by the low-energy constants introduced in eqs. (31)-(32) and estimated in Appendix B. The paper itself states in the final paragraph of Section V.C that these evaluations 'involve significant uncertainties, possibly up to factors of two' and that higher-order chiral contributions can be comparable to the leading-order terms. Because the weak-phase differences in eq. (40) are ratios of amplitudes whose numerators are linear in the LECs, and because some contributions have opposite signs (e.g., the coefficients of C_u in A and B in eq. (38)), a factor-of-two error in a dominant bag-model constant from eq. (B3) could suppress the predicted asymmetry below the observability threshold rather than merely rescale it. Please add a quantitative sensitivity analysis that varies the LECs in eqs. (B2)-(B3) over a conservative range and reports the resulting spread of A_Λ_CP and A_Ξ_CP; if the claimed order of magnitude is to be robust, the paper should demonstrate it explicitly.
  2. [Section V, eqs. (31)-(32)] The set of leading-order chiral operators realizing Q_u and Q_± is presented as a list of terms, but the completeness of this operator basis is not demonstrated. If additional independent operators with the same chiral transformation properties exist at leading order, they would contribute to the amplitudes in Appendix A with new undetermined constants and could alter the phase differences in eq. (40). Please provide a group-theoretic enumeration of the possible leading-order structures (or otherwise justify that the listed operators are exhaustive), and include this source of uncertainty in the sensitivity analysis requested above.
  3. [Section V.C, eq. (53) and Figures 5-7] The numerical exploration is anchored to a single representative point, |C_S,ss_dD| = 0.08/TeV, R_d/s = -0.04, |Rhat_d/s| = 1, and m_H = 1 TeV, but the corresponding darkon mass m_D is not specified. Because the relic-density and direct-detection constraints vary strongly with m_D (Fig. 1), and because the authors note that only narrow ranges of arg Y_sd yield large asymmetries, the scan needs more documentation to support the existence claim: the number of generated points, the random distributions, the acceptance rates, and the number (and coordinates) of surviving points with |A_Λ_CP| > 10^{-3} after imposing the A_Λ + A_Ξ constraint in eq. (43). Without this, the reader cannot assess whether the result is a robust prediction or a fine-tuned corner.
minor comments (4)
  1. [Eq. (37)] The display formula for A_CP is typeset as 'ACP = α + α / α − α', which is ambiguous; use overline-α and explicit parentheses to distinguish (α + \bar α)/(α − \bar α).
  2. [Figure 1 caption] In the left panel caption, 'dashed cures' should be 'dashed curves'.
  3. [Section V.C, sentence before eq. (53)] The text refers to 'the darkon- H interactions'; this should be 'darkon-H interactions'.
  4. [Figure 6 caption] The horizontal dashed and dotted lines are described only in the body of the text; the caption should briefly define them for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hyperon CP asymmetries are computed from scanned Yukawa couplings under independent DM, kaon, and collider constraints, and are not fitted to CP observables.

full rationale

The paper's derivation chain is non-circular. The THDM+D is matched to LEFT operators C_u and C_± (Eqs. 26-27), the operators are realized in leading-order chiral perturbation theory with low-energy constants estimated in Appendix B from factorization plus bag-model matrix elements, and the new hyperon amplitudes are computed in Appendix A. The weak-phase differences are extracted by the standard ratio of the new amplitude to the measured non-CP amplitudes A_exp and B_exp (Eq. 40), and then converted to A_CP using measured strong phases (Eq. 41). Crucially, no CP-asymmetry measurement enters the extraction or the parameter scan. The scan instead imposes independent constraints from DM relic density, PandaX-4T Migdal direct detection, ΔM_K, ε, ε'/ε, and Re A_0,2. The A_CP values are outputs, not inputs; the paper even shows that the direction in parameter space most sensitive to hyperon CP, C_-u, is unconstrained by the kaon measurements, which is a genuine prediction rather than a forced fit. The self-citations ([2], [27], [33], [79]) supply previously derived formulas and SM benchmarks, but these are standard, published, and externally checkable results, and none is invoked as a uniqueness theorem or as a substitute for the present calculation. The authors' own caveat that the chiral evaluations 'involve significant uncertainties, possibly up to factors of two' is a robustness/correctness limitation, not evidence of circularity. Overall, the central claim is self-contained against external data and does not reduce to its inputs by construction.

Assumptions & free parameters 8 free parameters · 10 assumptions · 2 invented entities

The central claim depends on a large set of tunable model parameters: a light darkon mass, a heavy Higgs mass, a portal coupling lambda_3, and complex Yukawa couplings Ydd, Yss, and Ysd. The large hyperon CP asymmetry additionally requires a specific phase of Ysd and relies on leading-order chiral low-energy constants estimated from factorization and the bag model. The model also assumes the type-III 2HDM structure, an exact Z2 symmetry, degenerate heavy scalars, Yds = 0, EFT matching at 1 TeV, and that the Belle II excess is real new physics. These are model assumptions rather than independently established facts.

free parameters (8)
  • m_D (darkon mass) = 0.4 to 0.9 GeV in scans
    Chosen below the K* threshold so that B to KDD and darkon annihilation to light mesons work; the mass is scanned, not predicted.
  • m_H (heavy Higgs mass) = 1 TeV degenerate H, A, H+
    Sets the overall scale of all new-physics operator coefficients and is chosen for convenience while keeping constraints satisfied.
  • lambda_3 (darkon-Higgs portal coupling) = |lambda_3| below 4 pi, with |lambda_1| much smaller than |lambda_3|
    Controls the darkon-quark and darkon-meson interactions; its magnitude is scanned and its phase is set real by convention.
  • Ydd (complex Yukawa coupling) = Scanned with |Ydd| below sqrt(4 pi)
    Contributes to C- and to the hyperon and kaon amplitudes; constrained by kaon mixing, kaon decays, and dark matter data.
  • Yss (complex Yukawa coupling) = Scanned with |Yss| below sqrt(4 pi)
    Sets the size of the darkon-strange couplings that fix the relic density and DM-nucleon scattering; tied to C_S,ss_dD.
  • Ysd (complex Yukawa coupling) = Scanned; |Ysd| around 0.02 to 0.1 and arg Ysd near 84 or 96 degrees for large A_CP
    The main source of the new weak phase; large hyperon asymmetries appear only in narrow phase windows around these values.
  • R_d/s and |hat R_d/s| ratios = R_d/s = -0.04 and |hat R_d/s| = 1 for the representative point
    These ratios summarize the relative sizes of Ydd, Yss, and Ysd and are chosen from figure 1 to satisfy relic density and direct detection constraints.
  • |C_S,ss_dD| relic-density scale = 0.08 per TeV for the representative point
    Fixed by the relic density requirement for the chosen darkon mass; this value also enters the DM-nucleon cross section.
assumptions (10)
  • ad hoc to paper The darkon D is a SM-gauge singlet stabilized by an exact Z2 symmetry.
    Introduced in Section II.A to guarantee dark matter stability; this is a model assumption with no independent experimental input.
  • domain assumption The scalar sector is a type-III two-Higgs-doublet model with alpha = -pi/2, beta = 0, v2 = 0, no Yu_2, degenerate heavy Higgses, and no heavy-Higgs couplings to leptons.
    Section II.A defines this simplified setup; it makes the light Higgs SM-like and suppresses tree-level H to WW and ZZ, but is a strong model restriction.
  • domain assumption The heavy Higgs H, rather than the SM-like h, is the dominant mediator between the darkon and quarks.
    This follows from choosing |lambda_1| much smaller than |lambda_3| in Section II.A, a condition chosen to avoid constraining the SM-like Higgs.
  • ad hoc to paper Yds is set to zero to remove short-distance Yds and Ysd contributions to kaon mixing.
    Stated in Section V before eq. (27); this simplification is needed for the stated kaon mixing treatment and excludes a potentially relevant parameter direction.
  • standard math Tree-level SMEFT and LEFT matching at a new-physics scale around 1 TeV, with QCD running factors eta_b = 1.8, eta_s = 2.3, eta_u = 4.7, and eta_d = 4.8, captures the relevant low-energy physics.
    Used in Sections II.B, II.C, and around eq. (34); higher-order matching and electroweak corrections are assumed negligible.
  • domain assumption The darkon mass lies between (m_K - m_pi)/2 and roughly 900 MeV, and relic density is set by darkon annihilation into light pseudoscalar mesons described by leading-order chiral perturbation theory.
    Section IV.A restricts the mass range and uses chiral perturbation theory at temperatures of tens of MeV; the lower-mass region is discarded because it conflicts with relic and direct detection requirements.
  • domain assumption The chiral realization of Qu and Qpm in eqs. (31)-(32), with low-energy constants estimated from factorization and the bag model in Appendix B, is adequate for the hyperon CP amplitudes.
    This is the most fragile premise for the central CP claim. The paper admits in Section V.C that uncertainties may be up to factors of two and that higher-order chiral contributions can be comparable to leading order.
  • domain assumption The Belle II B+ to K+ invisible excess is treated as a new-physics signal rather than a fluctuation.
    The Introduction notes the excess is 2.1 sigma above the standard model average and still needs confirmation; the model is built around interpreting it as real new physics.
  • standard math Empirical strong phase shifts and experimental hyperon A and B amplitudes are used to estimate the new weak-phase differences.
    Used in Section V.A around eqs. (36)-(40), following standard practice for hyperon CP estimates; the Xi strong phase difference has a 1.1 degree uncertainty that propagates into the result.
  • domain assumption The global SMEFT limits from ref. [88] apply to the combinations of Yukawa products in this model.
    Used in Section V.C to bound products such as |Yss Ydd|; it assumes the operator normalization and renormalization group treatment of ref. [88] match this model.
invented entities (2)
  • Darkon D, a real scalar singlet dark matter particle independent evidence
    purpose: Provides the invisible final state in B+ to K+ DD, explains the Belle II excess, and sets the dark matter relic abundance through annihilation to light mesons.
    The model gives a specific mass range below the K* threshold and spin-independent DM-nucleon couplings constrained by Migdal-effect searches, so the particle could in principle be seen in direct detection or in B to K plus invisible events. No positive signal exists yet, and the required couplings are tuned.
  • Heavy Higgs bosons H, A, and H+ of the type-III two-Higgs-doublet sector
    purpose: Mediate the darkon-quark operators and the four-quark flavor-changing operators that feed hyperon and kaon CP violation.
    They are constrained indirectly through kaon mixing, kaon decays, B meson processes, and global SMEFT fits, but the paper does not provide a direct discovery signature or predicted mass that would single them out in collider data.

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Pith. "Pith review of $B\to K{+}$invisible, dark matter, and $CP$ violation in hyperon decays." pith.science (2026). https://pith.science/paper/FVGLNG5Y

@misc{pith2026250209603,
  author       = {Pith},
  title        = {Pith review of: $B\to K+$invisible, dark matter, and $CP$ violation in hyperon decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVGLNG5Y}},
  note         = {Machine review of arXiv:2502.09603}
}
abstract

Recently the Belle II Collaboration has reported a measurement of the $B^+\to K^+\nu\bar\nu$ rate that is higher than the standard-model expectation. Since the emitted neutrinos are unobserved, the excess could be due to the $B^+$ decaying into a $K^+$ and a dark-matter pair. We entertain this possibility in a two-Higgs-doublet model supplemented with a real singlet scalar boson acting as the dark matter. This model also accommodates strangeness-changing interactions providing new sources of $CP$ violation which can affect hyperon and kaon nonleptonic transitions. We find that the resulting $CP$ violation in the hyperon sector can be significant, reaching the current empirical bounds, after taking into account constraints from kaon mixing and decay and from dark-matter relic-density data and direct searches including the Migdal effect. We demonstrate that the hyperon and kaon processes are complementary probes of this new-physics scenario. Its prediction for sizable hyperon $CP$ violation is potentially testable in ongoing experiments, such as BESIII, Belle II, and LHCb, and in next-generation ones like PANDA and at the Super Tau Charm Facility.

Figures

Figures reproduced from arXiv: 2502.09603 by the authors.

Figure 1
Figure 1. FIG. 1. Left: the values of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Leading-order diagrams for the new contributions to two-body nonleptonic (a) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Diagrams for long-distance contribution to kaon mixing with either both hollow squares or one [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Parameter space in Re [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Top: the ranges of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Distributions of [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

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Cited by 1 Pith paper

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