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First measurement of the CKM angle $\phi_3$ with $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$ decays

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Pith's one-line read This paper reports the first model-independent measurement of the CKM angle $\phi_3$ using $B^{\pm}\to D(K_S^0\pi^+\pi^-\pi^0)K^{\pm}$ decays, finding $\phi_3 = (5.7^{+10.2}_{-8.8}\pm 3.5 \pm 5.7)^{\circ}$, consistent with the world…

desk verdict A legitimate first measurement that opens a new D four-body channel for gamma/phi3, with uncertainties too large to move the world average but a thorough experimental job that deserves a serious referee. read the letter →

arxiv 1908.09499 v2 pith:SZ2EOIP2 submitted 2019-08-26 hep-ex

Belle Collaboration: P. K. Resmi , J. Libby , K. Trabelsi , I. Adachi , H. Aihara , S. Al Said , D. M. Asner , V. Aulchenko
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T. Aushev V. Babu I. Badhrees A. M. Bakich C. Beleño J. Bennett V. Bhardwaj B. Bhuyan T. Bilka J. Biswal A. Bozek M. Bračko M. Campajola D. Červenkov A. Chen B. G. Cheon H. E. Cho K. Cho Y. Choi S. Choudhury D. Cinabro S. Cunliffe N. Dash G. De Nardo F. Di Capua S. Di Carlo Z. Doležal T. V. Dong S. Eidelman D. Epifanov J. E. Fast T. Ferber B. G. Fulsom R. Garg V. Gaur N. Gabyshev A. Garmash A. Giri P. Goldenzweig B. Golob Y. Guan K. Hayasaka H. Hayashii W.-S. Hou K. Huang T. Iijima K. Inami G. Inguglia A. Ishikawa R. Itoh M. Iwasaki Y. Iwasaki W. W. Jacobs H. B. Jeon Y. Jin D. Joffe A. B. Kaliyar K. H. Kang G. Karyan T. Kawasaki C. Kiesling D. Y. Kim K. T. Kim S. H. Kim K. Kinoshita P. Kodyš D. Kotchetkov P. Križan R. Kroeger P. Krokovny T. Kuhr R. Kumar A. Kuzmin Y.-J. Kwon S. C. Lee Y. B. Li K. Lieret D. Liventsev P.-C. Lu T. Luo C. MacQueen M. Masuda T. Matsuda D. Matvienko M. Merola K. Miyabayashi R. Mizuk G. B. Mohanty H. K. Moon T. Nakano M. Nakao K. J. Nath M. Nayak M. Niiyama N. K. Nisar S. Nishida K. Nishimura S. Ogawa H. Ono Y. Onuki P. Pakhlov G. Pakhlova B. Pal S. Pardi H. Park T. K. Pedlar R. Pestotnik L. E. Piilonen E. Prencipe M. T. Prim M. Ritter M. Röhrken G. Russo D. Sahoo Y. Sakai S. Sandilya L. Santelj T. Sanuki V. Savinov O. Schneider G. Schnell C. Schwanda A. J. Schwartz Y. Seino K. Senyo M. E. Sevior V. Shebalin C. P. Shen J.-G. Shiu B. Shwartz E. Solovieva M. Starič Z. S. Stottler J. F. Strube T. Sumiyoshi M. Takizawa U. Tamponi K. Tanida F. Tenchini M. Uchida T. Uglov S. Uno Y. Usov R. Van Tonder G. Varner A. Vinokurova V. Vorobyev A. Vossen B. Wang C. H. Wang M.-Z. Wang X. L. Wang S. Watanuki E. Won S. B. Yang H. Ye Z. P. Zhang V. Zhilich V. Zhukova
This is my paper · ORCID
classification hep-ex
keywords CKManglephi3gammaB±→DK±decaysmodel-independentstrong-phasedifferenceKS0pi+pi-pi0CPviolationBelleexperiment
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

This paper reports the first model-independent measurement of the CKM unitarity-triangle angle $\phi_3$ using the four-body decay chain $B^{\pm}\to D(K_S^0\pi^+\pi^-\pi^0)K^{\pm}$, where $D$ denotes a $D^0$ or $\overline{D}^{0}$ meson. The analysis uses the full Belle data set of $772\times 10^{6}$ $B\overline{B}$ events and external strong-phase inputs measured at the charm threshold. It obtains $\phi_3 = (5.7^{+10.2}_{-8.8}\pm 3.5 \pm 5.7)^{\circ}$ and $r_B = 0.323 \pm 0.147 \pm 0.023 \pm 0.051$, with a $95\%$ confidence interval $(-29.7, 109.5)^{\circ}$ that is consistent with the world average. A sympathetic reader should take this as evidence that the four-body $D\to K_S^0\pi^+\pi^-\pi^0$ mode is a viable addition to the suite of $B\to DK$ decays used to pin down $\phi_3$, with the dominant uncertainties coming from statistics and from external strong-phase knowledge.

What carries the argument

The engine is the binned partial-rate identity built from Eqs. (2.5) and (2.6): for each phase-space bin $i$, the $B^{\pm}$ partial widths are proportional to $K_i + r_B^2 \overline{K}_i + 2\sqrt{K_i\overline{K}_i}(c_i x_{\pm} - s_i y_{\pm})$, where $K_i$ and $\overline{K}_i$ are flavour-tagged $D$ fractions, $c_i$ and $s_i$ are the external amplitude-weighted strong-phase averages, and $(x_{\pm},y_{\pm})$ encode $r_B$, $\delta_B$, and $\phi_3$. The analysis determines $K_i$ from $D^{*\pm}$-tagged events, extracts $(x_{\pm},y_{\pm})$ from a simultaneous likelihood fit to $B^{\pm}\to D K^{\pm}$ and the control mode $B^{\pm}\to D\pi^{\pm}$, corrects for efficiency and bin-migration using a migration matrix, and finally converts $z=(x_+,y_+,x_-,y_-)$ into $(\phi_3,r_B,\delta_B)$ with a frequentist confidence-interval construction. The machinery works because the binning keeps the strong-phase variation within each bin small enough that the external $c_i$, $s_i$ inputs capture the interference.

What would settle it

A measurement of the same mode with a much larger data set and independently determined strong-phase inputs — for example, by re-extracting $(c_i,s_i)$ from an order-of-magnitude larger charm-threshold sample and seeing whether the binned $B^{\pm}$ yields still converge to the same $\phi_3$ — would settle the question; a shift larger than the quoted uncertainties would show the external inputs do not transfer.

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Extended reading notes

Core claim

The central claim is that $\phi_3$ can be extracted from $B^{\pm}\to D(K_S^0\pi^+\pi^-\pi^0)K^{\pm}$ without resorting to a model of the charm-meson decay amplitude. The paper divides the five-dimensional $D$ phase space into nine exclusive bins chosen around resonances such as $\omega$, $K^{*}$, and $\rho$, and uses externally measured bin-averaged strong-phase parameters $c_i$, $s_i$ as input. From the binned $B^{\pm}\to D K^{\pm}$ yields it determines the Cartesian parameters $x_{\pm} = r_B\cos(\delta_B \pm \phi_3)$ and $y_{\pm} = r_B\sin(\delta_B \pm \phi_3)$, then converts them into $\phi_3$, $r_B$, and $\delta_B$ through a frequentist treatment. The quoted result $\phi_3 = (5.7^{+10.2}_{-8.8}\pm 3.5 \pm 5.7)^{\circ}$ is consistent with the world average within two standard deviations, and the paper notes a local likelihood minimum near $\phi_3 = 75^{\circ}$, $\delta_B = 155^{\circ}$, reflecting the known twofold ambiguity.

Load-bearing premise

The whole analysis assumes that the measurements of how the D meson's decay amplitude changes in phase across the nine regions, made with charm-meson data at a different collision energy, are still correct when used to interpret the B-meson data, with any slight mismatch in mass resolution absorbed by one extra smearing adjustment.

Editorial extensions

If this is right

  • The four-body final state $D\to K_S^0\pi^+\pi^-\pi^0$ becomes a usable, model-independent channel for $\phi_3$; its large branching fraction and resonance-rich phase space make it competitive with the three-body $K_S^0\pi^+\pi^-$ mode.
  • Combining this result with Belle's earlier model-independent $D\to K_S^0\pi^+\pi^-$ measurements shifts the combined central value to $\phi_3 = (74^{+13}_{-14})^{\circ}$, a modest improvement over $(78^{+14}_{-15})^{\circ}$ without it.
  • The quoted $95\%$ confidence interval $(-29.7, 109.5)^{\circ}$ contains the world average, so the measurement adds a tree-level $\phi_3$ constraint that is consistent with the standard model and offers no hint of new physics.
  • The three quoted uncertainties — statistical, experimental systematic, and external strong-phase — show where future gains lie: larger $B$ samples and better charm-threshold inputs both directly reduce the total error.

Reading between the lines

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

  • If the same binned machinery were run with a finer binning informed by an amplitude model, a shift in $\phi_3$ beyond the quoted total uncertainty would indicate that strong-phase variation inside the current nine bins is not fully captured by the bin-averaged inputs.
  • Because this analysis deliberately leaves $B^{\pm}\to D\pi^{\pm}$ out of the $\phi_3$ extraction, a natural extension is to check whether its measured $(x_{\pm},y_{\pm})$, currently consistent with zero CP violation, would shift the central value when included; that check is not performed here.
  • A cross-experiment transfer check would be to compare the $c_i,s_i$ inputs obtained from two independent charm-threshold data sets; consistency within uncertainties would validate the approach for other four-body channels, and any discrepancy would set the floor on the external uncertainty.
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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

1 major / 4 minor

Summary. The paper reports the first model-independent measurement of the CKM angle φ3 using B±→D(K_S^0 π^+π^-π^0)K± decays, based on the full Belle data set of 772×10^6 BB events. The analysis follows the binned Giri-Grossman-Soffer-Zupan method: the five-dimensional D→K_S^0π^+π^-π^0 phase space is divided into nine exclusive bins, the c_i/s_i strong-phase inputs are taken from an external CLEO-c measurement (ref. [13]), the K_i and \bar{K}_i fractions are measured from a Belle D*±−tagged sample, and the parameters (x±,y±) are extracted from a simultaneous fit to 36 B±→DK± and B±→Dπ± samples. The quoted results are φ3=(5.7^{+10.2}_{-8.8}±3.5±5.7)° and rB=0.323±0.147±0.023±0.051, with a 95% confidence interval for φ3 of (-29.7,109.5)°, consistent with the current world average. The paper also includes a combination with previous Belle multibody Dalitz analyses, giving φ3=(74^{+13}_{-14})°.

Significance. If the result holds, this paper establishes a new D-decay mode for model-independent φ3 measurements, introducing a nine-bin partition of a four-body phase space and demonstrating that the mode is viable despite the large statistical uncertainty. The analysis is technically careful: it addresses efficiency and migration corrections, uses a B→Dπ control sample for validation, performs a simultaneous fit over all bins and charge modes, lists a comprehensive systematic table, and uses Feldman-Cousins intervals with an explicit check of the Gaussian assumption. The paper also provides concrete reproducibility material, including bin definitions, efficiency tables, migration matrices, and fit projections. The measurement is statistics-limited and the strong-phase input contributes a sizable systematic component, so the result is best viewed as a proof-of-principle for a future Belle II/BESIII program; the authors state this interpretation clearly. I saw no circularity: the external c_i/s_i inputs and the flavour-tag fractions are independent of the target parameter φ3.

major comments (1)
  1. [Eq. (2.5)] As printed, the first term in Eq. (2.5) is K_i and the second is \bar{K}_i. For B−→DK−, the favoured amplitude produces \bar{D}^0, so the leading term should be \bar{K}_i and the suppressed term K_i. If the overline on the first K_i was lost in typesetting, this is a local rendering problem; however, if the formula was actually used with this ordering, the interpretation of x− and y− would be altered and the derivation in Section 2 would be inconsistent. Please confirm that the overlines in Eqs. (2.5) and (2.6) are correct in the manuscript and that the fit uses the correct ordering.
minor comments (4)
  1. [Section 7] The statement that the K_i and \bar{K}_i values are "in reasonable agreement" with ref. [13] is not quantified. Since the authors note a deviation larger than 3σ in bin 9, which contains only 1.2% of the data, please state whether replacing K_9 and \bar{K}_9 with the values from ref. [13] changes (x±,y±) by a negligible amount or include the numerical effect in the systematics discussion.
  2. [Figure 10] The caption says the combination is shown by the "solid blue curve," but three curves are drawn in the figure; please identify the curves unambiguously (e.g., blue for the combination, green for D→K_S^0π^+π^-, brown for D→K_S^0π^+π^-π^0).
  3. [Section 5.2] The resolution scale factors 1.13±0.02 and 1.09±0.02 are introduced for the ω-mass smearing in the B and D* samples, respectively; it would be helpful to state explicitly how these factors are varied in the "m_{πππ0} resolution" row of Table 9.
  4. [Section 3 / Table 2] The footnote about the statistical uncertainties of the constrained s_i values marked with an asterisk is useful; adding a similar note about the treatment of correlated uncertainties for the c_i values would make the propagation of the external inputs easier for the reader to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: phi3 is extracted from Belle yields using independent external strong-phase inputs from CLEO-c data, with no parameter defined in terms of the target.

full rationale

The derivation chain is self-contained at the level required for a circularity check. The external inputs c_i and s_i are taken from the published CLEO-c quantum-correlation analysis in ref. [13], a separate experiment at the psi(3770) resonance, and are not fitted to the Belle data. The flavour-tagged fractions K_i and K_i are measured from the Belle D* sample in Section 7, and the yields in B±->Dh± bins are fitted using eqs. (2.5) and (2.6). The target parameters (phi3, rB, deltaB) are then obtained from the resulting (x±, y±) via the Feldman-Cousins procedure in Section 10. None of the inputs is defined in terms of phi3, and no equation reduces the quoted phi3 value to an assumed input. The fact that two authors of the present paper also appear on ref. [13] does not create circularity, because the CLEO-c strong-phase measurement is a real external result that does not assume the CKM angle and is falsifiable outside the fitted Belle values. The concern that the CLEO-c systematic uncertainties may not transfer exactly to the Upsilon(4S) environment is a legitimate experimental systematic-risk issue, not a circularity, and the paper quotes a separate uncertainty for it. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported from the authors, and no ansatz smuggled in via citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities and no ad hoc free parameters. It relies on the established binned GGSZ formalism, on external strong-phase inputs measured at CLEO-c, and on MC-derived efficiency and migration corrections. The quantities phi3, rB, and deltaB are fitted outputs, not additional assumptions.

assumptions (4)
  • domain assumption The B± to DK± amplitude factorizes with rB and deltaB independent of the D decay phase-space point (Eq. 2.1).
    Central modelling assumption of the GGSZ formalism; if it failed, Eqs. (2.5) and (2.6) would be incorrect. Standard and theoretically well motivated for two-body B to DK decays.
  • domain assumption The CLEO-c strong-phase inputs c_i and s_i are valid for the same D to K_S0 pi+ pi- pi0 decay in the Belle environment after efficiency and migration corrections.
    Used in Sections 3 and 8; the paper does not re-measure them but propagates their uncertainties.
  • domain assumption No significant CP violation in D0 to K_S0 pi+ pi- pi0 decays, so the D amplitudes are related purely by the strong-phase difference.
    Implicit in the definition of c_i and s_i in Section 2 and in the CLEO-c extraction [13]; standard for charm strong-phase measurements.
  • domain assumption MC simulation describes selection efficiencies and bin migration, with resolution differences corrected by Gaussian smearing.
    Efficiency and migration corrections in Sections 5, 9, and Appendix A rely on signal MC; the model is validated with data but not proven.

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Cite this review

Pith. "Pith review of First measurement of the CKM angle $\phi_3$ with $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$ decays." pith.science (2026). https://pith.science/paper/SZ2EOIP2

@misc{pith2026190809499,
  author       = {Pith},
  title        = {Pith review of: First measurement of the CKM angle $\phi_3$ with $B^\pm\to D(K_\rm S^0\pi^+\pi^-\pi^0)K^\pm$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZ2EOIP2}},
  note         = {Machine review of arXiv:1908.09499}
}
abstract

We present the first model-independent measurement of the CKM unitarity triangle angle $\phi_3$ using $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$ decays, where $D$ indicates either a $D^{0}$ or $\overline{D}^{0}$ meson. Measurements of the strong-phase difference of the $D \to K_{\rm S}^0\pi^+\pi^-\pi^0$ amplitude obtained from CLEO-c data are used as input. This analysis is based on the full Belle data set of $772\times 10^{6}$ $B\overline{B}$ events collected at the $\Upsilon(4S)$ resonance. We obtain $\phi_3 = (5.7~^{+10.2}_{-8.8} \pm 3.5 \pm 5.7)^{\circ}$ and the suppressed amplitude ratio $r_{B} = 0.323 \pm 0.147 \pm 0.023 \pm 0.051$. Here the first uncertainty is statistical, the second is the experimental systematic, and the third is due to the precision of the strong-phase parameters measured from CLEO-c data. The 95% confidence interval on $\phi_3$ is $(-29.7,~109.5)^{\circ}$, which is consistent with the current world average.

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