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Radiative three-body D-meson decays in and beyond the standard model

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Nico Adolph, Joachim Brod, Gudrun Hiller

High Energy Physics - Phenomenology · Submitted 29 Sep 2020 · 68 references

Pith's verdict: First QCDF analysis of D→P1P2γ plus a full HHχPT treatment of the FCNC modes, with a genuinely useful AFB null-test—but the abstract's '~10^-3' for the Cabibbo-favored modes is not supported by the paper's own table for one of the two modes.

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arXiv ID2009.14212 (version 2)
Pith IDTMABXDB4, Pith's permanent record of this paper and its review
CitationGudrun Hiller, Joachim Brod, Nico Adolph. "Radiative three-body D-meson decays in and beyond the standard model." arXiv:2009.14212 (2020). https://pith.science/paper/TMABXDB4
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The paper

The paper claims

Radiative three-body D-meson decays are predicted to reach branching ratios of order $10^{-3}$ (K$\pi\gamma$) and $10^{-5}$ ($\pi^+\pi^-\gamma$, $K^+K^-\gamma$), with Dalitz-plot CP asymmetries up to $10^{-3}$ from new-physics dipole…

The reading

Radiative three-body charm decays $D\to P_1P_2\gamma$ have never been measured, and this paper predicts their rates and shapes from two QCD methods. The central numbers are branching ratios around $10^{-3}$ for the Cabibbo-favored $K\pi\gamma$ modes (the ones with the largest weak-interaction mixing factors) and around $10^{-5}$ for the singly Cabibbo-suppressed $\pi^+\pi^-\gamma$ and $K^+K^-\gamma$ modes, putting them in reach of high-luminosity $e^+e^-$ charm factories. The forward-backward asymmetry vanishes in leading-order QCD factorization but not in heavy-hadron chiral perturbation theory, so the measured distributions can identify which decay mechanism the standard model actually uses. In the suppressed modes, new-physics dipole operators allowed by present constraints can raise the $D\to\pi^+\pi^-\gamma$ and $D\to K^+K^-\gamma$ CP asymmetries at the resonance regions to a few times $10^{-3}$, far above the standard-model values around $10^{-4}$. The paper therefore proposes the SM-dominated $K\pi\gamma$ decays as standard candles to calibrate the QCD tools before using the flavor-changing neutral-current (FCNC) modes as new-physics probes.

What carries the argument

The central machinery is the pair of QCD amplitude frameworks together with the forward-backward asymmetry that separates them. In QCDF the amplitude is carried by $A^{\rm WA}_\pm = (G_F e/\sqrt{2})\, \tilde C\, f_D Q_u\, [f_{P_1P_2}(s)/(\lambda_D\, v\cdot k)]\, \sum_{q,q'} V_{cq}^* V_{uq'}$, where $\tilde C=(4/9)C_1+(1/3)C_2$ is a Wilson-coefficient combination, $\lambda_D\simeq\Lambda_{\rm QCD}$ is the nonperturbative annihilation parameter set to $0.1$ GeV, and $f_{P_1P_2}(s)$ is the vacuum-to-$P_1P_2$ form factor taken from electromagnetic pion/kaon form factors or from the dispersive $K\pi$ form factor. In HH$\chi$PT the amplitude instead contains explicit $s$-, $t$-, $u$-channel vector-meson poles and is matched to Low's theorem in the soft-photon region. The discriminating observable is $A_{FB}(s)$, the forward-backward asymmetry in the $P_2$-$\gamma$ angle; it vanishes at leading order in QCDF because the WA form factors depend only on $s$, and it is nonzero in HH$\chi$PT. For the BSM part, the amplitudes receive contributions proportional to $(C_7\pm C_7')$ times tensor form factors $a',b',h'$, which is what generates the enhanced CP asymmetries.

Extended reading notes

Core claim

At its core, the paper establishes a first leading-order QCD-factorization description of $D\to P_1P_2\gamma$ in which the standard-model amplitude is dominated by weak annihilation with the photon emitted from the light quark of the $D$ meson, and a re-derived heavy-hadron chiral perturbation theory description that includes the full $\rho$, $\omega$, $K^\ast$, $\phi$ resonance structure. The two approaches bracket the same order of magnitude for the branching ratios, but they differ characteristically in the distributions: because the QCDF form factors depend only on $s=m_{P_1P_2}^2$, the forward-backward asymmetry vanishes at leading order, while the $t$- and $u$-channel resonances of HH$\chi$PT generate a nonzero asymmetry. Measuring that asymmetry in the SM-dominated $D\to K\pi\gamma$ modes is thus proposed as a data-driven discriminator between the mechanisms. Once the framework is fixed, the same amplitudes predict CP asymmetries in $D\to\pi^+\pi^-\gamma$ and $D\to K^+K^-\gamma$ that stay below about $1.4\times 10^{-4}$ in the SM (for $K^+K^-\gamma$; the $\pi^+\pi^-\gamma$ asymmetry vanishes at this order) but can reach the $10^{-3}$ level for dipole Wilson coefficients saturating the current bounds, especially when the $(s,t)$ Dalitz plot is binned instead of integrated over $t$.

Load-bearing premise

The load-bearing premise is that, in the kinematic region where QCD factorization applies ($s\lesssim 1.5$ GeV$^2$), the standard-model amplitude is dominated by the weak-annihilation diagram with the photon emitted from the $D$ meson's light quark, all other emissions being suppressed by $\Lambda_{\rm QCD}/m_c$ or $\alpha_s$, and that the nonperturbative annihilation parameter $\lambda_D$ can be set to $0.1$ GeV; if those suppressions fail or $\lambda_D$ is much smaller, the quoted branching ratios (proportional to $(1/\lambda_D)^2$) and the vanishing forward-backward asymmetry at leading order no longer follow.

Editorial extensions

If this is right

  • The Cabibbo-favored rates are large enough for first measurements: order $10^{-3}$ for $D^0\to\pi^0 K^0\gamma$ and $D^+\to K^-\pi^+\gamma$ (with $E_\gamma\ge 0.1$ GeV), so Dalitz-plot and asymmetry studies are statistically feasible at current charm factories.
  • A measured nonzero $A_{FB}(s)$ in $D\to K\pi\gamma$ at $s\lesssim 1.5$ GeV$^2$ would rule out the leading-order QCDF mechanism and point to $t/u$-channel resonance dynamics; a null result would support the weak-annihilation picture.
  • Branching-ratio measurements of the SM-like modes yield an empirical value of $\tilde C/\lambda_D$, and with an independent determination of $\lambda_D$ from color-allowed annihilation modes, the QCDF predictions for the FCNC modes become calibrated.
  • CP asymmetries in the suppressed modes, especially $D\to K^+K^-\gamma$ around the $\phi$ resonance and $D\to\pi^+\pi^-\gamma$ around $\rho$, provide SM null tests: at most about $10^{-4}$ in the SM versus up to a few $10^{-3}$ with dipole new physics, provided Dalitz bins avoid the $t$-integrated cancellations.
  • Since the QCDF rates scale as $(0.1\text{ GeV}/\lambda_D)^2$, the quoted SM branching ratios are upper estimates; smaller measured rates would imply a smaller $\lambda_D$ or missing subleading contributions rather than immediately requiring new physics.

Reading between the lines

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

  • Beyond the paper, a binned fit of $d^2\Gamma/ds\, dt$ could separate the two QCD mechanisms more sharply than the $s$-projected forward-backward asymmetry, since leading-order QCDF amplitudes are $t$-independent while HH$\chi$PT contains $t/u$-channel poles.
  • An extension the paper leaves implicit is to apply the same framework to the neutral modes $D^0\to\pi^0\pi^0\gamma$ and $D^0\to K^0 K^0\gamma$, which have no charged-meson bremsstrahlung and may give cleaner dipole constraints.
  • The renormalization-group relation $C_7(m_c)\simeq 0.4(C_7(\Lambda)-C_8(\Lambda))$ connects the radiative-decay CP asymmetries to the hadronic $\Delta A_{CP}$; a joint fit, which the paper does not perform, could test whether a common dipole origin explains both.
  • A model-independent check of the QCDF input follows from the soft-photon endpoint: Low's theorem fixes $d\Gamma/ds \sim 1/(m_D^2-s)$ there, so a measurement of the residual rate after bremsstrahlung subtraction isolates the weak-annihilation form-factor contribution before any new-physics interpretation.
Original abstract

We study radiative charm decays $D \to P_1 P_2 \gamma$, $P_{1,2}=\pi,K$ in QCD factorization at leading order and within heavy hadron chiral perturbation theory. Branching ratios including resonance contributions are around $\sim 10^{-3}$ for the Cabibbo-favored modes into $K \pi \gamma$ and $\sim 10^{-5}$ for the singly Cabibbo-suppressed modes into $\pi^+ \pi^- \gamma, K^+ K^- \gamma$, and thus in reach of the flavor factories BES III and Belle II. Dalitz plots and forward-backward asymmetries reveal significant differences between the two QCD frameworks; such observables are therefore ideally suited for a data-driven identification of relevant decay mechanisms in the standard-model dominated $D \to K \pi \gamma$ decays. This increases the potential to probe new physics with the $D \to \pi^+ \pi^- \gamma$ and $D \to K^+ K^- \gamma$ decays, which are sensitive to enhanced dipole operators. CP asymmetries are useful to test the SM and look for new physics in neutral $|\Delta C|=1$ transitions. Cuts in the Dalitz plot enhance the sensitivity to new physics due to the presence of both $s$- and $t,u$-channel intermediate resonances.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,292 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.66 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study radiative charm decays $D \to P_1 P_2 \gamma$, $P_{1,2}=\pi,K$ in QCD factorization at leading order and within heavy hadron chiral perturbation theory. Branching ratios including resonance contributions are around $\sim 10^{-3}$ for the Cabibbo-favored modes into $K \pi \gamma$ and $\sim 10^{-5}$ for the singly Cabibbo-suppressed modes into $\pi^+ \pi^- \gamma, K^+ K^- \gamma$, and thus in reach of the flavor factories BES III and Belle II. Dalitz plots and forward-backward asymmetries reveal significant differences between the two QCD frameworks; such observables are therefore ideally suited for a data-driven identification of relevant decay mechanisms in the standard-model dominated $D \to K \pi \gamma$ decays. This increases the potential to probe new physics with the $D \to \pi^+ \pi^- \gamma$ and $D \to K^+ K^- \gamma$ decays, which are sensitive to enhanced dipole operators. CP asymmetries are useful to test the SM and look for new physics in neutral $|\Delta C|=1$ transitions. Cuts in the Dalitz plot enhance the sensitivity to new physics due to the presence of both $s$- and $t,u$-channel intermediate resonances.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study radiative charm decays $D \to P_1 P_2 \gamma$, $P_{1,2}=\pi,K$ in QCD factorization at leading order and within heavy hadron chiral perturbation theory. contribution 0.90 0.88 0.05 asserted 0.91
s1 Branching ratios including resonance contributions are around $\sim 10^{-3}$ for the Cabibbo-favored modes into $K \pi \gamma$ and $\sim 10^{-5}$ for the singly Cabibbo-suppressed modes into $\pi^+ \pi^- \gamma, K^+ K^- \gamma$, and thus in reach of the flavor factories BES III and Belle II. result 0.98 0.78 0.21 asserted 0.93
s2 Dalitz plots and forward-backward asymmetries reveal significant differences between the two QCD frameworks; such observables are therefore ideally suited for a data-driven identification of relevant decay mechanisms in the standard-model dominated $D \to K \pi \gamma$ decays. interpretation 0.56 0.78 0.24 asserted 0.95
s3 This increases the potential to probe new physics with the $D \to \pi^+ \pi^- \gamma$ and $D \to K^+ K^- \gamma$ decays, which are sensitive to enhanced dipole operators. interpretation 0.98 0.42 0.23 hedged possibility 0.71
s4 CP asymmetries are useful to test the SM and look for new physics in neutral $|\Delta C|=1$ transitions. background 0.48 0.08 0.09 asserted 0.85
s5 Cuts in the Dalitz plot enhance the sensitivity to new physics due to the presence of both $s$- and $t,u$-channel intermediate resonances. interpretation 0.37 0.41 0.09 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from simulation or computation 0.58.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The weak annihilation diagram for D → P1P2γ. The blue cross indicates the dominant photon emission from the light quark of the D meson. Photon emission from the other quark lines is suppressed by powers of λQCD/mc or αs. where Qu = 2/3 denotes the electric charge of the up-type quarks, and we decomposed P = v mD. The nonperturbative parameter λD ∼ ΛQCD is poorly known and thus source of large theoretical uncertainties. In the following we use λD = 0.1 GeV [4]. For the final states π +π −γ and K+K−γ, the remaining form factors f P1P2 (q,q0) (s) can be expressed in terms of the elect
Figure 2. SM Dalitz plots for the decays D → K 0 π 0γ (upper left), D → K−π +γ (upper right), D → π +π −γ (lower left) and D → K+K−γ (lower right) based on HHχPT at µc = mc. First, we study differential and integrated branching ratios in Section III A. In Section III B we propose to utilze a forward-backward asymmetry, defined below in Eq. (32), to help disentangling the resonance contributions to the branching ratios. This subsequently improves the NP sensitivity of the D → P +P −γ decays. We consider the U-spin link, exploited already for polarization-asymmetries in radiative charm decays…
Figure 3. The SM predictions for the differential branching ratios of the decays
Figure 4. The forward-backward asymmetry AFB(s) (32) as a function of s. The red bands contain only non-resonant contributions. The green, orange and light blue bands contain additional contributions of a specific resonance channel. The dark blue bands are the complete forward-backward asymmetries according to HHχPT. To leading order QCDF AFB(s) = 0. u c u q q 0 u u c u q q 0 u D0 V − 2 P + 1 D0 P − 2 V + 1
Figure 5. The dominant diagrams to D → V P(→ P P γ) in the λQCD/mc and αs expansion. At order α 0 s QCDF reproduces the naive factorization [9]. The diagrams are shown for charged final state mesons. For the final state with uncharged mesons the u and q quark have to be exchanged. U-spin approximation is expected to yield better results in ratios of observables (which arise already at lowest order in the U-spin limit), such that overall systematics drops out. Useful applications have been made for polarization asymmetries in D → V γ decays [33]. However, three-body radiative decays are…
Figure 6. The SM predictions for the differential branching ratios of the decays
Figure 7. Comparison of QCDF-based SM predictions of differential branching ratios for
Figure 8. As in Fig. 7 but for HH
Figure 9. The forward-backward asymmetry in the SM (blue band) and beyond for the decays
Figure 10. As in Fig. 9 but within QCDF (40)
Figure 11. Predictions for the CP asymmetries in D → π +π −γ and D → K+K−γ as a function of s, within the SM and beyond (using (40)), based on QCDF. For the BSM scenarios, we have set one coefficient C (0) 7 to 0 and the other one to 0.05, 0.2. The weak phase of C (0) 7 is φw = ±π/2. We performed a cut s ≤ 1.5 GeV2 to remain within the region where QCDF applies. cancellations. ACP is very small beyond the (P1P2)res peak due to the cancellation of the (P1γ)res and (P2γ)res contributions upon integration over t. To avoid this cancellation one could use the s￾and t-dependent CP asymmetry ACP(s,…
Figure 12. As in Fig. 11 but for HH
Figure 13. Dalitz plot of ACP(s, t) for D → π +π −γ (upper plots) and D → K+K−γ decays (lower plots) based on HHχPT. We have set one BSM coefficient, C7 or C 0 7 , to 0 and the other one to 0.1, with weak phase φw = π/2. We employed a cut s ≤ 2 GeV2 to avoid large bremsstrahlung contributions in the normalization. 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 s [GeV 2 ] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 t [G e V 2 ] ACP(D K + K ), SM 0.00020 0.00015 0.00010 0.00005 0.00000 0.00005 0.00010 0.00015 0.00020
Figure 14. Dalitz plot of ASM CP(s, t) for D → K+K−γ decays based on HHχPT for s ≤ 2 GeV2
Figure 15. The real and imaginary part of the electromagnetic form factor
Figure 16. The real and imaginary parts of the electromagnetic kaon form factors (B8) (left) as well as their
Figure 17. The real and imaginary part (left) of the
Figure 18. Feynman diagrams for the D → π +K−γ decay, which contribute to the parity-even form factors A and E. The diagrams for the decays D → π +π −γ and D → K+K−γ are obtained by adjusting the flavors. We have added the diagrams E1,2 and E2,2 (see [7]) to make the amplitude E gauge invariant for any choice of a. Additionally, for each of the diagrams A1,1, A1,2, A1,3, A2,2, A2,3, A2,4, E1,1, E1,2, E1,3, E2,1 and E2,3 there is another one where the photon is coupled via a vector meson
Figure 19. Feynman diagrams for the D → π +K−γ decay, which contribute to the parity-odd form factors B and D. The diagrams for the decays D → π +π −γ and D → K+K−γ are obtained by adjusting the flavors. D (d,s) 1 = − √ 2 fDfK fπ λ 0  1 v · k + ∆ + g v · p2 v · p1 + v · k  1 v · k + ∆ + 1 v · p1 + ∆ D (d,s) 2 = − 1 2 fDfK fπ λgv  gω 3m2 ω + gρ m2 ρ   1 v · k + ∆ + g v · p2 v · p1 + v · k  1 v · k + ∆ + 1 v · p1 + ∆ D (d,s) 3 = − r 2 mD fK(α1mD − α2v · p2)
Figure 20. Feynman diagrams contributing to the tensor current form factors

Figures reproduced from arXiv:2009.14212 by the authors.

References
How the literature uses it

10 arXiv papers cite this in 27 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 21 sentences without a confident label

Contests it 1 paper
  • More comments are given in [ref] . * The contributions of the diagrams $A_4,1^+$ and $C_4,1^+$ vanish in our calculation. * Missing global minus sign in $C_2^+$. * We believe that there are diagrams that have not been shown in Ref. [ref] : For each of the diagrams $A_1,2^+$, $A_1,3^+$, $A_3,3^+$, $A_3,4^+$, $E_1,2^0$ and $E_2,3^0$ there is another one in which the photon couples via a vector meson.

    Probing QCD dynamics and the standard model with $D_{(s)} \to P^+ P^0 \gamma$ decays · Differences with respect to CITEMARK35Z · contest 0.79

Builds on it 1 paper
Cites it as background 4 papers

What it cites · 46 sentences without a confident label

Builds on 2 papers
Uses the method or tool of 8 papers
Uses data or values from 13 papers
Compares with 2 papers
  • However, we obtain $C^0=0$, as in Ref. [ref] . * We reproduce $A_1^+$, but for $A_1^0$ we get an expression $∼(q·k)-M(v ·k + v ·q)$.

    Cabibbo Allowed D -> K pi gamma Decays · Differences with respect to CITEMARK73Z · 0.96

  • * We have an extra factor of $2$ in $D_3^0$. * We obtain a relative minus sign for each vector meson in a diagram; however, we get the same relative signs for $R_γ^0/+$ as given in Eqs. (24) and (25) [ref] .

    Vector and pseudoscalar charm meson radiative decays · Differences with respect to CITEMARK73Z · 0.99

Discussion

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