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$D\to P \ell^+\ell^-$ decays assisted by QCD light-cone sum rules

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A QCD-based calculation determines the long-distance D+→π+γ* amplitude and predicts a D+→π+ℓ+ℓ− branching fraction in the experimentally vetoed regions that is of the same order as the current upper bound.

desk verdict New LCSR-assisted dispersion method for D->P l+l-; the low-q^2 prediction is solid, but the high-q^2 comparison with LHCb rests on an openly unquantified resonance model. read the letter →

arxiv 2505.21369 v1 pith:IYVXSDJI submitted 2025-05-27 hep-ph hep-ex

Anshika Bansal , Alexander Khodjamirian , Thomas Mannel This is my paper
classification hep-phhep-ex
keywords charmraredecayslight-conesumrulesD→πℓ+ℓ−weakannihilationvectormesonresonancesU-spinsymmetryflavour-changingneutralcurrentsnonlocalformfactor
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 tries to establish that the dominant long-distance part of the rare decay D+→π+ℓ+ℓ− can be computed from QCD rather than modeled ad hoc. The amplitude is written as a D+→π+γ* transition followed by γ*→ℓ+ℓ−, and that transition is calculated at spacelike photon virtualities with QCD light-cone sum rules using pion distribution amplitudes. Matching the sum-rule result to a hadronic dispersion relation fixes the relative phases of the ρ, ω and φ contributions and controls heavier states. The output is a prediction for the lepton-pair mass spectrum whose integral over the two experimentally vetoed regions is (1.33+0.17−0.24)×10⁻⁸, compared with the current upper bound of 6.7×10⁻⁸. If the calculation is right, the short-distance c→uℓ+ℓ− operator O9 in the Standard Model contributes three orders of magnitude less than the long-distance part, so only a very large new-physics enhancement of C9 could be seen.

What carries the argument

The load-bearing object is the LCSR for the D+→π+γ* amplitude, Eq. (5.10), built from a vacuum-to-pion correlation function of the electromagnetic current, the weak effective operators, and a D-meson interpolating current. An artificial momentum k at the weak vertex makes the correlation function a 2→2 scattering amplitude, which allows a dispersion relation in the D-meson channel; after analytic continuation P²→m_D² and Borel transform, the pion twist-2 distribution amplitude enters through annihilation-topology diagrams, while d- and s-loop topologies largely cancel by the GIM mechanism. The resulting spacelike amplitude is fitted to a once-subtracted hadronic dispersion relation in q² whose poles are the ρ, ω, φ mesons with data-fixed residues, plus either a z-expansion or excited ρ′, φ′ and an effective ρ″ pole for heavier states. The fit supplies the unknown resonance phases and heavier-state parameters, and the fitted amplitude is then used in the physical region.

What would settle it

Measure the D+→π+μ+μ− branching fraction in the two vetoed regions of dilepton invariant mass (below the ρ and above the φ) with precision at or below 10⁻⁸. If the measured value deviates from 1.33×10⁻⁸ by much more than the quoted uncertainty, the LCSR-assisted dispersion relation's spectral model is wrong; alternatively, a precise measurement of D+→π+V′ nonleptonic rates for V′=ρ′,φ′ would directly test the extended resonance model.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the D+→π+γ* amplitude — the nonlocal form factor that dominates D+→π+ℓ+ℓ− through weak annihilation with virtual-photon emission — can be obtained from QCD light-cone sum rules at spacelike q², Eq. (5.10), and then continued into the physical timelike region through a hadronic dispersion relation. The fit determines the relative phases of the ρ, ω and φ resonance terms and a description of excited states, in two variants: a z-expansion valid up to q² = 1.2 GeV², and an extended resonance model covering the whole physical region. The resulting differential branching fraction, integrated over the vetoed regions, is 1.33+0.17−0.24 ×10⁻⁸, of the same order as the current upper bound. In the same framework, the Standard Model O9 short-distance contribution is at least three orders of magnitude smaller, effectively invisible in the rate.

Load-bearing premise

The load-bearing premise is that the hadronic spectral density above about 1 GeV² is saturated by the excited resonances ρ′(1450), φ′(1680) and an effective ρ″(1700) pole, with ω′ and ω″ neglected; the authors state that if this model is wrong, the high-q² prediction's systematic uncertainty is hard to quantify.

Editorial extensions

If this is right

  • The D+→π+μ+μ− rate in the two experimentally vetoed q² windows is predicted to be about 1.3×10⁻⁸, within reach of the existing 6.7×10⁻⁸ upper bound and testable by near-future data.
  • The Standard Model O9 c→uℓ+ℓ− contribution is numerically invisible in this decay, so any observed excess in D+→π+ℓ+ℓ− would require C9 enhanced by orders of magnitude, not a modest shift.
  • The U-spin equalities between D+→π+, D_s+→K+, D_s+→π+ and D+→K+ amplitudes hold at about the 20% level, giving control channels to check the method.
  • The same LCSR-assisted dispersion method applies to the Cabibbo-favoured modes D_s+→π+ℓ+ℓ− and D0→K̄0ℓ+ℓ− and can be extended to other D(s)→Pℓ+ℓ− modes.

Reading between the lines

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

  • If the predicted vetoed-region branching fraction is confirmed, the long-standing ambiguity in the relative phases of the ρ, ω, φ terms in D→Pℓ+ℓ− is resolved from QCD, which would also sharpen interpretations of charm flavour-changing neutral current searches.
  • A natural next test is to apply the same matching to D→π+π−ℓ+ℓ− with dipion distribution amplitudes; the existing experimental observation of that mode could constrain the z-expansion at low q².
  • The result suggests that additional measurements of nonleptonic D→πV′ amplitudes would be unusually valuable, since each excited-vector-meson amplitude directly anchors the high-q² tail of the dispersion relation.
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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 / 5 minor

Summary. This paper proposes and implements a new method to compute the long-distance-dominated rare decays D_(s)->P l+l- . The central object is the D+->pi+gamma* amplitude (2.9), calculated via a light-cone sum rule (5.10) at spacelike photon virtualities, in LO QCD and at leading twist, including both annihilation and (in the SCS case) d/s-loop topologies. The LCSR result in -8 GeV^2 < q^2 < -3 GeV^2 is fitted to a once-subtracted hadronic dispersion relation containing rho, omega, phi poles, phases, and a z-expansion for heavier states, yielding the parameters in (7.6). An alternative unsubtracted dispersion relation with an extended resonance model (7.8) extends the prediction to the full physical region. Binned branching fractions for D+->pi+mu+mu- are given in Table 6; the sum over the two LHCb regions is (1.33+0.17-0.24)x10^-8 versus the bound 6.7x10^-8, and the O9 contribution is found to be at least three orders of magnitude smaller. The paper also derives U-spin amplitude relations and applies the method to D_s+->pi+l+l- and D0->Kbar0 l+l-.

Significance. The method is a genuine and interesting methodological step: it turns the conventionally uncontrolled relative phases of the rho, omega, phi terms and the model of heavier states into parameters constrained by QCD sum rules in the spacelike region, and it produces definite, falsifiable predictions for q^2-binned branching fractions that can be compared with LHCb data. The technical derivation is detailed and reproducible: the Cutkosky-rule spectral densities of the annihilation and loop topologies are worked out in Appendices A-C, and the numerical LCSR amplitudes are provided as ancillary data files. The 10% internal consistency between the fitted dispersion models and the LCSR in the spacelike window, the agreement of the low-q^2 prediction with the QCD-factorization-based result of [10], and the demonstration of the O9 invisibility (Sec. 7.6) are concrete, useful results. The main caveat is that the headline comparison with the LHCb upper bound rests on the unvalidated extended-resonance model for the high-q^2 region, so the significance of that specific number should be assessed with caution.

major comments (3)
  1. [Secs. 6, 7.4-7.5] The headline comparison with the LHCb bound -- the sum over the two LHCb regions, (1.33+0.17-0.24)x10^-8, quoted at the end of Sec. 7.5 and echoed in the abstract -- depends for its high-q^2 part (1.25 GeV^2 < q^2 < (m_{D^+} - m_pi)^2) entirely on the extended-resonance model of Eq. (7.8). That model approximates the spectral density above 1 GeV^2 by rho'(1450), phi'(1680) and an effective rho'' pole, omits omega' and omega'' on the basis of a kappa_omega/kappa_rho = 1/3 suppression, and is fitted only in the spacelike window -8 GeV^2 < q^2 < -3 GeV^2; its use in the upper physical region is therefore an extrapolation across the rho', phi' and rho'' poles, in a region where the authors themselves state (Sec. 6) that the systematic uncertainty is 'hard to quantify.' The residues r_{rho'} and r_{phi'} are themselves indirect inputs (Appendix D combines a Dalitz-plot branching fraction with a pion form-factor fit and an SVZ sum rule), adding a further layer of model dependence. The quoted uncertainty of the LHCb-region sum therefore covers only parametric variations, not the model uncertainty. I request a sensitivity study -- e.g., turning on omega' and omega'' with kappa-suppressed residues, varying the effective rho'' pole parameters, and comparing with e+e--based spectral constraints -- and, depending on the outcome, either a numerical systematic error for the high-q^2 prediction or a restructured claim that separates the robust low-q^2 prediction (supported by both models, see Table 6) from the model-dependent high-q^2 one.
  2. [Sec. 7.5, Table 6] The binned uncertainties in Table 6 are built by one-by-one input variation with refits (Sec. 7.5), but they exclude the uncertainty of the fitted phases and z-coefficients themselves and they do not include a model-uncertainty term. The two dispersion-relation models disagree by more than their quoted error bars in several bins: Bin II gives 0.90+0.29-0.27 (z-expansion) versus 2.19+0.17-0.18 (extended resonance), and Bin I gives 1.36+0.70-0.63 versus 1.81+0.14-0.17. The 'consistent picture' claimed in Sec. 9 for q^2 <= 1.2 GeV^2 should therefore be quantified in terms of these differences; ideally the two parameterizations should be combined into a single uncertainty that reflects the model spread. At minimum, the text should state explicitly that the quoted errors are parametric-only and that the model spread is a separate, comparable source of uncertainty.
  3. [Secs. 5, 7.2-7.3] The statement that the fitted hadronic models agree with the LCSR within 10% in the spacelike window measures the internal consistency of the hadronic ansatz with the sum rule, not the absolute accuracy of the method. The LCSR itself is computed at LO in alpha_s and at leading twist-2 only (Sec. 5), with NLO perturbative and soft-gluon higher-twist corrections omitted and not numerically estimated; these corrections are therefore absent from all quoted errors in Tables 6 and 8. Since the extracted phases and z-coefficients (7.6) and all subsequent predictions inherit these unknown corrections, I ask for a semi-quantitative estimate of their size -- for instance, based on the known magnitude of NLO corrections in the D->pi LCSR discussed in Sec. 7.1, or on a twist-4 power-counting estimate -- or, failing that, an explicit statement that the error bars represent only parametric lower bounds on the total uncertainty.
minor comments (5)
  1. [Abstract, App. D, Table 7] Typos: 'one readily needs a enhancement' in the abstract; 'approximaitng' in the paragraph below Eq. (D.7); and an unclosed parenthesis in Table 7, row 'BR(D_s^+->pi^+V) V->mu+mu-' (the entry reads '1.421+/-0.411x10^-7').
  2. [Fig. 9] Figure 9 would be easier to interpret if the vetoed rho/omega/phi regions and the two LHCb q^2 regions of Table 6 were marked on the plot, since the binned values in Table 6 are the paper's main quantitative output.
  3. [Sec. 7.3, Eqs. (7.6), (7.11)] The absolute phases returned by the two fits, (7.6) and (7.11), look very different; the explanation that the overall phase is unphysical appears only in Sec. 7.4. It would help to state explicitly at (7.6) that only phase differences are meaningful, with phi_rho - phi_omega ~ 0 and phi_rho - phi_phi ~ pi being the physical pattern.
  4. [Sec. 8, Table 8] For the D0->Kbar0gamma* amplitude the fit deviates from the LCSR by up to 25%; the text notes this, but the corresponding branching fractions in Table 8 should carry the same caveat where they are presented, since the stated universality of the method partly rests on this application.
  5. [Secs. 2, 5] The amplitude is denoted both A(D+pi+gamma*)(q^2) (Eq. 2.9) and A(D+->pi+gamma*)(q^2) elsewhere; using one convention throughout, with a short notation guide, would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the D+ -> pi+ l+ l- prediction is an extrapolation of an LCSR-fitted dispersion model, not a refit of the target branching fraction.

full rationale

The paper's derivation chain is self-contained against the target observable. The central input is the LCSR amplitude A^(LCSR) in Eq. (5.10), computed at spacelike q^2 from pion distribution amplitudes and the D-meson interpolating current. The hadronic dispersion models in Eqs. (6.14) and (7.8) are fitted to this LCSR amplitude only in the spacelike window -8.0 GeV^2 < q^2 < -3.0 GeV^2 (Eqs. 7.5 and 7.10), minimizing the difference between the model and the LCSR result. The fitted parameters are the resonance phases, z-expansion coefficients, and the effective rho'' residue; none of these is adjusted to the D+ -> pi+ mu+ mu- branching fraction or to the LHCb upper bound. The binned predictions in Table 6 are obtained by evaluating the fitted dispersion relations in the timelike region, which is a genuine extrapolation. The O9 short-distance estimate uses the independent lattice form factor f_+^{D pi}(q^2) from Ref. [34], and the pion Gegenbauer moments are taken from a pion-form-factor analysis, not from this decay. The extended-resonance model of Sec. 7.4 carries a large, admittedly 'hard to quantify' systematic uncertainty, but model dependence and unquantified errors are not circularity. Some inputs come from the same research group (e.g., the pion DA moments in [25] and the D->pi LCSR framework in [24]), but these are external benchmarks or method references, not results that presuppose the D+ -> pi+ l+ l- branching fraction. No step reduces by construction to its own input, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 9 free parameters · 10 assumptions · 0 invented entities

The LCSR calculation rests on standard QCD sum-rule assumptions: local and semilocal quark-hadron duality, twist-2 truncation, and a chosen Borel window. The dispersion continuation adds modeling assumptions for the hadronic spectral density. The central prediction inherits these, plus the fitted phases and coefficients. No new physical entities are introduced; the effective pole r_eff is a parameterization, not a new particle.

free parameters (9)
  • Borel parameter M^2 = (2.0 ± 0.5) GeV^2
    Chosen for LCSR stability following two-point sum rules; varied in uncertainty analysis.
  • D-meson duality threshold s0^D = (5.5 ± 0.5) GeV^2
    Adjusted so that the mass sum rule (7.2) reproduces m_D^2 within 1% in the fit interval.
  • Ds-meson duality threshold s0^{Ds} = (6.5 ± 1.0) GeV^2
    Analogous threshold for the Ds LCSR.
  • z-expansion fit parameters: phases phi_V = phi_rho = 1.809, phi_omega = 1.804, phi_phi = -1.339 rad
    Fitted to the LCSR amplitude by minimizing Eq. (7.5); uncertainties not assigned.
  • z-expansion coefficients alpha_1, alpha_2 = alpha_1 = (-7.858 + 0.727 i) x 10^-2; alpha_2 = (9.548 + 2.184 i) x 10^-2
    Complex coefficients describing the heavy-state spectral integral; fitted to LCSR.
  • Extended-resonance phases and effective pole = phi_rho = phi_omega = 5.944, phi_phi = 2.797, phi_rho' = phi_phi' = 5.925, |r_eff| = 3.538 x 10^-2, Arg[r_eff] = 3.312
    Alternative fit parameters for the extended resonance model (7.8).
  • Hadronic model kinematics = q1^2 = -8.0 GeV^2, q2^2 = -3.0 GeV^2, q0^2 = -2.0 GeV^2, s_th = (1.60 ± 0.15) GeV^2, s_max = 1.2 GeV^2
    Chosen by hand to define the LCSR fit window and the reach of the z-expansion.
  • Gegenbauer moments of pion and kaon DAs = a2_pi(1.5 GeV) = 0.23 ± 0.04; a4_pi = 0.14 ± 0.04; a1_K = 0.09 ± 0.035; a2_K = 0.21 ± 0.12
    Taken from fits in [25] and related references; their shape uncertainties affect the LCSR.
  • Excited vector meson residues r_rho', r_phi' = |r_rho'| = (9.64 +2.92 -4.76) x 10^-3 GeV^2; |r_phi'| = (11.87 +4.74 -3.79) x 10^-3 GeV^2
    Estimated in Appendix D from BaBar pion form factor fit, branching fraction data, and an SVZ sum rule; needed only for the extended resonance model.
assumptions (10)
  • standard math Analyticity, unitarity, and dispersion relations for the nonlocal form factor; Cutkosky rules for OPE spectral densities.
    Standard QFT framework used throughout Sections 5-6 and Appendices A-B.
  • standard math CKM unitarity relations among lambda_d, lambda_s, lambda_b.
    Used in Section 2 to set lambda_s = -lambda_d in the GIM limit.
  • domain assumption Local quark-hadron duality in the D-meson channel: the OPE amplitude continued from spacelike P^2 to P^2 = m_D^2 approximates the full amplitude (Eq. 5.7).
    Invoked in Section 5; standard LCSR assumption, not independently proven.
  • domain assumption Semilocal quark-hadron duality: the heavier-state spectral density above s0 is replaced by the OPE spectral density in the D-meson channel.
    Used in deriving Eq. (5.10).
  • domain assumption Leading twist-2 and leading order OPE dominance: twist-3 terms vanish in the chiral limit and twist-4, NLO, and soft-gluon corrections are negligible at mu ~ m_c.
    Section 5 and 9; the pion DA is truncated at twist-2 with two Gegenbauer moments.
  • domain assumption The D+ to pi+ gamma* amplitude obeys an unsubtracted or once-subtracted dispersion relation in q^2 with spectral function saturated by rho, omega, phi below about 1 GeV^2.
    Section 6, Eqs. (6.8) and (6.10).
  • ad hoc to paper The integral over hadronic states heavier than phi is a smooth function expandable in z(q^2) and truncated at K=2 for q^2 < 1.2 GeV^2.
    Eq. (6.13); the z-expansion is truncated to three terms, not tested at higher orders.
  • ad hoc to paper For the high-q^2 region, the spectral density above s_th is saturated by rho'(1450), phi'(1680), and an effective rho'' pole, with omega' and omega'' neglected.
    Section 7.4 and Appendix D; this assumption is required for the high-q^2 LHCb bins and has an unquantified systematic uncertainty.
  • domain assumption GIM limit lambda_b = 0 for the dominant amplitude; O9 estimated separately with C9(mu) and lattice f+_Dpi.
    Section 2; valid because |lambda_b| ~ 10^-4 and all O(lambda_b) effects are neglected.
  • domain assumption Approximate U-spin symmetry for amplitude relations, with ~20% breaking from m_s - m_d.
    Section 4; used for relations (4.10)-(4.13).

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Cite this review

Pith. "Pith review of $D\to P \ell^+\ell^-$ decays assisted by QCD light-cone sum rules." pith.science (2026). https://pith.science/paper/IYVXSDJI

@misc{pith2026250521369,
  author       = {Pith},
  title        = {Pith review of: $D\to P \ell^+\ell^-$ decays assisted by QCD light-cone sum rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYVXSDJI}},
  note         = {Machine review of arXiv:2505.21369}
}
abstract

We suggest a new method to analyse the rare $D_{(s)}\to P \ell^+\ell^-$ decays ($P=\pi,K$), combining QCD light-cone sum rules (LCSR) with hadronic dispersion relations. As our main study case, we consider the $D^+\to\pi^+\ell^+\ell^-$ mode which attracts much of interest from the point of view of GIM cancellation and potential new sources of the FCNC $c\to u$ transitions. The hadronic amplitude of this decay is dominated by the combination of weak annihilation with the emission of a virtual photon. This $D^+\to \pi^+\gamma^*$ amplitude is calculated at spacelike photon virtualities, $-q^2 \gg \Lambda_{QCD}^2$, using LCSR with pion distribution amplitudes. The LCSR results are then fitted to the hadronic dispersion relation in the variable $q^2$. The fit allows us to determine relative phases of the $\rho$-,$\omega$- and $\phi$-meson terms and to estimate the contributions of heavier hadronic states. The latter are parameterized in two different ways: first, with a $z$-expansion and second, using a model with excited vector-meson resonances. The resulting $D^+\to\pi^+\ell^+\ell^-$ width obtained from the LCSR-assisted dispersion relation is not much smaller than the current upper bound measured by LHCb collaboration. In comparison with our result for the dominant $D^+\to \pi^+\gamma^*$ transition, the contribution induced by the short distance $c\to u \ell^+\ell^-$ quark transition generated by the effective $O_9$ operator in Standard Model turns out to be practically invisible. To detect its effect, one readily needs a enhancement of the coefficient $C_9$ from non-standard effects by many orders of magnitude. We also establish new $U$-spin symmetry relations between the amplitudes of $D\to P \ell^+\ell^-$ decays and apply our method to the Cabibbo-favoured modes $D^+_s\to\pi^+\ell^+\ell^-$ and $D^0\to\bar{K}^0\ell^+\ell^-$ which proceed via the same weak annihilation mechanism.

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Reviewed August 7, 2026 · model on record in the stance chip above.