Charm physics

Alexander Lenz; David Friday; Evelina Gersabeck; Maria Laura Piscopo

REVIEW 2 major objections 5 minor 1 cited by

Charm physics

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This review argues that the charm quark, awkwardly placed between light and heavy quarks, is the Standard Model's most sensitive testing ground, with the heavy quark expansion as its organizing tool.

desk verdict A solid, current pedagogical review of charm physics; the only real wrinkle is the HQE lifetime section, where 'no evident signal of breakdown' sits uneasily next to the admitted negative D+ lifetime. read the letter →

arxiv 2506.15584 v2 pith:TN3DVACI submitted 2025-06-18 hep-ph

David Friday , Evelina Gersabeck , Alexander Lenz , Maria Laura Piscopo This is my paper

classification hep-ph PACS 13.20.Fc14.40.Lb

keywords charmquarkheavyexpansionD0mixingCPviolationinlifetimesraredecaysleptonflavouruniversalityeffectiveHamiltonian

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 review chapter sets out to give a pedagogical and up-to-date account of charm physics, arguing that the charm quark is an ideal testing ground for the Standard Model precisely because it is awkward: heavy enough to organize decays in a systematic expansion, but light enough that every approximation can be challenged by data. It covers production, spectroscopy, lifetimes, $D^0$ mixing, CP violation and rare decays, and lays out the theoretical tools (effective Hamiltonians and the heavy quark expansion) alongside the experimental methods that measure them. A sympathetic reader takes away a map of what is established, what is still contested (notably the size of CP violation and the convergence of the heavy quark expansion), and where new physics would show up.

What carries the argument

The load-bearing tool is the heavy quark expansion (HQE), which expresses an inclusive decay width as an operator product expansion in $\Lambda_{\rm QCD}/m_c$, with dimension-three, -five and -six operators encoding the free-quark decay, kinetic and chromomagnetic effects, and spectator (Pauli-interference) effects. For mixing, the machinery is the mass/width splitting formalism with parameters $x$ and $y$ and the CP-violating phase $\phi_{12}$, together with GIM cancellations that suppress the Standard Model predictions; for exclusive decays, it is the effective Hamiltonian with Wilson coefficients, decay constants and form factors. The chapter uses this machinery to compare HQE predictions against lifetime and semileptonic data, and to explain why $D^0$ mixing and charm CPV are so hard to predict.

What would settle it

A precise measurement that disagrees with the HQE prediction for, say, the $D^+$ lifetime or the ratio of semileptonic widths $\Gamma(D_s^+\to X e^+\nu_e)/\Gamma(D^0\to X e^+\nu_e)$ beyond the quoted uncertainties, or a lattice QCD calculation of the dimension-six four-quark matrix elements that shifts the predicted lifetime pattern, would show the expansion is not converging.

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Extended reading notes

Core claim

The chapter's central claim is that the charm sector is a unique probe because it sits between the light- and heavy-quark regimes. The Standard Model description proceeds through a weak effective Hamiltonian and, for inclusive observables, the heavy quark expansion; despite $m_c\sim 1$ GeV being dangerously close to $\Lambda_{\rm QCD}\sim 400$ MeV, the expansion reproduces the measured lifetimes and semileptonic rates without obvious breakdown. On the experimental side, the chapter states that $D^0$-mixing is firmly established (no-mixing excluded at more than $10\sigma$) and that CP violation in charm decays has been observed via $\Delta A_{CP}\sim 10^{-3}$, about a factor of ten above naive SM expectations, leaving open whether this signals new physics or underestimated strong-interaction effects. It moreover documents null tests (rare decays, lepton flavour universality ratios, forbidden modes) that currently agree with the SM.

Load-bearing premise

The review's quantitative lifetime and mixing predictions assume the charm quark is heavy enough for the expansion in $\Lambda_{\rm QCD}/m_c$ to converge; since $m_c\sim 1$ GeV is only marginally above $\Lambda_{\rm QCD}\sim 400$ MeV, a breakdown would invalidate those HQE-based predictions.

Editorial extensions

If this is right

If the heavy quark expansion is trusted for charm, inclusive semileptonic charm data can deliver independent determinations of $|V_{cs}|$ and $|V_{cd}|$.
A confirmed value of $\Delta A_{CP}$ above SM expectations would motivate either new physics in the up-type sector or significant progress in non-perturbative methods for charm hadronic decays.
The established non-zero $x$ and $y$ mean mixing parameters and their CP-violating phase will continue to be precision targets for LHCb, Belle II and BESIII.
Rare and forbidden charm decays are positioned as null tests: observed rates above SM predictions would indicate new physics.
Quantum-correlated charm pairs at threshold provide strong-phase inputs that directly improve the accuracy of time-dependent mixing and CPV measurements at colliders.

Reading between the lines

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

If the HQE continues to hold under Belle II and LHCb precision, charm lifetimes could become a competitive way to extract CKM elements and to calibrate the expansion itself.
The current 2.1$\sigma$ exclusion of no-CP in charm mixing suggests that a first observation of CP violation in mixing may be within reach; such a discovery would point to sources beyond the CKM phase.
The BESIII strong-phase program will keep sharpening LHCb mixing and CKM-angle measurements, potentially making model-dependent amplitude analyses less necessary for those inputs.
A future decisive test of the chapter's implicit claim that charm's marginal heaviness is a feature would be a lattice QCD calculation of the dimension-six four-quark matrix elements that anchors the HQE lifetime predictions.

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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

You should know this is a review chapter, not new research; it's the updated version of a previous edition. What it does well: it's a clear, well-structured, current overview of charm physics through early 2025, with a good mix of theory and experiment. The sections on HQE, mixing, CPV, and rare decays are thorough and appropriately hedged. The treatment of the HQE's questionable convergence in charm is honest, including the m_c ~ 1 GeV vs Lambda_QCD ~ 400 MeV caveat.

The stress-test note lands. In Section 4.2 the authors say the HQE 'succeeds in reproducing the observed pattern' and 'there is no evident signal of breakdown'—two sentences before they note the predicted D+ lifetime can be negative, which is unphysical. That's a genuine internal tension. A negative width means the 1/m_c expansion is likely failing for that observable, not that it's fine. The review should either classify that as a breakdown signal or show quantitatively that the negative region is a negligible tail. It does neither. This weakens the credibility of the lifetime section, though it doesn't sink the whole review. The mixing section inherits some of the same doubt, but the paper does flag the theoretical difficulty there.

Also, as a review it introduces nothing new; novelty is low by design. The citation pattern is fine: self-citations to the authors' own HQE papers are backed by experimental comparisons.

Who it's for: graduate students and researchers entering charm physics, or anyone needing a rapid orientation. It deserves a serious referee for a review venue; the factual content should be double-checked, but it's clearly above the bar.

Recommendation: engage with it as a reference, and in the review process, ask the authors to reconcile the D+ lifetime statement. That's the one substantive point.

Referee Report

2 major / 5 minor

Summary. This manuscript is a review chapter on charm physics, written for an edited volume. It provides a pedagogical overview of the charm sector of the Standard Model: the history and production of charmed particles, experimental techniques at BESIII, Belle/Belle II and LHCb, the effective Hamiltonian, inclusive and exclusive decays, lifetimes, D0-D0bar mixing, CP violation, and rare charm decays. The chapter is intended to bring students and researchers up to date with the current theoretical and experimental status, and it deliberately highlights open issues such as the poor convergence of the heavy quark expansion for charm.

Significance. As a review, the manuscript does not claim new results, but it fills a useful role: it is comprehensive, up-to-date (including 2024-2025 results), and honest about unresolved problems. Its main strengths are the breadth of coverage, the clear pedagogical structure, and the explicit discussion of the theoretical challenges of charm physics (e.g., m_c ~ Lambda_QCD, large alpha_s, strong GIM cancellations). The manuscript would be a valuable reference for students and researchers entering the field, provided the internal tension in the assessment of the heavy quark expansion is resolved.

major comments (2)

[4.2] The passage 'Despite the large uncertainties, the HQE succeeds in reproducing the observed pattern and there is no evident signal of a possible breakdown of the theoretical framework. However, note that currently the HQE predicts also negative values for the lifetime of the D+ meson, which are clearly not physical' is internally inconsistent. A negative predicted lifetime implies a negative total width, which is exactly the kind of unphysical outcome expected if the 1/m_c expansion is breaking down, especially given the near-cancellation between the free-quark term Gamma_3 and the Pauli-interference contribution described in Section 3.1. The review should either explicitly state that the negative D+ width is a signal of a likely HQE breakdown (consistent with the m_c ~ 1 GeV vs Lambda_QCD ~ 400 MeV caveat of Section 3.1), or it should demonstrate quantitatively that the negative values correspond to a negligible tail of the uncertainty distribution. Without such a statement, a reader cannot determine whether the HQE is being validated or falsified by the D+ lifetime, and the subsequent discussion of D-mixing predictions in Section 6.1 inherits the same unresolved doubt.
[6.1] In the theoretical status subsection, the text states that the HQE prediction for y is y_HQE ~ 3.6e-7, four orders of magnitude below the experimental value of y, and that the GIM cancellations are extreme. This is a major failure of the same theoretical framework that Section 4.2 claims shows 'no evident signal of breakdown'. The two parts of the review should be cross-referenced and the earlier statement qualified so that the reader understands that the HQE's success is limited to lifetimes and semileptonic widths, while charm mixing is not described by the same expansion. As written, the review gives conflicting signals about the reliability of the HQE in charm, which undermines its usefulness as a reference.

minor comments (5)

[3.3] In the text below Eq. (20), the decay amplitude is described as factorising into the kaon decay constant f_D and a form factor, but the equation correctly uses f_K; this is a typo that should be corrected.
[3.1 and 4.1] The numerical values used for the charm quark mass and Lambda_QCD differ between the two sections (m_c ~ 1.5 GeV, Lambda_QCD ~ 0.5 GeV in Eq. (3); m_c ~ 1 GeV, Lambda_QCD ~ 400 MeV in Section 4.1). These are rough estimates, but the inconsistency is confusing and should be harmonized or explicitly acknowledged.
[2.4] The phrase 'consisting of a charm quark and a charm anti-quark' appears with an inconsistent hyphen; 'anti-quark' should be written consistently as 'antiquark' throughout.
[6.1] In Eq. (73), the uncertainties are written with an extra leading zero for some entries (e.g., x_CP = (0.397 ± 0.0544) x 10^-2); this should be formatted as 0.0544 or consistent with the other entries.
[5.2] In the caption of Table 10, 'see Eqs. (58), (58)' is a typo; the two references should be to distinct equations or the duplicate citation removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the review's HQE comparisons are checked against independent experimental data, and the admitted negative D+ lifetime is a validity concern rather than a self-referential step.

full rationale

This is a review chapter; it presents no new derivation whose output could coincide with its input. The HQE lifetime and semileptonic predictions shown in Figs. 34 and 35 are taken from the authors' earlier papers, so self-citation occurs, but the review does not use those citations to forbid alternatives or to define the observables. The predictions are compared with independent experimental measurements (Belle II, LHCb, CLEO, HFLAV), and the hadronic matrix element inputs are stated as coming from lattice QCD, HQET sum rules, constituent quark models, and B-sector fits, not from the charm lifetimes being 'predicted'. The negative D+ lifetime admitted in Sec. 4.2 is an internal consistency and validity concern about the 1/m_c expansion, not a case where a prediction reduces by construction to a fitted parameter. The charm-mixing section likewise reports that the HQE result y_HQE ~ 3.6e-7 is four orders below experiment, which is the opposite of forcing agreement. No equation is shown to be equivalent to its own input, no fitted parameter is renamed as a prediction, and the self-citations are supported by independent external data.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities. The assumptions listed are standard theoretical frameworks invoked in the review. The central 'claim' is the review's completeness and accuracy, which rests on the validity of these frameworks as applied to charm.

assumptions (3)

domain assumption The charm quark is heavy enough for the heavy quark expansion to be applicable, m_c >> Lambda_QCD.
Invoked in Section 4.1 to expand inclusive decay widths in powers of Lambda_QCD/m_c; the paper itself notes this assumption may not hold for charm.
standard math CKM unitarity holds.
Used to relate CKM elements and simplify expressions such as Eq. (83).
domain assumption SU(3)_F symmetry is a useful approximate symmetry for charm decays, with corrections of order m_s/Lambda_QCD.
Used in Section 5.3 to relate decay amplitudes and in the classification of charm states.

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Cite this review

Pith. "Pith review of Charm physics." pith.science (2026). https://pith.science/paper/TN3DVACI

@misc{pith2026250615584,
  author       = {Pith},
  title        = {Pith review of: Charm physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TN3DVACI}},
  note         = {Machine review of arXiv:2506.15584}
}

abstract

50 years after the discovery of the first charmed particle, charm physics continues to be an extremely lively field of research and a cornerstone in particle physics. The study of charm, with its unique properties, is characterised by many challenging but also exciting peculiarities, making it an ideal testing ground for Standard Model (SM) predictions and a very sensitive probe of new physics. This chapter is intended to provide a pedagogical introduction to the physics of the charm quark and to its current theoretical and experimental status. Specifically, it discusses the main features of the charm sector of the SM, the theoretical and experimental challenges that arise when dealing with the charm quark, and the methods used to study it. An overview, both from a theoretical and experimental perspective, of fundamental observables such as lifetimes of charm hadrons, $D^0$-meson mixing, charm charge-parity violation (CPV) and rare charm decays is also presented.

Figures

Figures reproduced from arXiv: 2506.15584 by the authors.

**Figure 1.** Quarks are bound via the strong interaction into hadrons. The most well-known hadron is probably the proton, consisting of two up quarks and one down quarks, as well as sea-quarks, here indicated by the ss¯ pair, and gluons, indicated by the curly red lines. The charm quark is the main constituent of the J/ψ resonance, the D mesons and the Λ + c baryon. This simultaneous discovery of the J/ψ opened the gates for a n… view at source ↗

**Figure 2.** Energy hierarchy of selected charmonia masses and of the most common thresholds for open charm production channels—both ground and excited states. In recent years, significant progress has been made, and the following phenomena have been experimentally established—all of which will be discussed in more detail in the next sections: 1. 2003 onwards: charm spectroscopy— For a long time only baryons and mesons were exp… view at source ↗

**Figure 3.** First combination of the mixing parameters x and y excluding the no-mixing hypothesis (0,0) with more than 5σ. Taken from [20]. The previous points illustrate how the physics of charm is a fast-advancing field of research with many interesting open questions. The remaining sections of this chapter aim, therefore, to provide a pedagogical guide to the physics of the charm quark and its current theoretical and experim… view at source ↗

Figures from the paper (58 more)

**Figure 4.** Figure 4: Visualisation of the Cabibbo matrix, which describes the coupling strength of the W boson to the first two generations of quarks. The existence of a fourth quark—the charm—to complete the second generation of quarks, was first theorised by Bjorken and Glashow in 1964 […

**Figure 5.** Figure 5: Leading Feynman diagrams with internal up (left) and charm (right) quarks describing the decay K 0 → µ +µ − . The contribution of the up quark alone would lead to a much higher decay probability than actually observed. The contribution from the charm quark largely comp…

**Figure 6.** Figure 6: Timeline of key theoretical and experimental milestones in charm physics. 2.2 Charm experiments: production and cross-sections After the discovery of the J/ψ particle, many new experiments were built at several different laboratories and a rich charm physics programme…

**Figure 7.** Figure 7: At the LHC, heavy-flavour quark pairs such as bb¯, explicitly shown, and cc¯ pairs are produced in the forward - or backward - direction (left). Therefore, the LHCb experiment, and its upgrade (right) are designed as a forward spectrometer. Taken from [48, 49] [PITH_F…

**Figure 8.** Figure 8: Experiments at electron-positron colliders: Belle II (left) and BESIII (right) - note that only the upper half of the latter is shown. Both detectors are hermetic with a 4π geometry. At Belle II the beam energies are asymmetric and particles are boosted in contrast to …

**Figure 9.** Figure 9: Example of charm-quark continuum production processes - prompt charm production - at electron-positron colliders. The charm anti-charm quarks will hadronise into QCD bound states. Examples are ground-state mesons like D 0 (D¯ 0 ), D + (D − ) and D + s (D − s ), baryons…

**Figure 10.** Figure 10: (Left) Secondary charm production at b-factories operating at the Υ(4S ) resonance: the Υ(4S ) decays into B +B − (and also B 0 d B¯0 d ) mesons, which subsequently decay into charmed hadrons. (Right) Illustration of the most common energy thresholds: the mass of the …

**Figure 11.** Figure 11: Production mechanism for quantum entangled charm meson pairs at the ψ(3770) threshold at electron positron colliders. The mass of the ψ(3770) resonance is just slightly higher than the mass of the D 0D¯ 0 pair [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]

**Figure 12.** Figure 12: Cross-sections at electron-positron colliders as a function of the centre of mass energy ECM or √ s for: (left) strange charm-meson pairs measured by the CLEO collaboration and taken from [72] - DsDs (red), D ∗ sDs (green) and D ∗ sD ∗ s (blue); (right) charm baryon a…

**Figure 13.** Figure 13: Examples of charm-quark production processes at hadron colliders. The charm anti-charm pair subsequently hadronises. for searches of exotic hadrons with hidden charm. Charm at hadron colliders: Charm quarks can also be produced at proton (anti-)proton colliders, see …

**Figure 15.** Figure 15: (Top) Production cross-sections of promptly produced D 0D¯ 0 pairs at Belle, at centre-of-mass energies ranging from the D 0D¯ 0 threshold to 5 GeV, with the vertical lines corresponding to the ψ(3770), ψ(4040), ψ(4160) and ψ(4415) resonances [110]. (Bottom) Different…

**Figure 16.** Figure 16: Example of Feynman diagram describing the strong decay of an excited D ∗+ meson into ground state neutral D 0 and charged pion π + states [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]

**Figure 17.** Figure 17: Tagging of prompt (left) and secondary (middle) and doubly tagged (right) charm decays in proton-proton collisions. Prompt decays (i.e. decays of promptly produced charm mesons) are usually tagged by the charge of the soft pion in the strong decay D ∗+ → D 0π + . Deca…

**Figure 18.** Figure 18: The prompt charm production from D ∗ gives access to large yields but it is important to carefully separate the contributions from secondary charm productions for the reasons outlined above. With muon-tagged charm decays, there are lower yields and higher levels of co…

**Figure 19.** Figure 19: Example of tagging used for charm production at threshold at e.g. BESIII. The information from the tag side is used to pin down the properties of the D meson decaying on the signal side. ∼ ◦ ∼ ◦ ∼ ◦ ∼ For charm particles produced at threshold in electron-positron coll…

**Figure 21.** Figure 21: The SU(4)F multiplets 20F with J P = 3/2 + (a), 20F with J P = 1/2 + (b) and 4F (c) of baryons composed of u, d, s and c quarks, where J and P denote the particle spin and parity, respectively. Baryons with one c quark are one level up from the lowest level, followed…

**Figure 22.** Figure 22: (Left) Several narrow excited Ωc resonances from [177], the contributions are described in the legend: the five narrow peaks in blue are excited Ωc resonances seen in an earlier version of this study, and the brown and red peaks correspond to newly discovered excited …

**Figure 23.** Figure 23: Besides the Higgs boson, 79 new particles have been discovered at the LHC, and 52 contain hidden or open charm. Taken from [190] with permission. See the Quantum Working Group Exotics hub [191] for similar interactive plots with results from all experiments [PITH_FUL…

**Figure 24.** Figure 24: Using Schrodinger’s cat to visualise the concepts of non-locality and quantum entanglement for the state (1 ¨ / √ 2) |D 0 ⟩|D¯ 0 ⟩ − |D¯ 0 ⟩|D 0 ⟩ . This entangled state can be created via the decay of a ψ(3770) resonance into two D mesons, denoted by D1 and D2. A …

**Figure 25.** Figure 25: Feynman diagram describing the decay of a muon via the weak interactions [PITH_FULL_IMAGE:figures/full_fig_p020_25.png]

**Figure 26.** Figure 26: Examples of Feynman diagrams describing the weak decay c → sdu¯ at tree-level (left) and at 2-loop in the strong coupling (right). which equals the total width Γ of the muon, reads - see e.g. [198] for an early reference: Γ(µ − → νµe − ν¯e) = G 2 Fm 5 µ 192π 3 f  …

**Figure 27.** Figure 27: By expanding the W-boson propagator at leading order in 1/m 2 W , the c → sdu¯ amplitude in the SM (full theory) (left) is matched into a local four-quark interaction in the weak effective theory (right). LL NLL NNLL NNNLL Tree 1 - - - 1-loop αs ln αs - - 2-loop α 2 s…

**Figure 28.** Figure 28: Penguin diagram describing the quark-level decay c → uqq¯, contributing to e.g. D 0 → K +K − (q = s) and D 0 → π +π − (q = d) [PITH_FULL_IMAGE:figures/full_fig_p022_28.png]

**Figure 29.** Figure 29: Feynman diagram describing the leptonic tree-level decay D + s → µ + νµ. Non-perturbative QCD interactions - shown in violet - are responsible for the binding of the D + s meson. fD (MeV) fDs (MeV) fDs / fD Ref. 212.0 (0.7) 249.9 (0.5) 1.1783 (0.0016) [209] [PITH_FUL…

**Figure 30.** Figure 30: Feynman diagram describing the semileptonic treelevel decay D + → K¯ 0 e + νe. Non-perturbative QCD effects - shown in violet - are responsible for the binding of the D + s and K¯ 0 mesons as well as for their interactions. Lattice QCD [209] LCSRs [210] f D→π + (0) 0…

**Figure 31.** Figure 31: Example of tree-level (left) and penguin (right) Feynman diagrams contributing to the non-leptonic decay D 0 → K +K − . Non-perturbative QCD effects - shown in violet - are responsible for the binding of all mesons as well as for their interactions [PITH_FULL_IMAGE:f…

**Figure 32.** Figure 32: Box diagrams responsible for the mixing between the D 0 meson and its anti-particle D¯ 0 . Analogous diagrams with up-type quarks inside the loop describe mixing of down-type mesons i.e. K 0 , B 0 d , and B 0 s . hadron decays make use of topological amplitude decompo…

**Figure 33.** Figure 33: Schematic representation of the HQE in Eq. (42). By assuming that the charm quark is heavy, the discontinuity of two- and one-loop diagrams obtained from the double insertion of the effective Hamiltonian in Eq. (41), is matched into local two- and four-quark operators…

**Figure 34.** Figure 34: Comparison of HQE predictions and experimental data for charmed mesons total decay widths (top left), semileptonic branching ratios (top right), lifetime ratios (bottom left) and ratios of semileptonic widths (bottom right), based on [240]. hierarchy τ(Ξ 0 c ) < τ(Λ +…

**Figure 35.** Figure 35: Comparison of HQE predictions and experimental data for charmed baryons lifetimes, based on [277]. 5 Exclusive charm hadron decays 5.1 Leptonic and semileptonic decays Studies of leptonic and semileptonic charm hadron decays provide invaluable information on the prope…

**Figure 36.** Figure 36: Missing mass squared distribution for the decays D + → µ + νµ, taken from [299]. The data are shown as black points with uncertainty bars. The signal distribution is indicated in blue. All background contributions are described in the legends of the figures [PITH_FU…

**Figure 38.** Figure 38: In BSM scenarios like the 2HDM, the leptonic decay D + s → µ + νµ would receive additional contributions compared to those allowed in the SM, such as due to a new charged Higgs boson. where rH depends on the parameters of the 2HDM models, such as the mass of the charg…

**Figure 39.** Figure 39: Definition of the variables used in four-body semileptonic decays as defined in [331] for the D + → K −π + e + νe mode. There are two rest frames, one for the lepton pair and one for the final state hadrons pair. θK is the angle between the pion and the D-meson direct…

**Figure 40.** Figure 40: Example of penguin (left) and box (right) diagrams contributing in the SM to the rare decay D 0 → µ +µ − [PITH_FULL_IMAGE:figures/full_fig_p034_40.png]

**Figure 41.** Figure 41: Examples of diagrams contributing to the rare decay D + → π +µ +µ − : FCNC transitions (left) and weak annihilation (right) [PITH_FULL_IMAGE:figures/full_fig_p034_41.png]

**Figure 42.** Figure 42: Example of diagram describing the contribution of intermediate vector meson resonances to the FCNC decay D + → π +µ +µ − . and for D → ωℓ+ νℓ in [318]. Moreover, studies of the four-body semileptonic decays, and of their S - and P-wave contributions, were recently pub…

**Figure 43.** Figure 43: SM differential branching fractions of the rare semileptonic decays D + → π +µ +µ − (left) and D + s → K +µ +µ − (right). In each plot, the blue band represents the short-distance (non-resonant) contribution, whereas the orange band the long-distance (resonant) one. T…

**Figure 44.** Figure 44: Branching fractions limits for rare and forbidden D + -meson decays. These are classified into decays which are FCNC (left panel), lepton flavour (LF) violating (middle panel) and lepton number (L) violating (right panel) (the white and grey areas are there to simply …

**Figure 45.** Figure 45: Lepton flavour universality principle. 5.2 LFU tests with leptonic and semileptonic decays Lepton flavour universality (LFU) is an intrinsic feature of the SM but also accidental. The principle behind it is that the leptons of all three families, i.e. e, µ, τ, interac…

**Figure 46.** Figure 46: LFU tests with semileptonic charm decays: the results for the ratio R are shown for different decay modes (here e.g. D + → π 0 τ + µ + ν indicates that the ratio R corresponds to B(D +→π 0 τ +ν) B(D+→π 0µ +ν) , and similarly for the other entries); the experimental …

**Figure 47.** Figure 47: Some of the topologies contributing to the CF, SCS, and DCS two-body D 0 decays. The CF and DCS proceed only through tree-level amplitudes. The SCS decays receive contributions from tree-level and penguin amplitudes, which have different strong and weak phases. To kee…

**Figure 48.** Figure 48: Three different Dalitz plots of the D 0 → KS π +π − decay from D ∗+ → D 0π + decays formed with the two-body invariant masses M2 KS π− , M2 KS π+ (left), M2 KS π− , M2 π+π − (middle), M2 π +π− , M2 KS π+ (right). To guide the eye, the vertical, horizontal and diagonal…

**Figure 49.** Figure 49: Three different Dalitz plot projections of the D 0 → KS π +π − decay from D ∗+ → D 0π + , formed with the two-body invariant masses M2 KS π− (top), M2 KS π+ (middle), M2 π+π − (bottom). The plots on the right use a log scale to highlight sensitivity to low fit fractio…

**Figure 50.** Figure 50: Both charm mixing and the Cabibbo suppression contribute to the di [PITH_FULL_IMAGE:figures/full_fig_p042_50.png]

**Figure 52.** Figure 52: Decay-time evolution of the ratio, R(t), of WS to RS yields (data points in black) with the projection of the mixing allowed (blue solid line) and no-mixing (blue dashed line) fits overlaid. Taken from [21]. In comparison to two-body analysis, multi-body analysis has…

**Figure 53.** Figure 53: Possible decay routes for the mode D 0 → K 0 S π +π − . If there is CP violation in the D 0 → D¯0 step, this could increase/decrease the ratio of CF to DCS decays [PITH_FULL_IMAGE:figures/full_fig_p043_53.png]

**Figure 54.** Figure 54: “Binning” of the D 0 → K 0 S π +π − Dalitz plot. Colours indicate the absolute value of the bin index b (each of the 8 colours on the vertical z axis corresponds to one bin in the strong phase difference). m 2 + refers to the invariant mass combination of KS π + and m…

**Figure 55.** Figure 55: CP-averaged yield ratios in bins of decay time. Ratios are calculated for bins, Ri from above and below the meridian of [PITH_FULL_IMAGE:figures/full_fig_p045_55.png]

**Figure 58.** Figure 58: HFLAV fit of the mixing parameters x and y in the case of CPV allowed. The no-mixing case (0,0) is excluded at more than 10σ. Taken from [305]. HFLAV [305] testing the no-CPV, i.e. |q/p| = 1 and ϕ = 0, and the CPV-allowed, i.e. |q/p| , 1 and ϕ , 0, hypotheses, respect…

**Figure 59.** Figure 59: Comparison of different tags for the D 0 → KS π +π − decay at BESIII: flavour tags (left), CP-even tags (middle), CP-odd tags (right). Taken from [466]. • Mixed CP content: these are D-meson decays in which the final state is not a CP eigenstate and has mixed CP conte…

**Figure 60.** Figure 60: The strong-phase difference parameters ci , si measured by the BESIII collaboration for the equal binning scheme shown in [PITH_FULL_IMAGE:figures/full_fig_p047_60.png]

**Figure 61.** Figure 61: Signal distributions of D 0 → K +K − (left) and D 0 → π +π − (right) decays. In the two top plots, the distributions of the invariant mass of D 0π correspond to promptly produced charm mesons, while in the two bottom plots, the distributions of the D 0 invariant mass …

**Figure 62.** Figure 62: Decay-time dependent asymmetries of D 0 → K +K − (left) and D 0 → π +π − (right) decays measured by LHCb. Taken from [511] [PITH_FULL_IMAGE:figures/full_fig_p052_62.png]

**Figure 63.** Figure 63: HFLAV plot showing the current status of the direct CPV, ∆a dir CP (along the vertical axis), and indirect CPV, a ind CP (along the horizontal axis), in charm decays, obtained combining measurements from LHCb (blue), Belle (green), BaBar (yellow) and CDF (pink) experi…

**Figure 64.** Figure 64: Central values and two-dimensional confidence regions in the (a d K−K+ , a d π −π + ) plane for the Run 1 measurement using 3 fb−1 (in black), and the legacy LHCb measurement using the full Run 1 and Run 2 samples corresponding to about 8.7 fb−1 (in red). The no-direc…

**Figure 65.** Figure 65: Mass distributions for the D + s candidates in three representative Dalitz plot bins (a,b,c), defined in the lower right subfigure. Fitting these distributions greatly reduces the error on N i when using the Miranda method. The Dalitz description here is slow for th…

**Figure 67.** Figure 67: Visualisation of the energy test method. The blue dots represent a sample n formed of D 0 mesons and the white dots a sample ¯n formed of D¯ 0 mesons. Searches of CPV among the samples are performed comparing the following three differences D 0 − D 0 (red), which eval…

**Figure 68.** Figure 68: (a) Examples of tree-level (left) and exchange (right) topologies contributing to AT in Eq. (118). (b) Example of penguin topology with internal q = d, s, b quark pair contributing to A q P in Eq. (118). Analogous diagrams with s → d contribute to D 0 → π +π − . the b…

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Nonperturbative Dynamics in D-meson Mixing
hep-ph 2025-08 conditional novelty 6.0 of 10

First calculation of dimension-11 and dimension-12 QCD condensate contributions to D-meson mixing gives xD = 1.27e-5, still below experiment.

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Reviewed August 6, 2026 · model on record in the stance chip above.

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