# Comptes Rendus
Physique

Evelina Gersabeck and Antonio Pich

## Tau and charm decays

Volume 21, issue 1 (2020), p. 75-92

Published online: 17 September 2020

Issue date: 17 September 2020

https://doi.org/10.5802/crphys.14

**Part of Special Issue:** A perspective of High Energy Physics from precision measurements

**Guest editors:** Stéphane Monteil (Clermont Université, CNRS/IN2P3, Clermont-Ferrand) and Marie-Hélène Schune (Université Paris-Saclay, CNRS/IN2P3, Orsay)

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_2.png" /></figure>

Les Comptes Rendus. Physique sont membres du
Centre Mersenne pour l'édition scientifique ouverte
www.centre-mersenne.org

e-ISSN : 1878-1535
A perspective of High Energy Physics from precision measurements
*La physique des Hautes Energies du point de vue des mesures de précision*

# Tau and charm decays

*Les désintégrations du τ et du charme*

Evelina Gersabeck*, a and Antonio Pich b

a University of Manchester, Oxford Road, Manchester, M13 9PL, UK

b IFIC, Universitat de València-CSIC, Catedrático José Beltrán 2, E-46980 Paterna, Spain.

*E-mails*: evelina.gersabeck@cern.ch (E. Gersabeck), pich@ific.uv.es (A. Pich).

**Abstract.** A summary of recent precise results in tau and charm physics is presented. Topics include leptonic and hadronic tau decays, lepton flavour and lepton number violation, charm mixing and CP violation, leptonic and semileptonic charm decays, rare decays and spectroscopy.

**Résumé.** Nous présentons une sélection de mesures précises et récentes des physiques du lepton τ et du quark charmé et leur interprétation. Nous examinons en particulier les désintégrations hadroniques et leptoniques du τ, les recherches de violation du nombre leptonique et du nombre leptonique par famille, le mélange des mésons neutres charmés et l'observation de la brisure de la symétrie CP, les désintégrations leptoniques et semileptoniques des hadrons charmés, leurs désintégrations rares et les récentes découvertes spectroscopiques.

**Keywords.** Tau and charm physics, $D^0$ mixing, CP violation.

**Mots-clés.** La physique du charme et du tau, Oscillation du $D^0$, Violation de CP.

## 1. Introduction

The τ is a third generation lepton that decays into quarks and leptons of the first two families. A precision study of its dynamical properties could then shed some light in our understanding of the flavour problem: why fermions are replicated in three sequential generations with identical properties, except the values of their masses. Moreover, the τ lepton is heavy enough to have a rich variety of hadronic decays, providing a clean laboratory to test QCD at low energies [1].

The charm is an up-type quark accessible to precision experiments. Being a member of the second family, it allows us to study the interplay of weak and strong interactions through its large diversity of weak decays: leptonic, semileptonic, Cabibbo favoured, Cabibbo suppressed,

<footnote>* Corresponding author.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** SM relation between the τ lifetime and $B_e \equiv B(\tau^- \to v_\tau e^- \bar{v}_e)$ (blue band), compared with the measured values (red cross). The band width reflects the current uncertainty on $m_\tau$.</figcaption></figure>

doubly Cabibbo suppressed, and rare decays that are very suppressed (radiative decays, flavour-changing neutral-current transitions) or even forbidden in the Standard Model (SM). The investigation of $D^0-\bar{D}^0$ mixing and CP-violation phenomena in the up-type sector is of enormous interest to test the CKM quark-mixing mechanism. The confinement properties of QCD can also be analysed through the spectroscopy of open charm mesons and charm baryons.

The τ mass, $m_\tau = (1776.86 \pm 0.12)$ MeV, is very close to the lightest charmed-particle mass, $m_{D^0} = (1864.83 \pm 0.05)$ MeV [2]. Therefore, both types of particles have similar production mechanisms at $e^+e^-$ colliders, making the physics of τ and c a common objective in many experiments. The electromagnetic (or $Z^0$ exchange) production cross section is usually maximised by running at some resonance peak that, moreover, decays to quantum correlated $\tau^+\tau^-$, $D^0\bar{D}^0$ or $D^+D^-$ pairs. The LEP-I experiments were running at the $Z^0$ peak, MARKIII (at SPEAR at SLAC), CLEO (at CESR-c) and BESIII (at BEPCII) utilise the $\psi(3770)$ resonance, and the Y(4S) is the default choice at the b-factories, Belle (at KEKB) and BaBar (at PEP-II). The charm cross-sections at low-energy $e^+e^-$ colliders are at the order few nb. The charm production at hadron colliders occurs in very asymmetric collisions due to the fact that the protons are no longer point-like particles. At hadron colliders experiments LHCb (at LHC) and CDF (at Tevatron), the cross-sections for prompt and secondary production (from B decays) are significantly higher: they can reach about 2 mb for prompt $D^0$ at 13 TeV [3].

## 2. Leptonic tau decays

The τ lepton decays through the emission of a virtual $W^-$ boson, i.e., $\tau^- \to v_\tau W^{-*} \to v_\tau X^-$ with $X^- = e^- \bar{v}_e, \mu^- \bar{v}_\mu, d\bar{u}, s\bar{u}$. In the SM, the charged-current interaction has a universal strength that can be precisely measured in the analogous decay of the muon, $\mu^- \to v_\mu e^- \bar{v}_e$. Therefore, the leptonic decay width of the τ can be predicted with high accuracy or, equivalently, one gets a relation between the τ lifetime and its leptonic branching ratio that is in excellent agreement with the experimental values [2], shown in Figure 1. The main uncertainty originates in the current experimental error on the τ mass ($\Delta m_\tau / m_\tau = 0.7 \times 10^{-4}$) because $\Gamma_{\tau \to e}$ is proportional to $m_\tau^5$.

Alternatively, one can use the leptonic decay widths of the τ and the µ to test the predicted flavour universality of the $W^\pm$ couplings, i.e., that $g_e = g_\mu = g_\tau \equiv g$. The ratio $\Gamma_{\tau \to \mu}/\Gamma_{\tau \to e}$ is sensitive to $|g_\mu/g_e|$, while $\Gamma_{\tau \to e}/\Gamma_{\mu \to e}$ tests $|g_\tau/g_\mu|$. Table 1 shows the current experimental
<table><caption>**Table 1.** Experimental determinations of the ratios $g_\ell/g_{\ell'}$ [1,2]</caption><thead><tr><th></th><th>Γ<sub>τ→μ</sub>/Γ<sub>τ→e</sub></th><th>Γ<sub>π→μ</sub>/Γ<sub>π→e</sub></th><th>Γ<sub>K→μ</sub>/Γ<sub>K→e</sub></th><th>Γ<sub>K→πμ</sub>/Γ<sub>K→πe</sub></th><th>Γ<sub>W→μ</sub>/Γ<sub>W→e</sub></th></tr></thead><tbody><tr><td>|g<sub>μ</sub>/g<sub>e</sub>|</td><td>1.0018 (16)</td><td>1.0021 (16)</td><td>0.9978 (20)</td><td>1.0010 (25)</td><td>0.996 (10)</td></tr><tr><td>|g<sub>τ</sub>/g<sub>μ</sub>|</td><td>1.0011 (15)</td><td>0.9962 (27)</td><td>0.9858 (70)</td><td>1.034 (13)</td><td></td></tr><tr><td>|g<sub>τ</sub>/g<sub>e</sub>|</td><td>1.0030 (15)</td><td>1.031 (13)</td><td></td><td></td><td></td></tr></tbody></table>

constraints, together with the most precise tests from leptonic $\pi$, K and $W^\pm$ decays. Charged-current universality has been successfully tested at low energies with a 0.15% precision. The direct leptonic decays of the $W^\pm$ suggest a slight excess of events in $W^- \to \tau^- \bar{v}_\tau$, implying a 2.5$\sigma$ deviation from universality that is not compatible with the one order of magnitude more accurate constraints from W-mediated decays. Better $W^\pm$ data would be welcome.

The Lorentz structure of the $\ell^- \to \nu_\ell \ell'^- \bar{v}_{\ell'}$ interaction can be analysed in a model-independent way. The most general, local, derivative-free, lepton-number conserving, four-lepton interaction Hamiltonian, consistent with locality and Lorentz invariance contains ten possible structures with their corresponding complex couplings $g_{e\omega}^n$, where $n = S, V, T$ denotes the type of interaction (scalar, vector, tensor) and the subindices label the left or right chiralities of $\ell$ ($\omega$) and $\ell'$ ($\epsilon$) [1]. Taking out a common global factor that is determined by the total decay rate, the couplings are normalised so that they satisfy $|g_{e\omega}^S| \le 2$, $|g_{e\omega}^V| \le 1$ and $|g_{e\omega}^T| \le 1/\sqrt{3}$. In the SM, $g_{LL}^V = 1$, while all other couplings are identically zero. Measuring the energy and angular distribution of the final charged lepton, complemented with polarisation information whenever available, it is possible to disentangle the contributions from the different operators.

In $\mu$ decay, where precise polarisation measurements have been performed of both $\mu$ and $e$, it has been experimentally proved that the bulk of the decay amplitude is indeed of the predicted $V-A$ type, $|g_{LL}^V| > 0.960$ (90% C.L.) [2] (one needs also information from the inverse transition $\nu_\mu e^- \to \mu^- \nu_e$), and upper bounds on all other couplings have been set. Owing to its much shorter lifetime, the analysis of the $\tau$ interaction is more challenging. It is still possible to get polarisation information about the initial $\tau$, through the correlated distribution of $\tau^+ \tau^-$ pairs produced in $e^+ e^-$ annihilation. However, the polarisation of the secondary charged lepton from the $\tau$ decay has never been measured. Since the data agree with the SM, there exist upper bounds on those couplings corresponding to an initial right-handed $\tau$ [2], but the Lorentz structure of a left-handed decaying $\tau$ remains undetermined.

### 3. Hadronic tau decays

A large set of kinematically-allowed semileptonic decays can be accessed with $\tau$ decay data. Contrary to $e^+ e^-$ annihilation that only tests the electromagnetic vector current, the decay $\tau^- \to \nu_\tau H^-$ probes the matrix elements of both vector and axial-vector currents between the vacuum and the given hadronic state $H^-$. Moreover, one can also disentangle the Cabibbo allowed ($\bar{d}u$) and suppressed ($\bar{s}u$) currents through the strangeness of the produced hadrons. The $\tau$ provides a very good data sample to investigate the dynamics of the QCD Goldstone bosons ($\pi, K, \eta$) in the resonance region, around 1 GeV.

For the lowest-multiplicity decays, $H^- = \pi^-, K^-$, the hadronic matrix elements are already known from $\pi^- \to \mu^- \bar{v}_\mu$ and $K^- \to \mu^- \bar{v}_\mu$, which makes possible to perform the universality tests in Table 1. One can also make a determination of $|V_{us}|$, but it is not yet competitive with those from $K \to \ell v$ and $K \to \pi \ell v$, owing to the currently larger uncertainties in $\tau^- \to K^- \nu_\tau$.
The $\pi^-\pi^0$, $\pi^-\tilde{K}^0$ and $\pi^0 K^-$ final states give us access to an interesting variety of vector form factors with relevant dynamical information. The decay into the odd G-parity state $\pi^-\eta$ is strongly suppressed in the SM with an expected branching fraction around $10^{-5}$ [4, 5]; its observation above this level would imply new physics incorporating second-class currents. With the large data samples collected at the B factories, differential decay distributions with three hadrons in the final state have become available and branching ratios into high-multiplicity 3- and 5-prong decays have been measured [6]. However, several inconsistencies are known to exist in some branching fraction measurements [2] and errors are still large in the Cabibbo suppressed modes. Thus, there is ample room for improvements at Belle II.

The inclusive hadronic width of the $\tau$ can be rigorously computed in QCD. Its Cabibbo allowed component can be expressed in the form [7]

$$R_{\tau, V+A} \equiv \frac{\Gamma(\tau^{-} \rightarrow \nu_{\tau} + \text{hadrons} [\bar{d}u])}{\Gamma(\tau^{-} \rightarrow \nu_{\tau} e^{-} \bar{\nu}_{e})} = N_{C} |V_{ud}|^{2} S_{\text{EW}}\{1 + \delta_{P} + \delta_{\text{NP}}\}, \quad (1)$$

with $N_C = 3$ the number of QCD colours and $S_{EW} = 1.0201 \pm 0.0003$ [8–10] the electroweak radiative corrections. The non-perturbative correction $\delta_{NP}$ is strongly suppressed by six powers of the $\tau$ mass [7] and, moreover, can be extracted from the invariant mass distribution of the final hadrons [11]. Detailed studies performed by ALEPH [12–16], CLEO [17] and OPAL [18] have confirmed that non-perturbative contributions are below 1%. The theoretical prediction of $R_{\tau, V+A}$ is then governed by the perturbative correction $\delta_P$ (~20%), which is known to $O(\alpha_s^4)$ [19] and is very sensitive to the strong coupling, making possible a quite accurate determination of $\alpha_s$ [7, 20, 21]. The main theoretical uncertainty originates in the unknown higher-order perturbative corrections [22, 23].

The most precise experimental determination, extracted from the ALEPH $\tau$ decay distributions, gives $\delta_{NP} = -0.0064 \pm 0.0013$ and $\alpha_s^{(n_f=3)}(m_\tau^2) = 0.332 \pm 0.005_{\text{exp}} \pm 0.011_{\text{th}}$ [24]. Taking as input the ALEPH value of $\delta_{NP}$, the strong coupling can be also determined from the total $\tau$ hadronic width (and/or lifetime); one gets $\alpha_s^{(n_f=3)}(m_\tau^2) = 0.331 \pm 0.013$ [1], in perfect agreement with the ALEPH result. An exhaustive phenomenological re-analysis of the ALEPH data has been recently performed, exploring all strategies previously considered in the literature and several complementary approaches. The results from all adopted methodologies are in excellent agreement, leading to a very robust and reliable value of the strong coupling [25]:

$$\alpha_s^{(n_f=3)}(m_\tau^2) = 0.328 \pm 0.013. \qquad (2)$$

After evolution up to the scale $M_Z$, it implies $\alpha_s^{(n_f=5)}(M_Z^2) = 0.1197 \pm 0.0015$, which agrees perfectly with the direct measurement at $s = M_Z^2$ from the $Z^0$ hadronic width, $\alpha_s^{(n_f=5)}(M_Z^2) = 0.1196 \pm 0.0030$ [26]. The comparison of these two determinations, provides a beautiful test of the predicted QCD running:

$$\alpha_s^{(n_f=5)}(M_Z^2)|_{\tau} - \alpha_s^{(n_f=5)}(M_Z^2)|_{Z} = 0.0001 \pm 0.0015_{\tau} \pm 0.0030_{Z}. \qquad (3)$$

The ratio of the inclusive $|\Delta S| = 1$ and $|\Delta S| = 0$ $\tau$ decay widths (normalised to the electronic width), $R_{\tau,S}/R_{\tau,V+A}$, provides a clean determination of $V_{us}$ [27, 28]. To a first approximation, the
<figure><img src="image_4.png" /><figcaption>Figure 2. Experimental upper limits on τ LFV and LNV branching ratios [32].</figcaption></figure>

experimental ratio directly measures $|V_{us}/V_{ud}|$. Taking into account the PDG value of $V_{ud}$ and the small SU(3)-breaking correction $\delta R_{\tau,th} = 0.242 \pm 0.033$ [29–31], one finds [32]

$$|V_{us}| = \left( \frac{R_{\tau,S}}{\frac{R_{\tau,V+A}}{|V_{ud}|^2} - \delta R_{\tau,th}} \right)^{1/2} = 0.2195 \pm 0.0019, \quad (4)$$

which is $2.9\sigma$ lower than the unitarity expectation $|V_{us}|^{uni} = \sqrt{1 - |V_{ud}|^2 - |V_{ub}|^2} = 0.2257 \pm 0.0009$. More precise measurements of the Cabibbo-suppressed $\tau$ branching fractions at Belle II are expected to clarify the current discrepancy [33].

## 4. Lepton flavour and lepton number violation in tau decays

The current experimental limits on neutrinoless lepton-flavour-violating (LFV) decays of the $\tau$ lepton are shown in Figure 2. Thanks to the large data samples collected at the B factories, sensitivities of a few times $10^{-8}$ have been achieved in many leptonic ($\tau \to \ell\gamma$, $\tau \to \ell'\ell^+\ell^-$) and semileptonic ($\tau \to \ell P^0$, $\tau \to \ell V^0$, $\tau \to \ell P^0 P^0$, $\tau \to \ell P^+ P'^-$) LFV decay modes. Competitive limits for some selected final states, such as $\tau \to 3\mu$, have been also set by LHCb. Belle II is expected to push these limits to the $10^{-9}$ level [33], increasing in a drastic way the sensitivity to new-physics scales.

Lepton-number violation (LNV) has been also searched for in $\tau$ decays. Very stringent upper limits in the range $(2.0-8.4) \times 10^{-8}$ (90% C.L.) have been set on the decay modes $\tau^- \to \ell^+ P^- P'^-$, with $\ell = e, \mu$ and $P, P' = \pi, K$ [34]. Worth mentioning is also the experimental limit $B(\tau^- \to \Lambda\pi^-) < 7.2 \times 10^{-8}$ (90% C.L.), which tests the violation of both lepton and baryon numbers [35].
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Left: The HFLAV averages for the mixing parameters $x$ and $y$; Right: The combination plot of $\Delta A_{CP}$ and $A_{\Gamma}$. The point of no CP violation (0,0) is shown as a filled circle [32].</figcaption></figure>

**5. Mixing and CP violation in charm decays**

The phenomenon of CP violation is one of the keys to uncover why there is an overwhelming amount of matter over antimatter in our Universe, as postulated by Sakharov in 1967 [36]. The CP symmetry applies to processes invariant under the combined transformation of charge conjugation (C) under which a particle is exchanged with its own anti-particle, and spatial inversion (parity, $P$). The SM accommodates CP violation in a single complex phase of the CKM matrix which only appeared after introducing the third family of quarks. The CP violation has been well established in decays of $K^0(\bar{s}d)$, $B^0(\bar{b}d)$, $B_s^0(\bar{b}s)$ and $B^+(\bar{b}u)$ mesons. Unlike for the $b$-system where we expect sizeable effects, for D meson decays, SM CP violation effects are expected to be tiny because of the smallness of the imaginary component of the CKM elements involved in the relevant processes. CP violation in charm was only recently observed in decays of $D^0(c\bar{u})$ to a pair of charged kaons and to a pair of charged pions [37] and there is no agreement whether its size is compatible with the SM predictions [38–42].<fnref n="1" />

Neutral charm mesons can periodically change into their antimatter counterparts and back – they oscillate. As a result, the mass eigenstates, with well-defined masses and lifetimes, are linear combinations of flavour eigenstates, with well-defined quark composition: $|D_{1,2}\rangle = p|D^0\rangle \pm q|\bar{D}^0\rangle$. Here, $q, p$ are complex numbers that are related by $p^2 + q^2 = 1$. The mixing parameters $x$ and $y$ are defined as $y \equiv \Delta\Gamma/(2\Gamma)$, where $\Delta\Gamma = \Gamma_2 - \Gamma_1$ is the width difference of the charm mesons, and $x \equiv \Delta m/\Gamma$, where $\Delta m$ is their mass difference. The mixing process is suppressed in charm because the corresponding box diagrams contain down-type quarks in the internal lines: owing to the GIM cancellation, each virtual quark contribution is proportional to $m_q^2$, and the heaviest quark, $b$ is not quite as heavy as the top quark (which plays an important role in the B meson mixing) and has very small couplings with $c$ and $u$ quarks. This leads to very slow oscillation of the D mesons, and to very small values of $x$ and $y$, experimentally challenging to measure. The current world averages are displayed in Figure 3 (left) [32]. Including the latest LHCb result [43], the parameter $x$ related to the mass difference between neutral charm-meson eigenstates is measured to be greater than 0 with a significance exceeding 3 standard deviations for the first time.

<footnote n="1">¹The theory predictions for charm mesons are particularly difficult because of the intermediate mass of the $c$ quark. Effective theories such as HQET (too light) and Chiral Perturbation Theory (too heavy) are not directly applicable.</footnote>
The amplitude for the decay of a D hadron to a final state $f$ can be expressed as $A_f$, and the amplitude of the charge-conjugated process as $\bar{A}_{\bar{f}}$. There are several ways to generate a CP asymmetry. Direct CP violation occurs for a non-zero asymmetry in the decay amplitudes. This type of CP violation depends on the decay mode, and it can involve either charged or neutral particles. In order for direct CP violation to be realised the amplitudes $A_f$ and $\bar{A}_{\bar{f}}$ require interference of at least two different processes with different weak and strong phases, defined as the phase which changes its sign under CP transformation (the weak phase), and the one that does not (the strong phase). The indirect CP violation comprises the CP violation in mixing processes incorporating neutral particles, and CP violation in the interference between mixing and decay amplitudes. CP violation in mixing takes place if the transition probability of particles to antiparticles compared to the reverse process differs, and occurs when $|q/p| \neq 1$. This type of CP violation is universal and does not depend on the decay mode. CP violation in the interference between mixing and decay amplitudes is present if the imaginary part of $\lambda \equiv p\bar{A}_{\bar{f}}/(qA_f)$ is non-zero.

In two-body singly-Cabibbo-suppressed (SCS) charm decays, the direct CP violation is searched through measuring the time-integrated CP asymmetry in the decay rates of the charm mesons ($f = K^+K^-, \pi^+\pi^-$)

$$A_{CP} \equiv \frac{\Gamma(D \to f) - \Gamma(\bar{D} \to \bar{f})}{\Gamma(D \to f) + \Gamma(\bar{D} \to \bar{f})}, \qquad (5)$$

where $\Gamma$ denotes the partial decay rate. Taking the difference of the asymmetries in two different final states, $\Delta A_{CP} = A_{CP}(D^0 \to K^+K^-) - A_{CP}(D^0 \to \pi^+\pi^-)$, has two advantages: the nuisance asymmetries originating from production and detection cancel, and the sensitivity to CP violation is enhanced as the CP asymmetries in these two channels are expected to be of similar magnitude but with opposite signs. $\Delta A_{CP}$ is mostly a measure of direct CP violation in charm. A study using the full Run 1 and Run 2 data of LHC yields $\Delta A_{CP}^{\text{exp}} = (-15.4 \pm 2.9) \times 10^{-4}$, with a significance of more than five standard deviations [37]. A range of new SM predictions for $\Delta A_{CP}$ [38–42] argue whether beyond the SM (BSM) physics is necessary to explain this result, or whether it originates in a mild non-perturbative enhancement due to rescattering effects or to the presence of a nearby $0^{++}$ resonance such as $f_0(1710)$. Some papers suggest that resolving this tension within an extension of the SM includes a flavour violating $Z'$ that couples only to $\bar{s}s$ and $\bar{c}u$ quarks [38]. The value of $\Delta A_{CP}$ together with other experimental data can then be used to make predictions on CP violation in several $D^0 \to PP$ and $D^0 \to VP$ channels [42, 44].

This is the first observation of CP violation in the charm sector, and so far the only one. Several measurements in other two-body decays have greatly improved the precision of the asymmetries for the decay modes $D^0 \to K_S^0 K_S^0, D^0 \to K_S^0 h^+, D^0 \to \eta' h^+, D^0 \to \pi^+ \pi^0, D^0 \to \pi^0 \pi^0, D^0 \to K_S^0 \pi^0, D^0 \to K_S^0 \eta^{(\prime)},$ etc. [32]. From the theoretical point of view, a promising two-body decay to probe for CP violating effects is $D^0 \to KK^*$ [45]. In addition, various model-dependent and model-independent techniques probe for CP violation in multibody decays but the experimental results are so far compatible with CP symmetry conservation [32].

The asymmetry between the effective lifetimes, $^2\bar{\Gamma}$, of mesons initially produced as $D^0$ and $\bar{D}^0$ and decaying into the CP-even final states $D \to hh$, where $h = K, \pi$, is a measure of indirect CP violation. The current best results from LHCb, combining the full Run 1 and Run 2 statistics [46], $A_\Gamma(KK) = (-4.4 \pm 2.3 \pm 0.6) \times 10^{-4}$ and $A_\Gamma(\pi\pi) = (2.5 \pm 4.3 \pm 0.7) \times 10^{-4}$ are compatible with no CP violation in charm mixing or the interference between mixing and decay. These are yet the

<footnote>$^2$The effective lifetime is the lifetime obtained from a single exponential fit to the decay-time distribution.</footnote>
most precise experimental measurements of CP asymmetries. The interplay between direct and indirect CP violation in two-body charm decays is presented in Figure 3 (right) combining the $\Delta A_{CP}$ and $A_Γ$ results.

Two methods are employed in establishing mixing in charm decays. Firstly, the flavour of the neutral D meson at production has to be determined. Usually this is done by the charge of the pion in the strong decay of $D^{*+} \to D^0\pi^+$. Alternatively, the flavour of the secondary charm decays from $\tilde{B} \to D^0\mu^-\nu_\mu X$ can be tagged by the charge of the muon. The second technique is used at the LHCb experiment only. Until 2012, the mixing in charm was established with more than 5 standard deviations significance only by a combination of three different experiments [47–50] done by HFLAV [32]. The mixing parameters can be extracted from the time-dependent ratio of $D^0 \to K^+\pi^-$ (also referred to as wrong-sign, WS) to $D^0 \to K^-\pi^+$ (also known as right-sign, RS) decay rates

$$R(t) = \frac{N_{WS}(t)}{N_{RS}(t)} \approx R_D + \sqrt{R_D} y' \frac{t}{\tau} + \frac{x'^2 + y'^2}{4} \left(\frac{t}{\tau}\right)^2. \qquad (6)$$

Here, $t/\tau$ is the decay time expressed in units of the average $D^0$ lifetime $\tau$, and $R_D$ is the ratio of doubly-Cabibbo-suppressed (DCS) to Cabibbo-favoured (CF) decay rates. Note that in (6), $x'$ and $y'$ are rotated by the strong phase difference between the CF and DCS amplitudes, $\delta$, compared to the $x, y$ parameters defined above. By observing a decay-time dependence of the ratio $R(t)$, and measuring the parameters $y'$ and $x'^2$, the LHCb experiment reported a first observation of $D^0 - \tilde{D}^0$ oscillations in a single measurement [51]. Since then, more precise measurements have been reported [52]. Following a similar strategy, $D^0 - \tilde{D}^0$ oscillations were observed in $D^0 \to K3\pi$ decays as well [53].

These measurements have been extremely useful in establishing that the neutral charm mesons oscillate. However, the golden mode for measuring the mixing parameters $x$ and $y$ without the strong phase rotation is $D^0 \to K_S^0\bar{h}h$. This final state is accessible both through decays of $D^0$ and $\tilde{D}^0$. The multiple CF and DCS interfering amplitudes enhance the sensitivity to $x$ and $y$. There are two techniques that can be employed to analyse these decays: a model-independent one where the decay-time evolution in bins of similar strong phase difference is studied, and a model-dependent one where the effective lifetimes of individual resonances are measured. The model-dependent technique where time-dependent amplitude analysis of self-conjugated decays allows for a direct measurement of $x$ and $y$, and a simultaneous search for CP violation in mixing, in the decay and in the interference between mixing and decay, was developed by the CLEO experiment [54], and was later extended by the BaBar and Belle experiments [55, 56]. The model-independent methods [57, 58] rely on external input for the strong phase differences between charm decay amplitudes from quantum correlated $D^0$ and $\tilde{D}^0$ produced at threshold at the $\psi(3770)$ resonance at CLEO [59] or BESIII experiments. The most precise results come from BESIII [60, 61].<fnref n="3" /> The latest LHCb measurement in [58], combined with previous measurements of the mixing parameters, yields the first evidence that the neutral charm meson masses are different, and $x$ is positive. An overview of how complex the state-of-art of the theory predictions for $x$ and $y$ is can be found in [63]. The mixing parameters can be extracted for $D^0$ and $\tilde{D}^0$ which probes CP violation in charm mixing. To date, all results are compatible with CP symmetry and agree with the SM predictions.

<footnote n="3">³The current best results are based on a data sample of about 3 fb⁻¹ while the collaboration considers increasing this data sample to 20 fb⁻¹. In addition to improving our knowledge on charm mixing, these measurements play an important role in reducing systematic uncertainties on determinations of the CKM angle $\gamma$ allowing for sub-degree precision [62].</footnote>
## 6. Leptonic and semileptonic charm decays

Purely leptonic charm decays $D_{(s)}^{+} \to \ell^{+}v_{\ell}$, where $\ell$ is a lepton, proceed through a $W^{\pm}$ annihilation diagram. In the SM at tree level, the decay width is given by

$$ \Gamma(D^{+} \to \ell^{+}v_{\ell}) = \frac{G_F^2 f_{D^{+}}^2}{8\pi} |V_{cd}|^2 m_{\ell}^2 m_{D^{+}} \left(1 - \frac{m_{\ell}^2}{m_{D^{+}}^2}\right)^2 \quad (7) $$

where $G_F$ is the Fermi coupling constant, $m_{\ell}$ is the lepton mass, and $m_{D^+}$ is the $D^+$-meson mass. All these quantities are known with a very good accuracy [2]. An experimental measurement of the decay width (or the branching fraction) allows for a determination of the product of the decay constant, $f_{D^+}$, and the CKM element $V_{cd}$. The unique tagging technique and the excellent performance of the BESIII detector allows to reconstruct these decays. BESIII measured $B(D^{+} \to \mu^{+}v_{\mu}) = (3.71 \pm 0.19 \pm 0.06) \times 10^{-4}$ [64]. This is the most precise result of this quantity to date, and determines $f_{D^+}|_{V_{cd}} = (45.75 \pm 1.20 \pm 0.39)$ MeV. The decay constant $f_{D^+}$ is obtained using as input the CKM matrix element $|V_{cd}| = 0.22520 \pm 0.00065$ from the global fit in the SM [65]. Alternatively, $|V_{cd}|$ is determined using $f_{D^+} = (207 \pm 4)$ MeV from lattice QCD (LQCD) as input. The results are: $f_{D^+} = (203.2 \pm 5.3 \pm 1.8)$ MeV, and $|V_{cd}| = 0.2210 \pm 0.0058 \pm 0.0047$.

Recently, the decay $D^+ \to \tau^+v_{\tau}$ was observed for a first time by the BESIII collaboration [66]. Its branching fraction was determined to be $B(D^+ \to \tau^+v_{\tau}) = (1.20 \pm 0.24 \pm 0.12) \times 10^{-4}$. Taking the world average result for $B(D^+ \to \mu^+v_{\mu}) = (3.74 \pm 0.17) \times 10^{-4}$, a test for lepton flavour universality (LFU) was reported:

$$ R(D^{+})_{\tau/\mu} = \frac{\Gamma(D^{+} \to \tau^{+}v_{\tau})}{\Gamma(D^{+} \to \mu^{+}v_{\mu})} = 3.21 \pm 0.64 \pm 0.43, \quad (8) $$

which is consistent with the SM expectation $R(D^+)_{\tau/\mu} = 2.67$.

The decays $D_s^+ \to \ell^+v_\ell$ have been studied by BESIII [67, 68], as well as by its predecessor CLEO [69], Belle [70] and BaBar [71] experiments. Analogously to the case above, the values $f_{D_s^+}$ and $|V_{cs}|$ were extracted and the most precise values are 252.9 ± 3.7 ± 3.6 MeV and 0.985 ± 0.014 ± 0.014, respectively [67]. These results are important to calibrate various theoretical predictions [72].

Similarly to $R(D^+)_{\tau/\mu}$, $R(D_s^+)_{\tau/\mu} = 10.2 \pm 0.5$ [73] is in agreement with the SM expectation. With a future sample of 20 fb⁻¹ of data at 3.773 GeV at BESIII, the precision on $R(D^+)_{\tau/\mu}$ will be statistically limited to about 8%. Increasing the data sample at 4.178 GeV to 6 fb⁻¹, the precision on $R(D_s^+)_{\tau/\mu}$ will be systematically limited to about 3%. The rate of the $D_{(s)}^+ \to e^+v_e$ decay is helicity suppressed by a factor $m_e^2$ and is beyond the sensitivity of the BESIII experiment.

The measurements of $V_{cs(d)}$ from purely leptonic decays are the most precise ones. The projections with 20 fb⁻¹ at BESIII [73] and 50 ab⁻¹ of data at Belle II [74] indicate that their precision could be improved by an order of magnitude. An alternative way to measure $|V_{cs(d)}|$ is through the differential rate of semileptonic decays $D^0 \to K^{\pm}(\pi^{-})\ell^{+}v_{\ell}$, which in the $m_{\ell} = 0$ limit takes the form

$$ \frac{d\Gamma}{dq^2} = \frac{G_F^2}{24\pi^3} |V_{cs(d)}|^2 |p_{K^-(\pi^-)}|^3 |f_+^{K^-(\pi^-)}(q^2)|^2, \quad (9) $$

where $p$ is the three-momentum of the $K(\pi)$ meson in the rest frame of the D meson, and $f_+(q^2)$ represents the hadronic form factors which depend on the four-momentum transfer between the D meson and the final hadron. The form factors parameterise the strong interaction effects and can be calculated in LQCD. As the uncertainties in the predictions of the form factors shrink [75–77], experimental validation of the results becomes increasingly important. In [78, 79], using final states with electrons, the form factors are studied by fitting the differential decay rates with different shape parameters predicted by the various models, and the compatibility between the data and the calculations is interpreted. The best fit results in [79] for the form-factors at
<figure><img src="image_6.png" /><figcaption>**Figure 4.** The current best 90% C.L. limits of rare $D^0$ decays. The different regions combine FCNC, LFV, LNV, BLN decays [32]. Similar summary plots for charged D mesons and charm baryons can be found in [32].</figcaption></figure>

$q^2 = 0$ are $f_+^K(0) = 0.7368 \pm 0.0026 \pm 0.0036$ and $f_+^\pi(0) = 0.6372 \pm 0.0080 \pm 0.0044$. In analogy to the leptonic decays, $|V_{cs(d)}|$ are obtained using form-factor predictions as an input [75, 80] yielding $|V_{cs}| = 0.9601 \pm 0.0033 \pm 0.0047 \pm 0.0239$ and $|V_{cd}| = 0.2155 \pm 0.0027 \pm 0.0014 \pm 0.0094$.

The LFU tests with semileptonic decays indicate no deviation from the SM: $R_{\mu/e} = \Gamma(D^+ \to \bar{K}^0 \mu^+ \nu_\mu) / \Gamma(D^+ \to \bar{K}^0 e^+ \nu_e) = 0.988 \pm 0.033$ [81] is consistent with the predicted value [82] within uncertainties; for $D^0 \to K^- \ell^+ \nu_\ell$, the corresponding ratio $R_{\mu/e} = 0.974 \pm 0.007 \pm 0.012$ also agrees with the SM [77, 83, 84]; similarly, the ratios using $D^{0(+)} \to \pi^- \ell^0 \nu_e$ decays, $R_{\mu/e}^0 = 0.922 \pm 0.030 \pm 0.022$ and $R_{\mu/e}^+ = 0.964 \pm 0.037 \pm 0.026$, are in agreement with the SM [84] within $1.7\sigma$ and $0.5\sigma$, respectively. All these measurements are currently statistically limited and will be significantly improved with 20 fb$^{-1}$ of data taken at 3.773 GeV in the future, at BESIII [73], and 50 ab$^{-1}$ of data which is being collected by Belle II [33].

## 7. Rare charm decays or searches for BSM particles

The studies of charm rare decays provide a unique probe of BSM physics in the flavour sector, complementary to studies in K and B systems. These comprise studies of lepton flavour violation (LFV), lepton number violation (LNV), baryon number violation (BNV), flavour-changing neutral-current (FCNC) transitions, vector-meson-dominated (VMD) and radiative decays. The expected rates of these processes vary vastly, from forbidden, FCNC ($10^{-16}$ to $10^{-9}$), VMD ($10^{-8}$ to $10^{-6}$) to the not-so-rare radiative decays ($10^{-5}$ to $10^{-4}$).

Examples for such processes can be seen on Figure 4, together with the limits from different experiments. The most stringent limit on FCNC decays comes from $D^0 \to \mu^+ \mu^-$ [85]; on LFV from $D^0 \to e^\pm \mu^\mp$ [86]; on LNV from $D_{(S)}^+ \to \pi^+ \mu^- \mu^-$ [87], and on BNV from $D^0 \to pe$ [88].

The FCNC processes are heavily suppressed in the SM. Short distance contributions to effective $c \to u$ transitions are rather small, therefore the branching fractions are dominated by long distance contributions. An enhancement of the predicted decay rates could signal the presence of new physics.

In the future, many of these limits will be pushed further by the precision experiments LHCb (Upgrade I and II) and Belle II. The large production cross sections will likely translate into world's best measurements. The BESIII experiment is complementary for decays which are difficult for LHCb and Belle, with missing neutrinos, neutral particles, etc., due to its unique capability to identify the flavour of the D meson at production in quantum correlated decays, e.g. decays such
as $D^0 \to \pi^0 \nu \bar{\nu}$ or $D^{\pm,0} \to \pi^{\pm,0} a$, where $a$ is a light pseudoscalar. A future Super $\tau$-charm factory in Novosibirsk (or China) could further help in pushing the limits for rare and forbidden charm decays.

The rarest charm decay observed to date is the $D^0 \to K^+ K^- \mu^+ \mu^-$, studied along with $D^0 \to \pi \pi \mu \mu$ decays. It is observed away from the known resonances $\rho^0/\omega, \phi, \eta$ and its branching fraction, $\mathcal{B}(D^0 \to K^+ K^- \mu^+ \mu^-) = (1.54 \pm 0.27 \pm 0.09 \pm 0.16) \times 10^{-7}$ [89], is in agreement with the SM prediction [90, 91]. The CP asymmetries in the non-resonance regions for these two decays were determined as well [92], $A_{CP}(D^0 \to \pi^+ \pi^- \mu^+ \mu^-) = (4.9 \pm 3.8 \pm 0.7)\%$ and $A_{CP}(D^0 \to K^+ K^- \mu^+ \mu^-) = (0 \pm 11 \pm 2)\%$, and also agree with the SM expectations, which in turn imposes constrains on several BSM models [90, 91, 93–97].

## 8. Charm spectroscopy

Charm spectroscopy provides an excellent ground to study the dynamics of light quarks in the environment of a heavy quark. The theoretical framework for analysing decays of hadrons with one heavy quark is the Heavy Quark Effective Theory (HQET), making use of the limit $m_Q \to \infty$. Many of the exited states predicted in the 80's have not yet been observed [98]. Two of the lowest-lying (1S) states and four (1P) orbital excitations of the open charm mesons are known [2], both for non-strange and strange mesons. Recent experimental effort allows to study the properties (masses, widths) and to determine the quantum numbers such as the total angular momentum and parity of some of the newly observed states. A prompt production of the excited states allows to establish whether a state is natural ($J^P = 0^+, 1^-, 2^+, \dots$) or unnatural ($J^P = 0^-, 1^+, 2^-, \dots$). A secondary production of charmed mesons allows for a full spin-parity analysis. The excited states $D_{s1}(2536)^{+}(1^{+})$, $D_{s2}^{*}(2573)^{+}(2^{+})$, $D_{s1}^{*}(2700)^{+}(1^{-})$ and $D_{s3}^{*}(2860)^{+}(3^{-})$ were observed by [99–103], yielding information on their properties, including spin-parity assignments. In addition to the states reported above, an enhancement around $D_{sJ}^{*}(3400)^{+}$ was seen [103]. The most recent results from charm meson spectroscopy report the resonance parameters, quantum numbers and partial branching fractions of the $D_1(2420)$, $D_1(2430)$, $D_0(2550)$, $D_1^{*}(2600)$, $D_2(2740)$ and $D_3^{*}(2750)$ resonances, which are measured for the first time in a four-body amplitude analysis of the $B^- \to D^{*+} \pi^- \pi^-$ decays [104].

Singly charmed baryons consist of one heavy charm quark and two light $(u, d, s)$ quarks. The large mass difference between the charm quark and the lighter ones justifies the usage of HQET. Excited $\Lambda_c, \Sigma_z$ and $\Xi_c$ states have been well studied [2]. This was not the case for the heaviest of them, the $\Omega_c$ baryon with quark content `css` and quantum numbers $J^P = 1/2^+$, until not long ago. The first observed spin-excited $\Omega_c^*$ state was seen in a decay $\Omega_c^* \to \Omega_c \gamma$ [105, 106], presumed to be a $J^P = 3/2^+$ state. The LHCb experiment has reported the discovery of five new excited $\Omega$ states decaying to $\Omega_c^{*0} \to \Xi_c K^-$, with $\Xi_c \to p K^- \pi^+$ [107]. These five new very narrow states (with widths $\le 10$ MeV) are $\Omega_c(3000)^0$, $\Omega_c(3050)^0$, $\Omega_c(3066)^0$, $\Omega_c(3090)^0$, and $\Omega_c(3119)^0$ (see Figure 5, left). In addition, a broad structure around 3188 MeV has been identified, which could be resolved with more data. Since these baryons were reconstructed using a two-body decay, their quantum numbers are not determined, and their masses are not compared to the theoretical predictions. A possible analysis of three-body final states can provide additional information. The first four of these states were confirmed using a smaller data sample by the Belle experiment, reporting the first observation of these states in $e^+ e^-$ colliders [108]. These baryons have been interpreted as bound states of a c-quark and a P-wave ss-diquark [109]. An alternative interpretation is noted in which the heaviest two states are 2S excitations with $J^P = 1/2^+$ and $3/2^+$, while the lightest three are those with $J^P = 3/2^-, 3/2^-, 5/2^-$ expected to decay via D-waves. The lattice predictions of their masses are summarised in [110]. A molecular model has also been suggested for the interpretation of these states [111].
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Left: The excited $\Omega_c(\text{css})$ states observed by the LHCb collaboration [107]; Right: The first observation of $\Xi_{cc}^{++} \rightarrow \Lambda_c^+ K^- \pi^+ \pi^+$ [112].</figcaption></figure>

The doubly charmed baryons are built of two c quarks and a lighter quark: these are $\Xi_{cc}^+(\text{ccd})$, $\Xi_{cc}^{++}(\text{ccu})$ and $\Omega_{cc}^+(\text{ccs})$. The first two form an isospin doublet with quantum numbers $J^P = 1/2^+$ and $L = 0$. Many predictions of their masses in the range 3500 to 3800 MeV [113–124] and lifetimes from 0.1 to 1.5 ps [121, 122] exist. For the isospin doublet, the difference in the mass is not expected to exceed 1 MeV [120].

The SELEX collaboration [125, 126] observed a peak which was interpreted as a $\Xi_{cc}^+$ baryon in the final states $\Lambda_c^+ K^- \pi^+$ and $pD^+ K^-$, with a measured mass of (3518.7 $\pm$ 1.7) MeV. The collaboration also reported a measurement of the $\Xi_{cc}^{++}$ meson mass to be 3460 MeV [127]. The mass difference is in conflict with the expected mass splitting of isospin doublets. The $\Xi_{cc}^+$ lifetime was experimentally measured to be less than 33 fs at the 90% C.L. which disagrees with the theoretical predictions. The $\Xi_{cc}^+$ observation has not been confirmed in searches performed at the FOCUS [128], BaBar [129], Belle [130], and LHCb [112, 131] experiments.

The doubly-charmed baryon $\Xi_{cc}^{++}$ was observed for the first time in the final state $\Xi_{cc}^{++} \rightarrow \Lambda_c^+ K^- \pi^+ \pi^+$ by the LHCb experiment [112] (see Figure 5). Its mass was determined as (3621.40 $\pm$ 0.72$^{+0.024}_{-0.27}$ $\pm$ 0.14) MeV [112], and its lifetime was measured to be (0.256 $\pm$ 0.022 $\pm$ 0.014) ps [132]. Since the first observation, $\Xi_{cc}^{++}$ was also observed in the final state of $\Xi_{cc}^{++} \rightarrow \Xi_c^+ \pi^+$ [133]. The production cross-section of $\Xi_{cc}^{++}$ was determined relative to that of $\Lambda_c$ baryons to be (2.22 $\pm$ 0.27 $\pm$ 0.29) $\times 10^{-4}$ [134]. Currently, other decay modes of $\Xi_{cc}^{++}$ are investigated, and the searches for $\Xi_{cc}^+$ (to confirm the SELEX result), and $\Omega_{cc}^+$ are ongoing.

Measurements of lifetimes play an important role in validating effective models such as HQET and can be used to search for deviations from the SM predictions. Recently, the LHCb experiment reported the most precise measurements of the lifetimes of the charm baryons $\Omega_c$, $\Lambda_c^+$, $\Xi_c^+$ and $\Xi_c^0$ [135, 136]. While the last three agree with previous measurements, the lifetime of $\Omega_c^0$ is about four times larger (see Figure 6). It has been argued that the expected lifetime hierarchy, due to the higher-order contributions discussed above, should be $\tau(\Xi_c^+) > \tau(\Lambda_c^+) > \tau(\Xi_c^0) > \tau(\Omega_c^0)$ [113, 137–141]. The current best measurements are inconsistent with this hierarchy. Possible interpretations of this deviation include constructive interference between the s quark in the $c \rightarrow sW^+$ transition in the $\Omega_c^0$ decay and the spectator s quark in the final state being smaller than expected, that the spin of the ss system plays a larger role, or that additional higher-order contributions in the HQET need to be considered. However, according to [138], $\Omega_c^0$ can be either the most short-living or the most long-living among charmed baryons. For doubly charmed
<figure><img src="image_8.png" /><figcaption>**Figure 6.** The tension in the charmed baryon lifetimes measurements [136].</figcaption></figure>

baryons, the expected hierarchy is $\tau(\Xi_{cc}^{++}) \gg \tau(\Omega_{cc}^{++}) \approx \tau(\Xi_{cc}^{++})$ [137] which is why $\Omega_{cc}^{++}$ and $\Xi_{cc}^{++}$ are more difficult to discover at LHCb.

## 9. Summary

The investigation of the $\tau$ lepton properties has provided many beautiful tests of the SM and strong constraints on new physics scenarios. Belle II will significantly improve the current sensitivity to LFV, LNV and CP-violating phenomena in $\tau$ decays. Moreover, its huge data sample should allow for a more accurate scrutiny of SM properties, such as lepton universality, Lorentz structure of the charged-current interaction, quark mixing and QCD in the non-perturbative regime. Meanwhile, the $\tau$ has also become a superb experimental tool in the search for new physics at the LHC. At long term, the TeraZ option of a future FCC-ee collider running at the $Z^0$ peak would produce an enormous data sample of $1.7 \times 10^{11} \tau^+\tau^-$ pairs in extremely clean kinematic (and background) conditions [142], opening a broad range of interesting opportunities in $\tau$ physics.

Charm physics covers a vast range of studies. In the past decade charm mixing and direct CP violation have been discovered. Intriguingly narrow excited $\Omega_c$ states have been seen and the doubly charmed baryons $\Xi_{cc}^{++}$ have been observed. The rarest charm decay's branching fraction measured is of the order $10^{-7}$. The LHCb experiment is currently undergoing its first major Upgrade. Several components of the detector will be replaced with new ones able to withstand the much higher rates and radiation doses in Run 3 of the LHC. A total of 300 fb$^{-1}$ of data is planned to be recorded. Belle II is currently taking data and has planned to collect a total of 50 ab$^{-1}$ of data. The BESIII experiment will increase its charm data samples at least by a factor 3. The longer term future of charm physics will be shaped by these three experiments and a possible Super Tau Charm factory.

## Acknowledgements

This work was supported by the Royal Society, UK [grant DH160214], by the Spanish Government and ERDF funds from the EU Commission [grant FPA2017-84445-P], and by the Generalitat Valenciana [grant Prometeo/2017/053].

## References

[1] A. Pich, "Precision tau physics", Prog. Part. Nucl. Phys. 75 (2014), p. 41-85.

[2] M. Tanabashi et al., "Review of particle physics", Phys. Rev. D 98 (2018), no. 3, article ID 030001.

[3] R. Aaij et al., "Measurements of prompt charm production cross-sections in pp collisions at $\sqrt{s}$ = 13 TeV", J. High Energy Phys. 2016 (2016), article ID 159.
[4] A. Pich, "Anomalous' $\eta$ production in tau decay", Phys. Lett. B 196 (1987), p. 561-565.

[5] R. Escribano, S. Gonzalez-Solis, P. Roig, "Predictions on the second-class current decays $\tau^- \to \pi^- \eta^{(l)} v_\tau$", Phys. Rev. D 94 (2016), no. 3, article ID 034008.

[6] A. J. Bevan et al., "The physics of the B factories", Eur. Phys. J. C 74 (2014), article ID 3026.

[7] E. Braaten, S. Narison, A. Pich, "QCD analysis of the tau hadronic width", Nucl. Phys. B 373 (1992), p. 581-612.

[8] W. J. Marciano, A. Sirlin, "Electroweak radiative corrections to tau decay", Phys. Rev. Lett. 61 (1988), p. 1815-1818.

[9] E. Braaten, C.-S. Li, "Electroweak radiative corrections to the semihadronic decay rate of the tau lepton", Phys. Rev. D 42 (1990), p. 3888-3891.

[10] J. Erler, "Electroweak radiative corrections to semileptonic tau decays", Rev. Mex. Fis. 50 (2004), p. 200-202.

[11] F. Le Diberder, A. Pich, "Testing QCD with tau decays", Phys. Lett. B 289 (1992), p. 165-175.

[12] D. Buskulic et al., "Measurement of the strong coupling constant using tau decays", Phys. Lett. B 307 (1993), p. 209-220.

[13] R. Barate et al., "Measurement of the spectral functions of axial-vector hadronic tau decays and determination of $\alpha_s(M_\tau^2)$", Eur. Phys. J. C 4 (1998), p. 409-431.

[14] S. Schael et al., "Branching ratios and spectral functions of tau decays: final ALEPH measurements and physics implications", Phys. Rep. 421 (2005), p. 191-284.

[15] M. Davier, A. Hocker, Z. Zhang, "The Physics of hadronic tau decays", Rev. Mod. Phys. 78 (2006), p. 1043-1109.

[16] M. Davier, S. Descotes-Genon, A. Hocker, B. Malaescu, Z. Zhang, "The determination of $\alpha_s$ from tau decays revisited", Eur. Phys. J. C 56 (2008), p. 305-322.

[17] T. Coan et al., "Measurement of $\alpha_s$ from tau decays", Phys. Lett. B 356 (1995), p. 580-588.

[18] K. Ackerstaff et al., "Measurement of the strong coupling constant $\alpha_s$ and the vector and axial vector spectral functions in hadronic tau decays", Eur. Phys. J. C 7 (1999), p. 571-593.

[19] P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn, "Order $\alpha_s^4$ QCD corrections to Z and tau decays", Phys. Rev. Lett. 101 (2008), article ID 012002.

[20] S. Narison, A. Pich, "QCD formulation of the tau decay and determination of $\Lambda_{\text{MS}}$, Phys. Lett. B 211 (1988), p. 183-188.

[21] E. Braaten, "The perturbative QCD corrections to the ratio R for tau decay", Phys. Rev. D 39 (1989), article ID 1458.

[22] F. Le Diberder, A. Pich, "The perturbative QCD prediction to $R_\tau$ revisited", Phys. Lett. B 286 (1992), p. 147-152.

[23] A. Pich, "Tau-decay determination of the strong coupling", SciPost Phys. Proc. 1 (2019), article ID 036.

[24] M. Davier, A. Höcker, B. Malaescu, C.-Z. Yuan, Z. Zhang, "Update of the ALEPH non-strange spectral functions from hadronic $\tau$ decays", Eur. Phys. J. C 74 (2014), no. 3, article ID 2803.

[25] A. Pich, A. Rodríguez-Sánchez, "Determination of the QCD coupling from ALEPH $\tau$ decay data", Phys. Rev. D 94 (2016), no. 3, article ID 034027.

[26] M. Baak, J. Cúth, J. Haller, A. Hoecker, R. Kogler, K. Mönig, M. Schott, J. Stelzer, "The global electroweak fit at NNLO and prospects for the LHC and ILC", Eur. Phys. J. C 74 (2014), article ID 3046.

[27] E. Gamiz, M. Jamin, A. Pich, J. Prades, F. Schwab, "$V_{us}$ and $m_s$ from hadronic tau decays", Phys. Rev. Lett. 94 (2005), article ID 011803.

[28] E. Gamiz, M. Jamin, A. Pich, J. Prades, F. Schwab, "Determination of $m_s$ and $|V_{us}|$ from hadronic tau decays", J. High Energy Phys. 01 (2003), article ID 60.

[29] E. Gamiz, M. Jamin, A. Pich, J. Prades, F. Schwab, "$|V_{us}|$ and $m_s$ from hadronic tau decays", Nucl. Phys. Proc. Suppl. 169 (2007), p. 85-89.

[30] A. Pich, J. Prades, "Strange quark mass determination from Cabibbo suppressed tau decays", J. High Energy Phys. 10 (1999), article ID 4.

[31] A. Pich, J. Prades, "Perturbative quark mass corrections to the tau hadronic width", J. High Energy Phys. 06 (1998), article ID 13.

[32] Y. S. Amhis et al., "Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2018", 2019, preprint, https://arxiv.org/abs/1909.12524.

[33] W. Altmannshofer et al., "The Belle II physics book", PTEP 2019 (2019), no. 12, article ID 123C01.

[34] Y. Miyazaki et al., "Search for lepton-flavor-violating and lepton-number-violating $\tau \to \ell h\bar{h}'$ decay modes", Phys. Lett. B 719 (2013), p. 346-353.

[35] Y. Miyazaki et al., "Search for lepton and baryon number violating tau-decays into $\bar{\Lambda}\pi^-$ and $\Lambda\pi^-$", Phys. Lett. B 632 (2006), p. 51-57.

[36] A. D. Sakharov, "Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe", Pisma Zh. Eksp. Teor. Fiz. 5 (1967), p. 32-35, Usp. Fiz. Nauk 161 (1991), no. 5, p. 61.

[37] R. Aaij et al., "Observation of CP violation in charm decays", Phys. Rev. Lett. 122 (2019), article ID 211803.

[38] M. Chala, A. Lenz, A. V. Rusov, J. Scholtz, "$\Delta A_{CP}$ within the Standard Model and beyond", J. High Energy Phys. 07 (2019), article ID 161.

[39] H.-N. Li, C.-D. Lü, F.-S. Yu, "Implications on the first observation of charm CPV at LHCb", 2019, preprint, https://arxiv.org/abs/1903.10638.
[40] Y. Grossman, S. Schacht, "The emergence of the $\Delta U = 0$ rule in charm physics", J. High Energy Phys. 07 (2019), article ID 20.

[41] A. Soni, "Resonance enhancement of Charm CP", 2019, preprint, https://arxiv.org/abs/1905.00907.

[42] H.-Y. Cheng, C.-W. Chiang, "Revisiting CP violation in $D \to PP$ and VP decays", Phys. Rev. D 100 (2019), no. 9, article ID 093002.

[43] R. Aaij et al., "Measurement of the mass difference between neutral charm-meson eigenstates", Phys. Rev. Lett. 122 (2019), article ID 231802.

[44] F. Buccella, A. Paul, P. Santorelli, "$SU(3)_F$ breaking through final state interactions and CP asymmetries in $D \to PP$ decays", Phys. Rev. D 99 (2019), no. 11, article ID 113001.

[45] U. Nierste, S. Schacht, "Neutral $D \to KK^*$ decays as discovery channels for charm CP violation", Phys. Rev. Lett. 119 (2017), no. 25, article ID 251801.

[46] R. Aaij et al., "Updated measurement of decay-time-dependent CP asymmetries in $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decays", Phys. Rev. D 101 (2020), no. 1, article ID 012005.

[47] B. Aubert et al., "Evidence for $D^0 - \bar{D}^0$ mixing", Phys. Rev. Lett. 98 (2007), article ID 211802.

[48] M. Starič et al., "Evidence for $D^0 - \bar{D}^0$ mixing", Phys. Rev. Lett. 98 (2007), no. 21, article ID 211803, [65 (2007)].

[49] T. Aaltonen et al., "Evidence for $D^0 - \bar{D}^0$ mixing using the CDF II detector", Phys. Rev. Lett. 100 (2008), article ID 121802.

[50] B. Aubert et al., "Measurement of $D^0 - \bar{D}^0$ mixing from a time-dependent amplitude analysis of $D^0 \to K^+ \pi^- \pi^0$ decays", Phys. Rev. Lett. 103 (2009), article ID 211801.

[51] R. Aaij et al., "Observation of $D^0 - \bar{D}^0$ oscillations", Phys. Rev. Lett. 110 (2013), no. 10, article ID 101802.

[52] R. Aaij et al., "Updated determination of $D^0 - \bar{D}^0$ mixing and CP violation parameters with $D^0 \to K^+ \pi^-$ decays", Phys. Rev. D 97 (2018), no. 3, article ID 031101.

[53] R. Aaij et al., "First observation of $D^0 - \bar{D}^0$ oscillations in $D^0 \to K^+ \pi^- \pi^+ \pi^-$ decays and measurement of the associated coherence parameters", Phys. Rev. Lett. 116 (2016), no. 24, article ID 241801.

[54] D. M. Asner et al., "Search for $D^0 - \bar{D}^0$ mixing in the Dalitz plot analysis of $D^0 \to K^0_{S(\bar{S})} \pi^+ \pi^-$", Phys. Rev. D 72 (2005), article ID 012001.

[55] T. Peng et al., "Measurement of $D^0 - \bar{D}^0$ mixing and search for indirect CP violation using $D^0 - K^0_S \pi^+ \pi^-$ decays", Phys. Rev. D 89 (2014), no. 9, article ID 091103.

[56] P. del Amo Sanchez et al., "Measurement of $D^0 - \bar{D}^0$ mixing parameters using $D^0 \to K^0_{S(\bar{S})} \pi^+ \pi^-$ and $D^0 \to K^0_{S(\bar{S})} K^+ K^-$ decays", Phys. Rev. Lett. 105 (2010), article ID 081803.

[57] R. Aaij et al., "Model-independent measurement of mixing parameters in $D^0 \to K^0_S \pi^+ \pi^-$ decays", J. High Energy Phys. 04 (2016), article ID 33.

[58] R. Aaij et al., "Measurement of the mass difference between neutral charm-meson eigenstates", Phys. Rev. Lett. 122 (2019), no. 23, article ID 231802.

[59] J. Libby et al., "Model-independent determination of the strong-phase difference between $D^0$ and $\bar{D}^0 \to K^0_{S,L} h^+ h^-$ ($h = \pi, K$) and its impact on the measurement of the CKM angle $\gamma/\phi_3$", Phys. Rev. D 82 (2010), article ID 112006.

[60] M. Ablikim et al., "Model-independent determination of the relative strong-phase difference between $D^0$ and $\bar{D}^0 \to K^0_{S,L} \pi^+ \pi^-$ and its impact on the measurement of the CKM angle $\gamma/\phi_3$", Phys. Rev. D 101 (2020), no. 11, article ID 112002.

[61] M. Ablikim et al., "Determination of strong-phase parameters in $D \to K^0_{S,L} \pi^+ \pi^-$", Phys. Rev. Lett. 124 (2020), no. 24, article ID 241802.

[62] R. Aaij et al., "Physics case for an LHCb Upgrade II - Opportunities in flavour physics, and beyond, in the HL-LHC era", 2018, preprint, https://arxiv.org/abs/1808.08865.

[63] A. A. Petrov, "Charm physics: theoretical review", eConf C030603 (2003), MEC05, [189 (2003)].

[64] M. Ablikim et al., "Precision measurements of $B(D^+ \to \mu^+ \nu_\mu)$, the pseudoscalar decay constant $f_{D^+}$, and the quark mixing matrix element $|V_{cd}|$, Phys. Rev. D 89 (2014), no. 5, article ID 051104.

[65] J. Beringer et al., "Review of Particle Physics (RPP)", Phys. Rev. D 86 (2012), article ID 010001.

[66] M. Ablikim et al., "Observation of the leptonic decay $D^+ \to \tau^+ \nu_\tau$", Phys. Rev. Lett. 123 (2019), no. 21, article ID 211802.

[67] M. Ablikim et al., "Determination of the pseudoscalar decay constant $f_{D_S^+}$ via $D_S^+ \to \mu^+ \nu_\mu$", Phys. Rev. Lett. 122 (2019), no. 7, article ID 071802.

[68] M. Ablikim et al., "Measurement of the $D_S^+ \to \ell^+ \nu_\ell$ branching fractions and the decay constant $f_{D_S^+}$", Phys. Rev. D 94 (2016), no. 7, article ID 072004.

[69] J. P. Alexander et al., "Measurement of $B(D_S^+ \to \ell^+ \nu)$ and the decay constant $f_{D_S^+}$ From 600 pb$^{-1}$ of $e^\pm$ Annihilation Data Near 4170 MeV", Phys. Rev. D 79 (2009), article ID 052001.

[70] A. Zupanc et al., "Measurements of branching fractions of leptonic and hadronic $D_S^+$ meson decays and extraction of the $D_S^+ \to K^0_S \pi^+ \pi^-$ meson decay constant", J. High Energy Phys. 09 (2013), article ID 139.

[71] P. del Amo Sanchez et al., "Measurement of the absolute branching fractions for $D_S^- \to \ell^- \bar{\nu}_\ell$ and extraction of the decay constant $f_{D_S^-}$", Phys. Rev. D 82 (2010), article ID 091103, [Erratum: Phys. Rev. D 91, (2015), no. 1, 019901].
[72] S. Aoki et al., "FLAG review 2019", Eur. Phys. J. C 80 (2020), no. 2, article ID 113.

[73] M. Ablikim et al., "Future physics programme of BESIII", Chin. Phys. C 44 (2020), no. 4, article ID 040001.

[74] A. J. Schwartz, "Semileptonic and leptonic charm meson decays at Belle II", 2019, preprint, https://arxiv.org/abs/1902.07850.

[75] H. Na, C. T. H. Davies, E. Follana, J. Koponen, G. P. Lepage, J. Shigemitsu, "D→π,ℓν semileptonic decays, $|V_{cd}|$ and $2^{nd}$ row unitarity from lattice QCD", Phys. Rev. D 84 (2011), article ID 114505.

[76] R. Li et al., "D meson semileptonic decay form factors at $q^2 = 0$", PoS LATTICE2018 (2019), article ID 269.

[77] V. Lubicz, L. Riggio, G. Salerno, S. Simula, C. Tarantino, "Scalar and vector form factors of $D→π(K)ℓν$ decays with $N_f = 2 + 1 + 1$ twisted fermions", Phys. Rev. D 96 (2017), no. 5, article ID 054514, [Erratum: Phys. Rev. D 99, (2019), no. 9, 099902].

[78] J. P. Lees et al., "Measurement of the $D^0→π^-e^+ν_e$ differential decay branching fraction as a function of $q^2$ and study of form factor parameterizations", Phys. Rev. D 91 (2015), no. 5, article ID 052022.

[79] M. Ablikim et al., "Study of dynamics of $D^0→K^-e^+ν_e$ and $D^0→π^-e^+ν_e$ decays", Phys. Rev. D 92 (2015), no. 7, article ID 072012.

[80] P. Ball, "Testing QCD sum rules on the light-cone in $D→(π,K)ℓν$ decays", Phys. Lett. B 641 (2006), p. 50-56.

[81] M. Ablikim et al., "Improved measurement of the absolute branching fraction of $D^+→\bar{K}^0μ^+ν_μ$", Eur. Phys. J. C 76 (2016), no. 7, article ID 369.

[82] J. G. Korner, G. A. Schuler, "Exclusive semileptonic heavy meson decays including lepton mass effects", Z. Phys. C 46 (1990), article ID 93.

[83] N. R. Soni, J. N. Pandya, "Decay $D→K^+(s)ℓ^+ν_ℓ$ in covariant quark model", Phys. Rev. D 96 (2017), no. 1, article ID 016017, [Erratum: Phys. Rev. D 99 (2019), no. 5, 059901].

[84] L. Riggio, G. Salerno, S. Simula, "Extraction of $|V_{cd}|$ and $|V_{cs}|$ from experimental decay rates using lattice QCD $D→π(K)ℓν$ form factors", Eur. Phys. J. C 78 (2018), no. 6, article ID 501.

[85] R. Aaij et al., "Search for the rare decay $D^0→μ^+μ^−$", Phys. Lett. B 725 (2013), no. 1-3, p. 15-24.

[86] R. Aaij et al., "Search for the lepton-flavour violating decay $D^0→e^+μ^∓$", Phys. Lett. B 754 (2016), p. 167-175.

[87] R. Aaij et al., "Search for $D^+_s→π^+μ^+μ^−$ and $D^+_s→π^−μ^+μ^+$ decays", Phys. Lett. B 724 (2013), no. 4-5, p. 203-212.

[88] P. Rubin et al., "Search for $D^0→\bar{p}e^+$ and $D^0→pe^−$", Phys. Rev. D 79 (2009), article ID 097101.

[89] R. Aaij et al., "Observation of $D^0$ meson decays to $π^+π^−μ^+μ^−$ and $K^+K^−μ^+μ^−$ final states", Phys. Rev. Lett. 119 (2017), article ID 181805.

[90] L. Cappiello, O. Cata, G. D'Ambrosio, "Standard Model prediction and new physics tests for $D^0→h^+h^−ℓ^+ℓ^−(h=π, K: ℓ = e, μ)$", J. High Energy Phys. 04 (2013), article ID 135.

[91] S. De Boer, G. Hiller, "Null tests from angular distributions in $D→P_1 P_2 ℓ^+ℓ^−$, $ℓ = e, μ$ decays on and off peak", Phys. Rev. D 98 (2018), no. 3, article ID 035041.

[92] R. Aaij et al., "Measurement of angular and CP asymmetries in $D^0→π^+π^−μ^+μ^−$ and $D^0→K^+K^−μ^+μ^−$ decays", Phys. Rev. Lett. 121 (2018), article ID 091801.

[93] S. Fajfer, S. Prelovsek, "Effects of littlest Higgs model in rare D meson decays", Phys. Rev. D 73 (2006), article ID 054026.

[94] I. I. Bigi, A. Paul, "On CP asymmetries in two-, three- and four-body D decays", J. High Energy Phys. 03 (2012), article ID 21.

[95] A. Paul, A. De La Puente, I. I. Bigi, "Manifestations of warped extra dimension in rare charm decays and asymmetries", Phys. Rev. D 90 (2014), no. 1, article ID 014035.

[96] S. Fajfer, N. Košnik, "Prospects of discovering new physics in rare charm decays", Eur. Phys. J. C 75 (2015), no. 12, article ID 567.

[97] S. de Boer, G. Hiller, "Flavor and new physics opportunities with rare charm decays into leptons", Phys. Rev. D 93 (2016), no. 7, article ID 074001.

[98] S. Godfrey, K. Moats, "Properties of excited charm and charm-strange mesons", Phys. Rev. D 93 (2016), no. 3, article ID 034035.

[99] J. P. Lees et al., "Dalitz plot analyses of $B^0→D^−D^0K^+$ and $B^+→\bar{D}^0D^0K^+$ decays", Phys. Rev. D 91 (2015), no. 5, article ID 052002.

[100] J. Brodzicka et al., "Observation of a new D(s)J meson in $B^+→\bar{D}^0D^0K^+$ decays", Phys. Rev. Lett. 100 (2008), article ID 092001.

[101] B. Aubert et al., "Study of D(s)J decays to D*K in inclusive $e^+e^−$ interactions", Phys. Rev. D 80 (2009), article ID 092003.

[102] B. Aubert et al., "Observation of a new $D_s$ meson decaying to DK at a mass of 2.86 GeV/c²", Phys. Rev. Lett. 97 (2006), article ID 222001.

[103] R. Aaij et al., "Study of $D_{sJ}^+$ mesons decaying to $D^{*+}K_S^0$ and $D^{*0}K^+$ final states", J. High Energy Phys. 02 (2016), article ID 133.

[104] R. Aaij et al., "Determination of quantum numbers for several excited charmed mesons observed in $B^−→D^{*+}π^−π^−$ decays", Phys. Rev. D 101 (2020), no. 3, article ID 032005.
[141] M. A. Shifman, M. B. Voloshin, "Preasymptotic effects in inclusive weak decays of charmed particles", *Sov. J. Nucl. Phys.* **41** (1985), 120, [Yad. Fiz. **41** (1985), 187].

[142] A. Abada et al., "FCC physics opportunities", *Eur. Phys. J. C* **79** (2019), no. 6, article ID 474.