The recent measurement of the tau -> Ks pion nu_tau spectrum by the Belle Collaboration provides the frame to set forth a theoretical description of the decay, which is based on the contributing vector F_+(s) and scalar F_0(s) form factors. We show that a good representation of data is obtained through the use of form factors calculated within resonance chiral theory and constrained by dispersion relations and short-distance QCD. Hence we obtain a determination of K*(892) parameters and the low-energy parameterization of F_+(s).
In this work, we study the lepton number violating tau decays via two intermediate on-shell Majorana neutrinos $N_j$ into two charged pions, and a charged lepton $τ^{\pm} \to π^{\pm} N_j \to π^{\pm} π^{\pm} \ell^{\mp}$. We consider the scenario where the heavy neutrino masses are within $0.5$ GeV $\leq M_N \leq 1.5$ GeV. We evaluated the possibility to measure the modulation of the decay width along the detector length for these processes at taus factories, such as Belle II. We study some realistic conditions which could lead to the observation of this phenomenon at futures $τ$'s factories.
We report the recent results of a search for lepton-flavor-violating tau decays and a search for CP violation in tau to nu Ks pi using a large data sample accumulated with the Belle detector at the KEKB asymmetric-energy e^+e^- collider. The sensitivity to these modes is significantly improved compared to previous experiments.
We report on the recent results of searches for lepton-flavor-violating tau decays as well as lepton-number-violating B decays with the world-highest data samples accumulated at the Belle.
Searches for lepton flavor violation in tau decays are unambiguous signatures of new physics. The branching ratios of tau leptons at the level of 10^-10 - 10^-9 can be probed with 50 ab^-1 of electron-positron annihilation data being collected by the Belle II experiment at the world's highest luminosity accelerator, the SuperKEKB, located at the High Energy Accelerator Research Organization, KEK, in Tsukuba, Japan. Searches with such expected sensitivity will either discover new physics or strongly constrain several new physics models.
The predictions of an isospin model are compared with the branching ratios of the 5 and 6 pion decays of the tau lepton. In both cases, the isospin model suggests that the tau favours decays in which there is an omega resonance. Recent measurements of such tau decays confirm this hypothesis. If the decay of the tau to 7 pions also proceeds through an intermediate omega, then the isospin model predicts that the branching ratio of the tau to seven charged pions should be small when compared with other 7 pion decays. New limits on this mode appear to support this argument.
A theoretical description of the differential decay spectrum for the decay tau- -> nu_tau K_S pi-, which is based on the contributing K pi vector and scalar form factors F_+^{K pi}(s) and F_0^{K pi}(s) being calculated in the framework of resonance chiral theory (R$χ$T), additionally imposing constraints from dispersion relations as well as short distance QCD, provides a good representation of a recent measurement of the spectrum by the Belle collaboration. Our fit allows to deduce the total branching fraction B[tau- -> nu_tau K_S pi-] = 0.427 +- 0.024 % by integrating the spectrum, as well as…
Within the assumption of Left-Handed (LH) New Physics (NP), we review the relations between $\mathcal{B}(B\to K^{(\ast)} τ^+τ^-)$ and $\mathcal{B}(B\to K^{(\ast)} ν\bar ν)$ for several Beyond the Standard Model (BSM) scenarios, commonly considered to explain the Lepton flavor Universality (LFU) violation observed in charged and neutral-current semileptonic $B$ decays. We employ the latest $R_{D^{(\ast)}}$ world averages that include the recent LHCb measurement and assess the possibility of simultaneously explaining the $B$-anomalies without spoiling current bounds on di-neutrino and di-tau mod…
We analyse tau -> (3 pions) nu_tau decays within the framework of the resonance chiral theory of QCD. We have worked out the relevant Lagrangian that describes the axial-vector current hadronization contributing to these processes, and the new coupling constants that arise have been constrained by imposing the asymptotic behaviour of the corresponding spectral function within QCD. Hence we compare the theoretical framework with the experimental data, obtaining a good quality fit from the ALEPH spectral function and branching ratio. We also get values for the mass and on-shell width of the a_1(…
The pion-photon transition form factor (TFF) provides strong constraints on the pion distribution amplitude (DA). We perform an analysis of all existing data (CELLO, CLEO, BaBar, Belle) on the pion-photon TFF by means of light-cone pQCD approach in which we include the next-to-leading order correction to the valence-quark contribution and estimate the non-valence-quark contribution by a phenomenological model based on the TFF's limiting behavior at both $Q^2\to 0$ and $Q^2\to\infty$. At present, the pion DA is not definitely determined, it is helpful to have a pion DA model that can mimic all …
Tau decays into hadrons foresee the study of the hadronization of vector and axial-vector QCD currents, yielding relevant information on the dynamics of the resonances entering into the processes. We analyse tau -> (3 pion) nu_tau decays within the framework of the Resonance Chiral Theory, comparing this theoretical scheme with the experimental data, namely ALEPH spectral function and branching ratio. Hence we get values for the mass and on-shell width of the a_1(1260) resonance, and provide the structure functions that have been measured by OPAL and CLEO-II.
It is noted that the low-energy behavior of the pion-photon transition form factor $F_{πγ}(Q^2)$ is sensitive to the transverse distribution of the pion wavefunction, and its high-energy behavior is sensitive to the longitudinal one. Thus a careful study on $F_{πγ}(Q^2)$ can provide helpful information on the pion wavefunction precisely. In this paper, we present a combined analysis of the data on $F_{πγ}(Q^2)$ reported by the CELLO, the CLEO, the BABAR and the BELLE collaborations. It is performed by using the method of least squares. By using the combined measurements of BELLE and CLEO Colla…
We extract constraints on the pion distribution amplitude from available data on the pion-photon transition form factor in the framework of light-cone sum rules. A pronounced discrepancy $(2.7-3)σ$ between the Gegenbauer expansion coefficients extracted from the CELLO, CLEO, and Belle experimental data relative to those from BaBar is found. Predictions for the pion-photon transition form factor are presented by employing a pion distribution amplitude obtained long ago from QCD sum rules with nonlocal condensates. These predictions comply with the Belle data but disagree with those of BaBar bey…
We study the expected sensitivity at Belle and Belle II for four-body $τ^\mp \to X^\pm l^\mp l^\mp ν_τ$ decays where $l=e$ or $μ$ and $X=π$, $K$, $ρ$ and $K^*$ mesons. These decay processes violate the total lepton number ($|ΔL|=2$ ) and they can be induced by the exchange of Majorana neutrinos. In particular, we consider lifetimes in the accessible ranges of $τ_N$ = 5, 100 ps and extract the limits on $|V_{\ell N}|^2$ without any additional assumption on the relative size of the mixing matrix elements. For an integrated luminosity collected of 1 ab$^{-1}$ at Belle, we found significant sensit…
We compute branching ratios and invariant mass distributions of the tau decays into four pions. The hadronic matrix elements are obtained by starting from the structure of the hadronic current in chiral limit and then implementing low-lying resonances in the different channels. Reasonable agreement with experiment is obtained both for the $τ\to ν_τ + (4 π) $ decay rates and the $e^+e^- \to (4 π) $ cross sections. Furthermore we supply an interface to use our matrix elements within the Tauola Monte-Carlo program.
Recent experimental data on tau decays are used to reconstruct the difference in hadronic spectral densities with vector and axial-vector quantum numbers. The saturation of Das-Mathur-Okubo and Weinberg sum rules is studied. Two methods of improving convergence and decreasing errors are applied, and good agreement with the predictions of current algebra and chiral perturbation theory is observed. The resulting value of the pion polarisability is (2.64 +/- 0.36) 10^-4 fm^3.
In this paper we calculate the power corrections to the pion transition form factor within the framework of perturbative QCD approach on the basis of $k_T$ factorization. The power suppressed contributions from higher twist pion wave functions and the hadronic structure of photon are investigated. We find that there exists strong cancellation between the two kinds contributions, thus the total power corrections considered currently are very small, and the prediction of the leading power contribution with joint resummation improved perturbative QCD approach is almost unchanged. This result conf…
The two-pion contribution to hadronic vacuum polarization can be extracted from tau decay data when isospin violating and radiative corrections are taken into account. When the dominant corrections are applied to the photon-inclusive decay of the tau into two pions, one obtains a shift (-12.0 +/- 2.6).10^{-10} for the anomalous magnetic moment of the muon. The shift appears to be too small to reconcile the determinations of hadronic vacuum polarization from existing tau and e+ e- data. The reliability of electromagnetic corrections in the photon-inclusive tau decay is examined.
Based on the S-matrix approach, we introduce a modified formula for the $π^{\pm}$ electromagnetic form factor which describes very well the experimental data in the energy region $2m_π \leq \sqrt{s} \leq 1.1$ GeV. Using the CVC hypothesis we predict $B(\tpp) = (24.75 \pm 0.38)\% $, in excellent agreement with recent experiments.
The phenomenon of neutrino flavor oscillations motivates searches for sterile neutrinos in a broad range of masses and mixing-parameter values. A sterile neutrino $N$ that mixes predominantly with the $τ$ neutrino is particularly challenging experimentally. To address this challenge, we propose a new method to search for a $ν_τ$-mixing with $N$ lighter than the $τ$ lepton. The method uses the large $e^+e^-\toτ^+τ^-$ samples collected at $B$-factory experiments to produce the $N$ in $τ$-lepton decays. We exploit the long lifetime of a sterile neutrino in this mass range to suppress background a…
We evaluate the model-dependent piece of O(alpha) long-distance radiative corrections to tau^- \to π^- π^0ν_τ decays by using a meson dominance model. We find that these corrections to the di-pion invariant mass spectrum are smaller than in previous calculations based on chiral perturbation theory. The corresponding correction to the photon inclusive rate is tiny (-0.15%) but it can be of relevance when new measurements reach better precision.
We study the effect of rho^0-gamma mixing in e^+e^- to pi^+pi^- and its relevance for the comparison of the square modulus of the pion from-factor |F^(e)_pi|^2, as measured in e^+e^- annihilation experiments, and |F^(tau)_pi|^2 the corresponding quantity obtained after accounting for known isospin breaking effects by an isospin rotation from the tau-decay spectra. After correcting the tau data for the missing rho-gamma mixing contribution, besides the other known isospin symmetry violating corrections, the pi pi I=1 part of the hadronic vacuum polarization contribution to the muon g-2 are full…
Present experimental datas have shown a 3.8$σ$ level discrepancy with the standard model in $\overline{B}\to D^{(*)}τ\barν_τ$. Some new physics models have been considered to explain this discrepancy possibly with new source of the CP violation. In this paper, we construct CP violating observables by using multi-pion decays in $B\to Dτν_τ$, and estimate sensitivity of these observables to generic CP violating operators. We also discuss possibilities of CP violation in leptoquark models and in 2HDM of type-III.
Tau decays into hadrons have a twofold interest: On the one hand, they are a clean environment for studying the hadronization of the left-handed current of QCD, while, on the other side, provide relevant dynamical information of the resonances that mediate these processes. Within an effective field theory-like framework, namely Resonance Chiral Theory, we analyse the decays ot the tau into K K pion modes and compare the results with CLEO and BaBar data. In this way, we provide bounds on the couplings entering our Lagrangian and predict the corresponding spectral functions. As a main result -an…
We consider the bremsstrahlung and model-dependent contributions to the radiative decay $τ^- \to π^-π^0ν_τγ$ in the context of a meson dominance model. We focus on several observables related to this decay, including the branching ratio and the photon and di-pion spectra. Particular attention is paid to the sensitivity of different observables upon the effects of model-dependent contributions and of the magnetic dipole moment of the $ρ^-(770)$ vector meson. Important numerical differences are found with respect to results obtained in the framework of chiral perturbation theory.
We study possible so far unaccounted isospin breaking effects in the relation between the pion form factor as determined in e^+e^- experiments and the corresponding quantity obtained after accounting for known isospin breaking effects by an isospin rotation from the tau--decay spectra. In fact the observed 10% discrepancy in the respective pion form factors may be explained by the isospin breaking which is due to the difference between masses and widths of the charged and neutral rho mesons. Since the hadronic contribution to the muon anomalous magnetic moment can be calculated directly in ter…
A state-of-the-art analysis of the pion-photon transition form factor is presented based on an improved theoretical calculation that includes the effect of a finite virtuality of the quasi-real photon in the method of light-cone sum rules. We carry out a detailed statistical analysis of the existing experimental data using this method and by employing pion distribution amplitudes with up to three Gegenbauer coefficients $a_2, a_4, a_6$. Allowing for an error range in the coefficient $a_6\approx 0$, the theoretical predictions for $γ^*γ\toπ^0$ obtained with nonlocal QCD sum rules are found to b…
In this work, we perform a detailed study of the $τ^-\toπ^-π^0γν$ decay process within the resonance chiral theory. We pay special attention to the triple-product asymmetry in the $τ^-\toπ^-π^0γν$ process. The minimal resonance chiral Lagrangian and the odd-intrinsic parity resonance operators are simultaneously included to calculate the decay amplitudes. Various invariant-mass distributions in the $π^-π^0$, $π^-γ$ and $π^0γ$ systems are studied and they reveal different resonance dynamics. We further predict the intriguing nonzero triple-product asymmetry distributions, which may provide usef…
We discuss the decay tau^- -> nu_tau pi^- pi^0 gamma in terms of a model with the correct low-energy structure and with the relevant resonance degress of freedom. The nontrivial radiative dynamics becomes visible for large photon momenta only. We use the model to calculate electromagnetic corrections for the two-pion contribution to hadronic vacuum polarization extracted from photon-inclusive two-pion decays. The corrections are insensitive to the details of the model and depend on the pion form factor only. Putting all relevant isospin violating corrections together, we obtain a shift Delta a…
It is believed that one can extract more accurate information of the pion distribution amplitude from the pion-photon transition form factor (TFF) due to the single pion in this process. However the BABAR and Belle data of the pion-photon TFF have a big difference for $Q^2\in [15,40]$ GeV$^2$, and at present, the pion DA can not be definitely determined from the pion-photon TFF. it is crucial to find the right pion DA behavior and to determine which data is more reliable. In this letter, we perform a combined analysis of the most existing data of the processes involving pion by using a general…
The contribution of the rho^{\pm} vector meson to the tau -> pi pi nu gamma decay is considered as a potential source for the determination of the magnetic dipole moment of this light vector meson. In order to keep gauge-invariance of the whole decay amplitude, a procedure similar to the fermion loop-scheme for charged gauge bosons is implemented to incorporate the finite width effects of the rho^{\pm} vector meson. The absorptive pieces of the one-loop corrections to the propagators and electromagnetic vertices of the rho^{\pm} meson and W^{\pm} gauge boson have identical forms in the limit o…
We review current issues in exclusive semileptonic tau decays. We present the formalism of structure functions, and then discuss predictions for final states with kaons, for decays into four pions and for radiative corrections to the decay into a single pion. Implications for testing the standard model at low energies are discussed.
We show that the a_1-rho-pi Lagrangian is a decisive element for obtaining a good phenomenological description of the three-pion decays of the tau lepton. We choose it in a two-component form with a flexible mixing parameter sin(theta). In addition to the dominant a_1-> pho pi intermediate states, the a_1->pi sigma ones are included. When fitting the three-pion mass spectra, three data sets are explored: (1) ALEPH 2005 pi-pi-pi+ data, (2) ALEPH 2005 pi-pi0pi0 data, and (3) previous two sets combined and supplemented with the ARGUS 1993, OPAL 1997, and CLEO 2000 data. The corresponding confiden…
We provide an overview of recent determinations of the strong coupling, $α_s$, from hadronic $τ$ decays. We contrast two analysis methods, the "truncated-OPE approach" and the "DV-model approach," highlighting the assumptions going into each of these. We argue that a detailed study based on ALEPH data shows the truncated-OPE approach to be quantitatively unreliable, while the DV-model approach passes all tests. New data for hadronic $τ$ decays from Belle and Belle-II could provide more stringent tests of the DV-model approach, and thus potentially lead to a more precise value of $α_s$ from had…
A quasi two-body QCD factorization is used to study the D0 --> K0S pi+ pi decays. The presently available high-statistics Dalitz plot data of this process measured by the Belle and BABAR Collaborations are analyzed together with the tau- --> K0S pi- nu_tau decay data. The total experimental branching fraction is also included in the fits which show a very good overall agreement with the experimental Dalitz plot density distributions. The branching fractions of the dominant channels compare well with those of the isobar Belle or BABAR models. We show that the branching fractions corresponding t…
An improved description of four-pion production in electron-positron annihilation and in tau lepton decays is presented. The model amplitude is fitted to recent data from BaBar which cover a wide energy range and which were obtained exploiting the radiative return. Predicting tau decay distributions from e+e- data and comparing these predictions with ALEPH and CLEO results, the validity of isospin symmetry is confirmed within the present experimental errors. A good description of two- and three-pion sub-distributions is obtained. Special emphasis is put on the predictions for omega pi (-> pi+p…
This article summarizes recent developments in $B\to D^{(\ast)}τν$ decays. We explain how to extract the tau lepton's production properties from the kinematics of its decay products. The focus is on hadronic tau decays, which are most sensitive to the tau polarizations. We present new results for effects of new physics in tau polarization observables and quantify the observation prospects at BELLE II.
tau -> (3 pions) nu_tau decays are analysed within the framework of the resonance effective theory of QCD. We work out the relevant Lagrangian that describes the axial-vector current hadronization contributing to these processes, in particular the local a_1(1260)-rho(770)-Goldstone interactions. The new coupling constants are constrained by imposing the asymptotic behaviour of the corresponding spectral function within QCD. Hence we compare the theoretical framework with the experimental data, obtaining a good quality fit from the ALEPH spectral function and branching ratio. We also get values…
We calculate the processes e+ e- -> 4 pions and tau -> nu_tau + 4 pions to O(p^4) in the low-energy expansion of the standard model. The chiral amplitudes of O(p^4) can be extended via resonance exchange to energies around 1 GeV. Higher-order effects have been included in the form of omega, a_1 and double rho exchange and by performing a resummation of the pion form factor. The predicted cross sections and the branching ratios BR(rho -> 4 pions) are in good agreement with available data.
In the two body hadronic tau decays, such as tau->to K pi (eta)nu, vector mesons play important role. Belle and Babar measured hadronic invariant mass spectrum of tau -> K pi nu decay. To compare the spectrum with theoretical prediction, we develop the chiral Lagrangian with vector mesons in Kimura:2012nx. We compute the form factors of the hadronic tau decay taking account of the quantum corrections of Nambu Goldstone bosons. We also show how to renormalizethe divergence of the Feynman diagrams with arbitrary number of loops and determine the counterterms within one loop using background fiel…
The role of the a_1(1420) resonance in the three-pion decays of the tau lepton is investigated using a phenomenological model. For all data before 2008, roughly equal fit quality is achieved when the basic a_1(1260) resonance is supplemented with either a_1(1640) or a_1(1420). However, two recent and more precise data sets require resonances with masses that are not very far from that of a_1(1420). This suggests that the axial-vector resonance that accompanies the a_1(1260) in the three-pion decays of the tauon is a_1(1420), not a_1(1640), as believed up to now. More data are needed to demonst…
We study a model-independent parameterization of the vector pion form factor that arises from the constraints of analyticity and unitarity. Our description should be suitable up to s^(1/2) ~ 1.2 GeV and allows a model-independent determination of the mass of the rho(770) resonance. We analyse the experimental data on tau^- -> pion^- pion^0 nu_tau and e^+ e^- -> pion^+ pion^- in this framework, and its consequences on the low-energy observables worked out by chiral perturbation theory. An evaluation of the two pion contribution to the anomalous magnetic moment of the muon, a_{mu}, and to the fi…
We reconsider Muskhelishvili-Omnès (MO) dispersive representations of photon-photon scattering to two pions, motivated by the very high statistics results recently released by the Belle collaboration for charged as well as neutral pion pairs and also by recent progress in the determination of the low-energy $ππ$ scattering amplitude. Applicability of this formalism is extended beyond 1 GeV by taking into account inelasticity due to $K\bar{K}$ . A modified MO representation is derived which has the advantage that all polynomial ambiguities are collected into the subtraction constants and have s…
The Monte Carlo for lepton pair production andtau decays consist of KKMC for lepton pair production, tauola for tau lepton decays and photos for radiative corrections in decays. An effort for adaptation of the system for precision data being collected at the Belle II experiment included simulation of additional light lepton pairs. Extension to processes where lepton pair is produced through narrow resonances, like dark photon or dark scalar phi resonances, was straight forward. Modified programs versions are available in stand-alone format from gitlab repository or through the basf2 system of …
We calculate the transition form factor of the neutral pion where one photon is virtual and another photon is real in the model where the light constituent quark mass and the quark-pion vertex are taken to be momentum independent. Radiative corrections to the lowest order triangle quark Feynman amplitude are calculated. The resummation of the lowest radiative corrections to the virtual photon vertex is done by applying the Sudakov exponential hypothesis. Using fitting parameters, the quark mass and the strong coupling constant, the results on the pion transition form factor are compared with e…
Status of tau lepton decay Monte Carlo generator TAUOLA is reviewed. Recent efforts on development of new hadronic currents are presented. Multitude new channels for anomalous tau decay modes and parametrization based on defaults used by BaBar collaboration are introduced. Also parametrization based on theoretical considerations are presented as an alternative. Lesson from comparison and fits to the BaBar and Belle data is recalled. It was found that as in the past, in particular at a time of comparisons with CLEO and ALEPH data, proper fitting, to as detailed as possible representation of the…
We extend our recent works on the two-pion $S$-wave resonance contributions to the kaon-pion ones in the $B$ meson hadronic charmonium decay modes based on the perturbative QCD approach. The $S$-wave $Kπ$ time-like form factor in its distribution amplitudes is described by the LASS parametrization, which consists of the $K^*_0(1430)$ resonant state together with an effective range nonresonant component. The predictions for the decays $B\rightarrow J/ψKπ$ in this work agree well with the experimental results from $BABAR$ and Belle Collaborations. We also discuss theoretical uncertainties, indic…
Belle Collaboration reported a new observed value of K*-(892) mass by studying tau --> K_S pi nu_tau decay, which is significantly different from the current world average value given by Particle Data Group 2006. Motivated by this new data, we revisit the issue on the K*0(892)-K*+(892) mass splitting. Our theoretical estimation favors the new measurement by Belle Collaboration. Therefore further experimental efforts are urgently needed to improve our understanding of these issues.
The study of the $τ$-lepton decays into hadrons has contributed to a better understanding of non-perturbative QCD and light-quark meson spectroscopy, as well as to the search of new physics beyond the Standard Model. The two- and three-meson decay modes, considering only those permitted by the Standard Model, are the predominant decays and together with the one-pion mode compose more than $85\%$ of the hadronic $τ$-lepton decay width. In this note we review the theoretical results for these modes implemented in the Monte Carlo event generator TAUOLA and present at the same time a comparison wi…
We accomplish the complete two-loop computation of the leading-twist contribution to the photon-pion transition form factor $γ\, γ^{\ast} \to π^0$ by applying the hard-collinear factorization theorem together with modern multi-loop techniques. The resulting predictions for the form factor indicate that the two-loop perturbative correction is numerically important. We also demonstrate that our results will play a key role in disentangling various models of the twist-two pion distribution amplitude thanks to the envisaged precision at Belle II.
Measurements of tau lepton, because of its long lifetime, large mass and parity sensitive couplings lead to broad physics interest. From the perspective of high-energy experiments such as at LHC, knowledge of tau lepton properties offers an important ingredient of new physics signatures. From the perspective of lower energies, tau lepton decays constitute an excellent laboratory for hadronic interactions. At present,hundreds of millions of tau decays have been amassed by both Belle and BaBar experiments. It is of utmost importance to represent such data in a form as useful for general applicat…
Recent progress in the determination of |V_us| employing strange hadronic tau-decay data are reported. This includes using the recent OPAL update of the strange spectral function, as well as augmenting the dimension-two perturbative contribution with the recently calculated order alpha_s^3 term on the theory side. These updates result in |V_us| = 0.2220 +- 0.0033, with the uncertainty presently being dominated by experiment, and already being competitive with the standard extraction from K_e3 decays and other new proposals to determine |V_us|. In view of the ongoing work to analyse tau-decay d…
We study the $τ^- \to ν_τ π^{-}π^{0}\ell^{+}\ell^{-}$ ($\ell=e,\,μ$) decays, which are $O(α^2)$-suppressed with respect to the dominant di-pion tau decay channel. Both the inner-bremsstrahlung and the structure- (and model-)dependent contributions are considered. In the $\ell=e$ case, structure-dependent effects are $\mathcal{O}(1\%)$ in the decay rate, yielding a clean prediction of its branching ratio, $2.3\times10^{-5}$, measurable with BaBar or Belle(-II) data. For $\ell=μ$, both contributions have similar magnitude and we get a branching fraction of $(1.6\pm0.3)\times10^{-7}$, reachable b…
