We review the status of mixing and lifetimes of $b$-hadrons. We will show that $ΔΓ/ ΔM$, $a_{sl}$ and $ϕ$ are better suited to search for new physics effects than $ΔM$ alone, because of our poor knowledge of the decay constants. The theoretical precision in the determination of $Γ_{12} / M_{12}$ - which contains all information on $ΔΓ/ ΔM$, $a_{sl}$ and $ϕ$ - can be tested directly by investigating the lifetimes of $b$-hadrons, because both quantities rely on the same theoretical footing. In particular we will also present a numerical estimate for the lifetime of the $Ξ_b$-baryon.
The investigation of mixing phenomena and lifetime in neutral B meson systems provides an important testing ground for standard model flavour dynamics. Spectroscopic parameters has been used to calculate the pseudoscalar decay constant and predictions of mixing mass parameters and lifetimes of the B$_{d}$ and B$_{s}$ mesons.
A review of the results on B^0- \bar {\rm B^0} mixing and b-lifetimes obtained by CDF, LEP and SLD collaborations is presented with special emphasis on B^0_s- \bar {{\rm B}^0_s} mixing.
We review recent experimental results on hadronic decays and lifetimes of hadrons containing b and c quarks. We discuss charm counting and the semileptonic branching fraction in B decays, the color suppressed amplitude in B and D decay, and the search for gluonic penguins in B decay.
The status of heavy quark expansions for charm and beauty lifetime ratios is reviewed. Taking note of the surprising semiquantitative success of this description for charm hadrons I interprete the new data on $τ(D_s)$ and re-iterate the call for more precise measurements of $τ(Ξ_c^{0,+})$ and $τ(Ω_c)$. A slightly larger $B^-$ than $B_d$ lifetime is starting to emerge as predicted; the largest lifetime difference in the beauty sector, namely in $τ(B_c)$ vs. $τ(B)$ has correctly been predicted; the problem posed by the short $Λ_b$ lifetime remains. The need for more accurate data also on $τ(B_s)…
The problem of the hierarchy of lifetimes of charmed hadrons is reviewed. The QCD-based theory of preasymptotic effects in inclusive weak decays dating back to the beginning of the eighties is now entering its mature phase. Combining recent and old results we argue that the observed hierarchy reflects most intimate features of the hadronic structure. The problem of a wide spread of lifetimes of charmed hadrons is addressed. We speculate on what is to be expected from QCD to provide the observed pattern. A number of predictions is given for the hierarchy of lifetimes in the family of the beauti…
Hadron elementary particle measurements have long revealed an alpha-dependence in the long-lived threshold-state lifetimes, together with an alpha-related mass structure, where alpha ~ 1/137 is the fine structure constant. A detailed lifetime analysis shows that of the 156 particles with well-determined lifetimes, the 120 shortest lifetimes have a continuum of values, but the 36 longer lifetimes occur in alpha-spaced groups which are sorted into s, c, b quark flavors. Guided by these lifetime results, we deduce an alpha-generated set of universal mass quanta, based on the electron ground state…
We review current status of theoretical predictions of lifetimes of heavy hadrons in heavy-quark expansion. We present a calculation of subleading 1/m_b corrections to spectator effects in the ratios of beauty hadron lifetimes. We find that these effects are sizable and should be taken into account in systematic analyses of heavy hadron lifetimes. In particular, the inclusion of 1/m_b corrections brings into agreement the theoretical predictions and experimental observations of the ratio of lifetimes of Lambda_b-baryon and B_d meson. We obtain τ(B_u)/τ(B_d) = 1.09 \pm 0.03, τ(B_s)/τ(B_d)= 1.00…
We re-examine constraints from the evidence for observation of the lifetime difference in $D^0$-${\bar D}^0$ mixing on the parameters of supersymmetric models with R-parity violation (RPV). We find that RPV SUSY can give large negative contribution to the lifetime difference. We also discuss the importance of the choice of weak or mass basis when placing the constraints on RPV-violating couplings from flavor mixing experiments.
A theoretical treatment for the weak decays of heavy-flavour hadrons has been developed that is genuinely based on QCD. Its methodology as it applies to total lifetimes and the underlying theoretical issues are sketched. Predictions are compared with present data. One discrepancy emerges: the beauty baryon lifetime appears to be significantly shorter than expected. The ramifications of those findings are analyzed in detail.
We re-examine constraints from the recent evidence for observation of the lifetime difference in D^0 - \bar{D}^0 mixing on the parameters of supersymmetric models with R-parity violation (RPV). We find that RPV SUSY can give large negative contribution to the lifetime difference. We also discuss the importance of the choice of weak or mass basis when placing the constraints on RPV-violating couplings from flavor mixing experiments.
We present the summary of the Working Group on lifetimes, mixing and weak mixing phases in charm and beauty mesons at the CKM 2010 workshop. In the past year or so good progress was achieved on both experimental and theoretical sides. While this yields improvement in our understanding of neutral meson mixing, further work is necessary to achieve the highest possible precision in order to investigate current hints for deviations between experiment and standard model predictions. With the recent LHC startup we see bright prospects for the near term future for huge improvements.
B-hadron physics plays a fundamental role to test and improve our understanding of flavor dynamics within and beyond the Standard Model. Of particular phenomenological interest are beauty hadron lifetime ratios and, width differences and semileptonic CP-asymmetries in B_d and B_s systems. We discuss their theoretical predictions which, in the last years, have been improved thanks to accurate lattice calculations of operator matrix elements and perturbative computations of Wilson coefficients.
We find that in the nonmanifest left-right (LR) model, an important LR mixing effect was not considered in the paper by E. Golowich, S. Pakvasa, and A. Petrov, Phys. Rev. Lett. 98, 181801 (2007).
New physics contribution to the lifetime difference in $D^0 - \bar{D}^0$ mixing is re-examined within the non-manifest Left-Right Symmetric Model. Diagrams with one of $ΔC = 1$ transitions, mediated by a propagator with $W_L - W_R$ mixing, are revisited. While these diagrams are believed to give the dominant contribution, compatible with the experimental data, it is shown that due to GIM cancelation, such diagrams are negligible in sum. Thus, Left-Right Symmetric Model contribution to the lifetime difference in $D^0 - \bar{D}^0$ mixing is about two orders of magnitude less than actual experime…
We develop perturbative QCD factorization theorems for inclusive b-hadron decays, in which radiative corrections characterized by the hadronic scale, the b-hadron mass, and the W boson mass are absorbed into a heavy hadron distribution function, a hard b quark decay amplitude, and a "harder" function, respectively. Double logarithmic corrections associated with a light energetic final-state quark, which appear at kinematic end points, are absorbed into a jet function. Various large logarithms contained in the above functions are summed to all orders, leading to the evolution factors among the …
I discuss the heavy quark expansion for the inclusive widths of heavy-light hadrons, which predicts quite well the experimental ratios of B_q meson lifetimes. As for $Λ_b$, current determinations of ${\cal O}(m_b^{-3})$ contribution to $τ(Λ_b)$ do not allow to explain the small measured value of $τ(Λ_b)/τ(B_d)$. As a final topic, I discuss the implications of the measurement of the B_c lifetime.
We present a calculation of 1/m^2_b corrections to the lifetime differences of B_s mesons ΔΓ_s in the heavy-quark expansion. We find that they are small to significantly affect ΔΓ_s and present the result for lifetime difference including non-perturbative 1/m_b and 1/m_b^2 corrections. We also analyze the generic ΔB = 1 New Physics contributions to ΔΓ_s and provide several examples.
The lifetime splitting between the B^+ and B_d^0 mesons has recently been calculated in the next-to-leading order of QCD. These corrections are necessary for a reliable theoretical prediction, in particular for the meaningful use of hadronic matrix elements computed with lattice QCD. Using results from quenched lattice QCD we find tau(B^+)/tau(B^0_d)=1.053 +/- 0.016 +/- 0.017, where the uncertainties from unquenching and 1/m_b corrections are not included. The lifetime difference of heavy baryons Xi_b^0 and Ξ_b^- is also discussed.
We discuss (i) the evaluation of the expectation values of four-quark operators assuming that the heavy quark expansion for $b$ sector converges at the third order in $1/m_Q$, and (ii) the estimation of the duality breaking short distance nonperturbative corrections to the parton decay rate. We finally point out the implications of the result obtained for the assumption of quark-hadron duality in heavy quark expansion.
We show that in a confining hidden valley model where the lightest hidden particles are dark hadrons that have mass splittings larger than $\mathcal{O}(0.1)$ GeV, if the lightest dark hadron is either stable or decays into Standard Model (SM) hadrons/charged leptons during the big-bang nucleosynthesis (BBN), at least one of the heavier dark hadrons needs to decay into SM particles within $\mathcal{O}(10)$ nanosec. Once being produced at collider experiments, this heavier dark hadron is likely to decay within $\mathcal{O}(1)$ meter distance, which strengthens the motivation of searching for lon…
We give an updated summary of the topics covered in Working Group I of the 2nd Workshop on the CKM Unitarity Triangle, with emphasis on the results obtained since the 1st CKM Workshop. The topics covered include the measurement of |V_{ub}|, of |V_{cb}| and of non-perturbative Heavy Quark Expansion parameters, and the determination of b-hadron lifetimes and lifetime differences.
Over the last few years a theoretical treatment for the weak decays of heavy-flavour hadrons has been developed that is genuinely based on QCD. Its methodology is described as it applies to total lifetimes, and the underlying theoretical issues are discussed. Theoretical expectations are compared with present data. One discrepancy emerges: the beauty baryon lifetime appears to be significantly shorter than predicted. The ramifications of those findings are analyzed in detail, and future refinements are described.
In this talk we review the theoretical status of the lifetime ratios of heavy hadrons and of the B-mixing quantities $ΔM_s$, $ΔΓ_s$ and $ϕ_s$. While $ΔM_s$ and $ΔΓ_s$ suffer from large uncertainties due to the badly known decay constants, the ratio $ΔΓ_s / ΔM_s$ can be determined with almost no non-perturbative uncertainties, therefore it can be used to look for possible new physics effects.
We present a calculation of $1/m_b^2$ corrections to the lifetime differences of $B_s$ mesons in the heavy-quark expansion. We find that they are small to significantly affect $ΔΓ$ $(B_s)$ and present the result for lifetime difference including non-perturbative $1/m_b$ and $1/m_b^2$ corrections. We also analyze the generic $ΔB$ = 1 New Physics contributions to the lifetime difference of $B_s$ mesons and provide several examples
The first general analysis of New Physics contributions to the D0-anti-D0 lifetime difference (equivalently Delta Gamma_D} is presented. The extent to which New Physics (NP) contributions to |Delta C|=1 processes can produce effects in Delta Gamma_D, even if such NP contributions are undetectable in the current round of D0 decay experiments, is studied. New Physics models which do and do not dominate the lifetime difference in the flavor SU(3) limit are identified. Specific examples are provided.
The preasymptotic effects originating from the four-quark operators are disentangled from the other contributions by considering appropriate combinations of inclusive decay rates. Under the assumption of the isospin and heavy-quark symmetry, a set of relations connecting charmed and beauty decays is obtained without invoking specific models. The results are compared with other approaches and confronted with experiment.
We review recent experimental results on lifetimes and hadronic decays of hadrons that contain $c$ and $b$ quarks. The theoretical implications of these results are also considered. An understanding of hadronic decays of heavy quarks is required to interpret the CP violating asymmetries in $B$ decays that will be observed in experiments planned for the near future.
The lifetime differences of bottom hadrons are known to be properly explained within the framework of heavy quark effective field theory(HQEFT) of QCD via the inverse expansion of the dressed heavy quark mass. In general, the spectrum around the endpoint region is not well behaved due to the invalidity of $1/m_Q$ expansion near the endpoint. The curve fitting method is adopted to treat the endpoint behavior. It turns out that the endpoint effects are truly small and the explanation on the lifetime differences in the HQEFT of QCD is then well justified. The inclusion of the endpoint effects mak…
Inclusive bottom hadron decays are analyzed based on the heavy quark effective field theory (HQEFT). Special attentions in this paper are paid to the b\to u transitions and nonspectator effects. As a consequence, the CKM quark mixing matrix elements |V_ub| and |V_cb| are reliably extracted from the inclusive semileptonic decays B\to X_u e νand B\to X_c e ν. Various observables, such as the semileptonic branch ratio B_SL, the lifetime differences among B^-, B^0, B_s and Λ_b hadrons, the charm counting n_c, are predicted and found to be consistent with the present experimental data.
We describe a theoretical approach to compute inclusive widths of heavy hadrons, based on a systematic expansion in the inverse powers of the heavy quark mass. The method reproduces quite well the experimental ratios of $B$ mesons lifetimes. As for $τ(Λ_b)/τ(B_d)$, we present a QCD sum rule calculation of the $O(m_b^{-3})$ contribution to $τ(Λ_b)$; we conclude that such correction is not responsible of the small experimental value of $τ(Λ_b)/τ(B_d)$.
A detailed, quantitative re-examination of the effect of hadron dynamics on baryon decay, modeled in terms of Skyrme-field tunneling, indicates that any hadronic suppression should be quite mild. This appears to be another illustration of the `Cheshire-cat' phenomenon, that variation of the apportionment between description of the nucleon as a bag of quarks and description as a Skyrme field configuration has little influence on many nucleon properties. Perhaps the largest remaining uncertainty in evaluating the decay rate has to do with the overlap between a specified quark-antiquark configura…
Inclusive heavy-flavour decays can be described through $1/m_Q$ expansions derived from QCD with the help of an operator product expansion. I sketch their methodology and apply them first to semileptonic B decays; $|V(cb)|$ can be extracted from $Γ_{SL}(B)$ and $\bar B \to l νD^*$ with the result: $|V(cb)|_{incl} = 0.0413 \pm 0.0016_{exp} \pm 0.002_{theor}$, $|V(cb)|_{excl} = 0.0377 \pm 0.0016_{exp} \pm 0.002_{theor}$. The lifetimes of charm and beauty hadrons are discussed. The charm lifetimes are predicted/reproduced as well as could be expected. Predictions on B meson lifetimes agree with a…
The heavy quark expansion implemented through an operator product expansion provides us with a treatment of inclusive decays of beauty and charm hadrons that is genuinely derived from QCD,though it requires one additional assumption, namely that of 'local' quark-hadron duality. Subtleties in the application of factorization to hadronic expectation values are pointed out. The observed pattern in the charm lifetime ratios is reproduced in a semi-quantitative manner. The ratio $τ(Λ_b)/τ(B_d)$ cannot be pushed significantly below 0.9 -- unless one invokes a new hitherto unknown paradigm for evalua…
A phenomenological analysis of lifetimes of bottom and charmed hadrons within the framework of the heavy quark expansion is performed. The baryon matrix element is evaluated using the bag model and the nonrelativistic quark model. We find that bottom-baryon lifetimes follow the pattern $τ(Ω_b)\simeqτ(Ξ_b^-)>τ(Λ_b)\simeqτ(Ξ_b^0)$. However, neither the lifetime ratio $τ(Λ_b)/τ( B_d)$ nor the absolute decay rates of the $Λ_b$ baryon and $B$ mesons can be explained. One way of solving both difficulties is to allow the presence of linear $1/m_Q$ corrections by scaling the inclusive nonleptonic widt…
In this work we discuss the theoretical status for the study of the lifetime of heavy hadrons. After presenting some introductory topics like the effective weak Hamiltonian and the heavy quark effective theory (HQET), we describe the construction of the heavy quark expansion (HQE), which constitutes the theoretical framework to systematically compute the total decay width of heavy hadrons, in terms of an expansion in inverse powers of the heavy quark mass. The structure of the HQE is discussed in detail, and the computation of the lowest dimensional contributions, explicitly outlined. Particul…
Combined results on b-hadron lifetimes, b-hadron production rates, B^0_d - \bar{B^0_d} and B^0_s -\bar{B^0_s} oscillations, the decay width difference between the mass eigenstates of the B^0_s -\bar{B^0_s} system, and the values of the CKM matrix elements | V_cb | and \Vert V_ub | are obtained from published and preliminary measurements available in Summer 99 from the ALEPH, CDF, DELPHI, L3, OPAL and SLD Collaborations.
We present the theoretical status of the lifetimes of weakly decaying heavy hadrons containing a bottom or a charm quark, and discuss the current predictions, based on the framework of the Heavy Quark Expansion (HQE), for both mesons and baryons. Potential improvements to reduce the theoretical uncertainties are also highlighted.
We update the Standard-Model predictions for several quantities related to $B_s - \bar B_s$ and $B_d - \bar B_d$ mixing. The mass and width differences in the $B_s$ system read $ ΔM_s^{\rm SM} = (17.3 \pm 2.6) ps^{-1}$ and $ΔΓ_s^{\rm SM}= (0.087 \pm 0.021) ps^{-1}$, respectively. The CP asymmetries in flavour-specific decays are $a_{\rm fs}^{s,\rm SM} = (1.9 \pm 0.3) \cdot 10^{-5}$ and $a_{\rm fs}^{d,\rm SM} = - (4.1 \pm 0.6) \cdot 10^{-4}$. We further critically discuss the sensitivity of $ΔΓ_d$ to new physics and the uncertainties in the relation between $ΔΓ_s$ and the branching fraction of …
Long distance contribution to the $K^0-\bar K^0$ mixing is taken into account consistently and the corrections to the naive duality result represented by the famous box diagram are found to be small. Estimates are given in the leading order of the chiral perturbation theory.
A new method for calculation of the mixing angle between the hadrons within QCD sum rules is proposed. In this method, the mixing is expressed in terms of quark and gluon degrees of freedom. As an application, the detailed calculation of the mixing angle between heavy cascade baryons Xi_Q and Xi_Q', Q=c,~b is presented and it is found that the mixing angle between Xi_b (Xi_c) and Xi_b' (Xi_c') is given by theta_b = 6.4^\circ \pm 1.8 ^\circ (theta_c = 5.5^\circ \pm 1.8^\circ).
Recent measurements of lifetimes of some $b$-flavored hadrons are presented and interpreted in the context of theoretical models, especially the Heavy Quark Expansion. Decay widths and decay width differences in the $B_s^0-\overline{B}_s^0$ system are discussed from studies of decays into the final states $J/ψK^+K^-$, $J/ψπ^+π^-$, $D_s^+D_s^-$, $K^+K^-$, and $D_s^{\pm}π^{\mp}$. Lifetime measurements of the baryons $Λ_b^0$, $Ξ_b^-$, $Ξ_b^0$, and $Ω_b^-$ are also shown.
We present a calculation of subleading 1/m_b corrections to spectator effects in the ratios of beauty hadron lifetimes in heavy-quark expansion. We find that these effects are sizable and should be taken into account in systematic analyses of heavy hadron lifetimes. In particular, the inclusion of 1/m_b corrections brings into agreement the theoretical predictions and experimental observations of the ratio of lifetimes of Λ_b-baryon and B_d meson. We obtain τ(B_u)/τ(B_d) = 1.06 \pm 0.02, τ(B_s)/τ(B_d)= 1.00 \pm 0.01, τ(Λ_b)/τ(B_d)= 0.86 \pm 0.05.
Lifetime differences of heavy hadrons can be consistently computed in heavy-quark expansion. The leading effects appear through spectator interactions at order 1/m_b^3. We compute a well-defined subset of 1/m_b^4 corrections to the lifetime ratio of Lambda_b baryon and B_d meson. We find that these corrections are large and should be taken into account in the systematic analysis of heavy hadron lifetimes. We claim that they could shift the ratio τ_{Λ_b}/τ_{B_d} by as much as -4.5%, significantly reducing the discrepancy between the theoretical predictions and experimental observations.
The color mixing of mesons propagating in a nucleus is studied with the help of a color-octet Pomeron partner present in the two-gluon model of the Pomeron. For a simple model with four meson-nucleon channels, color mixings are found to be absent for pointlike mesons and very small for small mesons. These results seem to validate the absorption model with two independent color components used in recent analyses of the nuclear absorption of $J/ψ$ mesons produced in nuclear reactions.
We discuss neutral B_{s,d} meson mixing and b-hadron lifetimes from the perspective of recent lattice calculations. In particular, we consider matrix elements which can be combined with measured ΔM_{s,d} to constrain |V_{td}|, the lifetime ratios τ(Λ_b)/τ(B_d) and τ(B^-)/τ(B_d) and the lifetime difference, ΔΓ_s/Γ_s, in the neutral B_s meson system.
Hadronic resonances, having very short lifetime, like $\rm{K}^{*0}$, can act as useful probes to understand and estimate lifetime of hadronic phase in ultra-relativistic proton-proton, p--Pb and heavy-ion collisions. Resonances with relatively longer lifetime, like $ϕ$ meson, can serve as a tool to locate the QGP phase boundary. We estimate a lower limit of hadronic phase lifetime in Cu--Cu and Au--Au collisions at RHIC, and in pp, p--Pb and Pb--Pb collisions at different LHC collision energies. Also, we obtain the effective temperature of $ϕ$ meson using Boltzmann-Gibbs Blast-Wave function, w…
Measurements of the effective lifetimes of $B_s$-meson decays, which only require untagged rate analyses, allow us to probe the width difference $ΔΓ_s$ and the CP-violating phase $ϕ_s$ of $B^0_s$-$\bar B^0_s$ mixing. We point out that the dependence of the effective lifetime on non-linear terms in $ΔΓ_s$ allows for a determination of $ϕ_s$ and $ΔΓ_s$ given a pair of $B_s$ decays into CP-even and CP-odd final states. Using recent lifetime measurements of $B^0_s\to K^+K^-$ and $B^0_s\to J/ψf_0(980)$ decays, we illustrate this method and show how it complements the constraints in the $ϕ_s$-$ΔΓ_s$…
Interplay between axial anomaly and quark-hadron duality in the presence of strong mixing is considered. The anomaly sum rule for meson transition form factors based on the dispersive representation of axial anomaly and quark-hadron duality in octet channel is analyzed. The comparison of this sum rule to the experimental data on $η$ and $η'$ mesons transition form factors shows that the interval of duality in this channel is rather small, contradicting the usual understanding of quark-hadron duality. The same values of interval of duality are supported by considering the two-point correlator i…
We provide an extensive study of the lifetimes of singly charmed baryons and mesons, within the heavy quark expansion with all known corrections included. A special attention is devoted to the choice of the charm mass and wavefunctions of heavy baryons. We give our predictions for lifetimes, lifetime ratios, and semileptonic branching ratios of singly charmed baryons. Our results accommodate the experimentally-favoured hierarchy of singly charmed baryon lifetimes \begin{eqnarray*} τ\left(Ξ_c^{0}\right) < τ\left(Λ_c^{+}\right)< τ\left(Ω_c^{0}\right) < τ\left(Ξ_c^{+}\right)\, \end{eqnarray*} in …
We investigate how coupling of valence q qbar to meson pairs can modify the properties of conventional q qbar and hybrid mesons. In a symmetry limit the mixing between hybrids and conventional q qbar with the same J^PC is shown to vanish. Flavor mixing between heavy and light q qbar due to meson loops is shown to be dual to the results of gluon mediated pQCD, and qualitatively different from mixing involving light flavors alone. The validity of the OZI rule for conventional q qbar and hybrid mesons is discussed.
We compute the next-to-leading order QCD corrections to spectator effects in the lifetime ratios of beauty hadrons. With respect to previous calculations, we take into account the non vanishing value of the charm quark mass. We obtain the predictions tau(B+)/tau(Bd) = 1.06 +- 0.02, tau(Bs)/tau(Bd)= 1.00 +- 0.01 and tau(Lambdab)/tau(Bd) = 0.90 +- 0.05, in good agreement with the experimental results. In the case of tau(Bs)/tau(Bd) and tau(Lambdab)/tau(Bd), however, some contributions, which either vanish in the vacuum insertion approximation or represent a pure NLO corrections, have not been de…
We review recent measurements of heavy B hadron states including masses and lifetimes of the Bc meson as well as excited B states (B**, Bs**). We discuss properties of the Bs meson such as lifetime, lifetime difference delta_Gamma/Gamma and CP violation in Bs -> J/Psi Phi decays. We also summarize new measurements of the masses and lifetimes of bottom baryons including the Lambda_b baryon, the Sigma_b baryon states as well as the Xi_b and Omega_b baryons.
$B_d$ and $B_s$ meson systems play a fundamental role to test and improve our understanding of Standard Model flavor dynamics. The mixing parameters $Δm_{d,s}$ represent important constraints in the unitarity triangle analysis. Their theoretical estimates require non-perturbative calculations of B-meson decay constants and B-parameters; accurate results, recently obtained from the lattice, are reviewed. Other phenomenologically interesting quantities are the beauty hadron lifetime ratios and, the width differences and CP-violation parameters in $B_d$ and $B_s$ systems. We discuss their theoret…
It has been realized for a long time that knowing the ηand η' wave functions in terms of quark and gluon components probes our understanding of non-perturbative QCD dynamics. Great effort has been given to this challenge -- yet no clear picture has emerged even with the most recent KLOE data. We point out which measurements would be most helpful in arriving at a more definite conclusion. A better knowledge of these wave functions will significantly help to disentangle the weight of different decay subprocesses in semi-leptonic decays of D^+, D_s^+ and B^+ mesons. The resulting insights will be…
