Revisiting Inclusive Decay Widths of Charmed Mesons

Determining for the first time the Darwin operator contribution for the non-leptonic charm-quark decays and using new non-perturbative results for the matrix elements of $\Delta C=0$ four-quark operators, including eye-contractions, we present a comprehensive study of the lifetimes of charmed mesons and inclusive semileptonic decay rates as well as the ratios, within the framework of the Heavy Quark Expansion (HQE). We find good agreement with experiment for the ratio $\tau(D^+)/\tau(D^0)$, for the total $D_s^+$-meson decay rate, for the semileptonic rates of all three mesons $D^0$, $D^+$ and $D_s^+$, and for the semileptonic ratio $\Gamma_{sl}^{D^+}/\Gamma_{sl}^{D^0}$. The total decay rates of the $D^0$ and $D^+$ mesons are underestimated in our HQE approach and we suspect that this is due to missing higher-order QCD corrections to the free charm quark decay and the Pauli interference contribution. For the $SU(3)_F$ breaking ratios $\tau (D_s^+) / \tau (D^0) $ and $\Gamma_{sl}^{D_s^+}/\Gamma_{sl}^{D^0} $ our predictions lie closer to one than experiment. This might originate from the poor knowledge of the non-perturbative parameters $\mu_G^2$, $\mu_\pi^2$ and $\rho_D^3$ in the $D^0$ and $D_s^+$ systems. These parameters could be determined by experimental studies of the moments of inclusive semileptonic $D$ meson decays.


Introduction
Lifetimes of charm mesons are determined experimentally very precisely [1] 1 and show a pattern which is clearly less monotonous than in the b-sector, with values spreading over a rather large range. Moreover, also inclusive semileptonic branching fractions have been measured [1], and recently an update for the D + s -meson has been released by the BESIII Collaboration [3]. A summary of the current experimental status is presented in Table 1. While in the bottom sector, the approximation that the  (2) τ (D q )/τ (D 0 ) 1 2.54 (2) 1.20 (1) Br(D q → Xe + ν e )[%] 6.49 (11) 16.07 (30) 6.30 (16) Γ(D q → Xe + ν e ) Γ(D 0 → Xe + ν e ) 1 0.977 (26) 0.790(26) Table 1: Status of the experimental determinations of the lifetime and the semileptonic branching fractions of the lightest charmed mesons (D q ∈ D 0 , D + , D + s ). All values are taken from the PDG [1] apart from the semileptonic D + s -meson decays which were recently measured by the BESIII Collaboration [3]. meson decay can be described in terms of the free b-quark decay is experimentally well accommodated, for the charm system this is poorly justified. A systematic way to study this assumption is provided by the heavy quark expansion (HQE) -see Refs. [4,5] for early references or Ref. [6] for a recent review, according to which the inclusive decay width of a meson containing a heavy charm quark can be written as  In the present work we will try to shed further light into the question, whether the expansion parameters α s (m c ) and Λ QCD /m c are small enough in order to ensure a meaningful convergence of the HQE. The Particle Data Group [1] quotes, for the pole and MS mass of the charm quark, the values while the dependence of the strong coupling on both the charm scale and the loop order (obtained using the RunDec package [7]) is shown in Table 2. In our numerical analysis we use the 5-loop running of the strong coupling. While the determination of the MS mass is theoretically well founded, that  Table 2: Numerical values of the strong coupling α s evaluated at different scales and loop order, obtained using the RunDec package [7].
of the pole mass seems to be affected by a potential breakdown of perturbation theory. On the other side, the pole mass is the natural expansion parameter of the HQE. The relation between the two mass schemes, up to third order in the strong coupling, reads [8][9][10] 48 GeV and express everything in terms of the pole mass. A further possibility would be to consider the expansion in Eq. (1.4) to be an asymptotic one, whose smallest correction appears at order α 2 s , which is where we stop the expansion. In this case we get the pole mass value from PDG, m Pole c = 1.67 GeV. We did numerical tests for this large value of the charm quark mass and the results for decay rates are roughly 70 − 90% larger than the values obtained using m Pole c = 1.48 GeV. Since we expect this enhancement to be compensated by missing NNLO corrections to the non-leptonic decay rates, we will not separately present results for m Pole c = 1.67 GeV.
2. Express the c-quark mass in terms of the MS mass [11], taking m c (m c ) = 1.27 GeV [1], and expand consistently up to order α s . Because of the dependence on the fifth power of the charm-quark mass, in this case Γ 3 is affected by a large correction 5 × (4/3)(α s /π).
3. Express the c-quark mass in terms of the kinetic mass [12,13]. The kinetic scheme has been introduced in order to obtain a short distance definition of the heavy quark mass which allows a faster convergence of the perturbative series and is still valid at small scales µ ∼ 1 GeV. The relation between the kinetic scheme and the MS and Pole schemes can be found, up to N 3 LO corrections, in Ref. [14]. At order α s one has where µ cut is the Wilsonian cutoff separating the perturbative and non-perturbative regimes.
Using m c (m c ) as an input, the authors of Ref. [14] obtain Comparing with Eq. (1.8) it follows that the kinetic scheme might be preferred to the MS scheme if the term in the round brackets of Eq. (1.9) would give a suppression factor. For µ cut = 1 GeV and m Kin c = 1.2 GeV, this is not the case, while using lower values i.e. µ cut < 1 GeV, the convergence of the series could be improved, however this would bring in an additional uncertainty due to the closeness to the non-perturbative scale Λ QCD . In our numerical analysis we will investigate the kinetic scheme with µ cut = 0.5 GeV. From Ref. [14] we take the following value m kin c (0.5 GeV) = 1.363 GeV , (1.12) obtained for consistency at NLO in α s and using as an input m c (m c ).

Effective Hamiltonian and HQE
The non-leptonic decay of a charm quark c → q 1q2 u (q i = u, d, s) is governed by the effective ∆C = 1 Hamiltonian (see e.g. Ref. [69]) where λ q 1 q 2 = V * cq 1 V uq 2 and λ b = V * cb V ub are the CKM factors, C i (µ 1 ) denote the Wilson coefficients at the renormalisation scale µ 1 ∼ m c , Q q 1 q 2 1,2 are tree-level ∆C = 1 operators 3 while Q j , j = 3...6 are penguin operators, which can only arise in the singly Cabibbo suppressed decays c → ssu and c → ddu or in further suppressed pure penguin decays like c → uūu. Values of the Wilson coefficients at different scales are shown in Table 3 both at NLO-QCD and LO-QCD.  We see that the Wilson coefficients of the penguin operators are very small, additionally their contributions are also strongly suppressed by the CKM factor λ b λ q 1 q 2 . Therefore, in our analysis, we neglect the effect of the penguin operators, given the current limited theoretical accuracy in the charm sector.
The complete effective Hamiltonian describing all possible c-quark decays is a sum of non-leptonic, semileptonic and radiative contributions, namely where H NL eff is given in Eq. (2.1), , while H rare eff describes decays like D → π + − , whose branching fraction is much smaller than those corresponding to tree-level transitions. Hence, in the following we neglect rare decays and we do not show an explicit expression for H rare eff . The total decay width of a heavy D meson with mass m D and four-momentum p µ D can be written as where PS denotes the phase space integration and we have summed over all possible final states X into which the D meson can decay. Eq. (2.6) can be related, via the optical theorem, to the discontinuity of the forward scattering matrix element of the time ordered product of the double insertion of the effective Hamiltonian, i.e.
with the transition operator In the framework of the HQE, the four-momentum of the decaying c-quark is parametrised in terms of "large" and "small" components as where v µ = p µ /m D denotes the four-velocity of the D-meson and k µ → iD µ , where D µ is the covariant derivative with respect to the background gluon field, accounting for the residual interaction of the c-quark with the light degrees of freedom, i.e. soft gluons and quarks. At the same time, the heavy charm quark field is redefined as to remove the large fraction of the c-quark momentum. Using Eqs. (2.9) and (2.10), Γ(D) in Eq. (2.7) can be expanded in the small quantity D µ /m c ∼ Λ QCD /m c , leading to the series in Eq. (1.1), for more details, see e.g. Ref. [70] or Ref. [49] for a more recent treatment. The result is schematically shown in Fig. 1. The first diagram on the top line of Fig. 1, corresponding to the limit m c → ∞, represents the decay of a free c-quark, while power corrections due to the interaction of the heavy quark with soft gluons and quarks are described respectively by the second and third diagrams on the top line of Fig. 1. Finally, before discussing the individual terms in Eq. (1.1) separately, it is worth emphasizing that the field c v is related to the effective heavy quark field h v , introduced in the framework of the HQET (see e.g. Ref. [71]), by

Dimension-three Contribution
The leading term in Eq. (1.1), Γ 3 , can be schematically written as where we define and we introduce the dimensionless mass parameter z q = m 2 q /m 2 c . Note than we neglect the neutrino as well as the electron, the up and down quarks masses, i.e. z ν = z e = z u = z d = 0, while z s = 0 = z µ . The first two terms in c 3 in Eq. (2.12) correspond to the semileptonic modes c → se + ν e and c → sµ + ν µ , while the third term to the Cabibbo favoured decay c → sud. The ellipsis stand for CKM suppressed contributions. The dependence on the ∆C = 1 Wilson coefficients is absorbed in the combination N a = 3 C 2 1 + 3 C 2 2 + 2 C 1 C 2 . The behaviour of N a as function of the renormalisation scale, both at LO-and NLO-QCD, is shown in Table 4    cancellations in N a that might lead to numerical instability. The phase-space function f (a, b, c) in Eq. (2.12) describes the effect of the final state masses. In the case of one massive particle, it reduces to the well-known expression 14) which shows that the contribution due to the finite s-quark mass is small. The analytic expression of f (a, b, c) for two different masses in the final state, can be found e.g. in the Appendix of Ref. [72]. By including also NLO-QCD corrections, Γ 3 can be schematically presented as where a summation over all the modes is implicitly assumed. At NLO, the expressions for C 3,11 , C 3,22 and C 3,SL are taken from Ref. [19], where the computation was done for three different final state masses, hence we can easily use these results for all c-quark decay modes. For the coefficient C 3,12 we use Ref. [22] for the c → sdu, c → dsu and c → ddu decay channels, while the result of Ref. [26] is used in the case of final state with two massive s-quarks, c → ssu. Neglecting final state masses and approximating |V ud | 2 ≈ 1 the following expression was determined in 1991 [20], i.e.
Pole (m c = 1.  Table 5: 3 using different schemes for the c-quark mass. The uncertainties are obtained by varying the renormalisation scale µ 1 between 1 GeV and 3 GeV. The first term on the r.h.s. of Eq. (2.16) stems from semileptonic decays and the next two terms from non-leptonic channels. For non-leptonic b-quark decays the NLO corrections are negative, while for charm quarks decays the third term will dominate over the second one and the correction becomes positive. Moreover, there is a sizable enhancement of the α s -corrections in the non-leptonic b-quark decays due to finite charm quark mass effects [21][22][23]26] -the corresponding increase in charm quark decays is much less pronounced as m 2 The numerical values for Γ 3 both in LO-and NLO-QCD, for different c-quark mass schemes are shown in Table 5. The range of NLO-QCD values from 1.3 ps −1 to 1.5 ps −1 for the free charm-quark decay at NLO-QCD, is in good agreement with the experimental determinations in Table 1. Moreover we observe small (< 5%) corrections due to a non-vanishing strange quark mass. Interestingly the NLO-QCD result is affected by strong cancellations. We in fact observe a suppression of the nonleptonic contribution because of the opposite sign between the NLO corrections to the diagrams describing QCD corrections to the upper left diagram of Fig. 1 and the QCD corrections intrinsic to the ∆C = 1 Wilson coefficients. A further cancellation is then present between the semileptonic and the non-leptonic modes. This behaviour can be nicely read of the result in the Pole scheme: Expressing the pole mass in terms of a short distance mass like the MS scheme, an additional large NLO correction arises from the conversion factor of m 5 c , which is the origin of the large shift between the LO and the NLO values in the MS-, the kinetic and the 1S-schemes, see Table 5. We find e.g. in the MS scheme The corrections due to the mass conversion also make the overall semileptonic NLO term in the MS scheme positive.
To get a first indication of the behaviour of the QCD series for the decay rate at higher orders, we briefly discuss here the NNLO [35] and NNNLO [37] corrections for the semileptonic b-quark decay and the preliminary NNLO-QCD corrections for the non-leptonic b-quark decay [39]. In the Pole mass scheme [37] the semileptonic decay rate gets large negative corrections, and in the MS-scheme one finds [37] sizable positive corrections -driven by the conversion of the quark mass from the Pole scheme to the MS-scheme and indicating again the importance of higher order perturbative corrections. For the semileptonic charm quark decay one finds even larger corrections 4 , e.g. in the Pole mass scheme which clearly spoils the perturbative approach and makes the use of different quark mass schemes mandatory.
Regarding the NNLO-QCD corrections to the non-leptonic decay rates, Ref. [39] presents a partial result (not resumming the large logarithms, neglecting effects of the operator Q q 1 q 2 2 and assuming a vanishing charm quark mass) for the b-quark. In the pole mass scheme the authors obtain (2.22) It is interesting to note that from the coefficient of the α 2 s term, 67.1, a contribution of 54.7 stems from not summing large logarithms of the form ln(M W /m b ) and ln 2 (M W /m b ). Using Eq. (2.19) and the fact that, in the approximations of Ref. [39] the ratio between non-leptonic and semileptonic rate is equal to 3 at LO-QCD, yields (2.23) For non-leptonic charm-quark decays the logarithms become even larger and we find that the coefficient of the α 2 s term increases from 52.3 to 91.2, which clearly indicates the necessity of summing the large logarithms. Neglecting final state masses seems to be well justified in the charm system. In order to further estimate the effect of neglecting the operator Q q 1 q 2 2 , we set in our code C 1 = 1 and C 2 = 0 and we get in the Pole scheme All in all we conclude that, higher order corrections seem to be crucial for a reliable determination of Γ 3 . Despite being conceptually very interesting, the result of Ref. [39] is not useful for phenomenological applications and a full NNLO determination of the non-leptonic decay rate using the effective Hamiltonian would be highly desirable.

Dimension-five Contribution
The first corrections to the free charm-quark decay arise at order 1/m 2 c and describe the effect of the kinetic and the chromomagnetic operators. Their matrix elements are parametrised by the two non-perturbative inputs µ 2 π and µ 2 G , i.e.
Both the operators receive a contribution from the expansion of the dimension-three matrix element D(p)|c v c v |D(p) [70]. However, the chromomagnetic operator receives further contributions due to the expansion of the short distance coefficient c 3 and of the quark-propagator in the external gluon field [41,42,49] -see the second diagram on the top line of Fig. 1. Hence, at order 1/m 2 c , we can schematically write (2.28) The coefficient of the kinetic operator is related to the dimension-three contribution 5 , c µπ = −c (0) 3 /2, and the chromomagnetic coefficient c G can be decomposed as where again a summation over all the modes is assumed. The individual contributions C G,nm for nonleptonic modes can be found e.g. in the Appendix of Ref. [49]. In the latter reference, the coefficients of the chromomagnetic operator were determined for the non-leptonic B-meson decays, however, since there are no IR-divergences at this order, it is straightforward to obtain the corresponding results for the charm-sector, namely by replacing m b → m c , m c → m s , etc. For the semileptonic decay c → sµ + ν µ , the expression for two different mass parameters z s = 0 = z µ can be found in the Appendix of Ref. [72]. 6 By neglecting the strange and muon masses and by considering only the dominant CKM modes, the result for c G becomes very compact, i.e.
(2.30) 5 Since the dimension-5 contribution for non-leptonic modes is known only at LO in QCD, we use the dimension-three coefficient c3 just at LO-QCD for cµ π . 6 Since ms ≈ mµ ≈ 100 MeV mc, in principle one can safely set zs = zµ and use the non-leptonic expressions for the semileptonic modes, e.g. c → ssu for c → sµνµ by setting Nc = 1, C1 = 1, C2 = 0.    Table 6: Comparison of the coefficients c SL G , c NL G , and c G = c SL G + c NL G for different values of the renormalisation scale µ 1 at LO and "NLO", setting for reference m c = 1.5 GeV.
Because of the large coefficient in front of C 1 C 2 and of its negative value, Eq. (2.30) can be affected by cancellations. In Fig. 3 we plot c G in Eq. (2.29), as a function of the renormalisation scale µ 1 while in Table 6 we list the numerical result for some reference values of µ 1 . For c G a change of sign occurs in the region between 1 and 2 GeV -leading to a large uncertainty due to scale variation. Note, that the "NLO" result shown in Fig. 3 and Table 6 only includes QCD corrections due to the ∆C = 1 Wilson coefficients. A complete calculation of the NLO-QCD corrections to c G is still missing (these corrections are only known for the semileptonic case) and would be very desirable in order to reduce the huge scale dependence. The numerical values of the non-perturbative parameters µ 2 π and µ 2 G will be discussed in Sections 3.2 and 3.1.

Dimension-six Two-Quark Operator Contribution
By determining higher order 1/m c corrections in the expansion of the quark-propagator, in the expansion of the the matrix elements of mass dimension-three and mass dimension-five and in the expansion of the corresponding short-distance coefficients, see e.g. Refs. [41,42,49,73] for details, one finds the dimension-six contribution to Γ(D), which can be compactly written as with the matrix element of the Darwin operator given by 7 The coefficient c ρ D can be decomposed into including both non-leptonic and semileptonic contributions. For B-mesons decays, the non-leptonic coefficients were computed recently in Refs. [49][50][51]. In order to determine the corresponding expressions for the charm system, some subtleties have to be considered. In b-quark decays, one assumes m b ∼ m c Λ QCD , and the coefficient of the Darwin operator for the semileptonic b → c ν decays is a finite function of ρ = m 2 c /m 2 b , which however diverges in the limit ρ → 0, i.e. in correspondence of the b → u ν transitions. This is due to the fact that, the radiation of a soft gluon off a massless quark leads to IR singularities at dimension-six. In non-leptonic b-quark decays, one has to further deal with the emission of a soft gluon from the internal light u-, d-, and s-quark lines. The corresponding IR divergences are of the form log(m q /m b ), for q = u, d, s, and are removed by taking into account the mixing between the four-quark operators with external q quarks and the Darwin operator under renormalisation, for details see e.g. Refs. [49,50,74,75]. Because of m c m s ∼ Λ QCD , it follows that one cannot trivially generalise the results from the b-to the c-sector, i.e. by only replacing m b → m c , m c → m s , etc., since there are further contributions due to the mixing of four-quark operators with external s-quarks which must be additionally included. Specifically, this leads to a modification of the coefficients proportional to C 2 1 and C 1 C 2 . Using the same procedure as discussed in Ref. [49], we have recomputed the coefficients of the Darwin operator required for the study of D-meson decays. The analytic expressions for C ρ D ,nm , including the full s-quark mass dependence, however finite in the limit m s → 0, are presented in Appendix B for all non-leptonic modes. To obtain the corresponding expression for C ρ D ,SL , it is sufficient to set in the results for the non-leptonic decays N c = 1, C 1 = 1, C 2 = 0 and z s = z µ for the c → sµ + ν µ mode. In particular, we confirm the results in Ref. [53].
Again, by neglecting the strange and muon masses and by considering only the dominant CKM modes, one finds It is interesting to note that in this combination all terms have the same sign and no cancellations arise. In Fig. 4 we show the dependence of the function c ρ D in Eq. (2.33) on the renormalisation scale µ 1 and in Table 7 we quote the numerical result for some reference values of µ 1 . The determination of the matrix element of the Darwin operator will be discussed in Section 3.3.

Dimension-six Four-Quark Operator Contribution
The perturbative coefficients in Eq. (1.1) considered so far are independent of the spectator quark in the D meson, in fact its effect appears only in the corresponding matrix elements of the dimension-five and dimension-six operators. Starting at order 1/m 3 c , there are also one-loop contributions, c.f.Γ 6 in Eq. (1.1), in which the spectator quark is directly involved. These correspond respectively to    Table 7: Note that compared to the terms discussed above, these contributions imply a phase space enhancement factor of 16π 2 . The corresponding ∆C = 0 four quark operators of dimension-six are 9 : where T A is a colour matrix and a summation over colour indices is implied. The parameterisation of the matrix elements of the operators in Eqs. (2.35) -(2.38) in QCD is given in Appendix C. However, by evaluating the matrix elements in the framework of the HQET, one obtains the following set of operators, i.e. 10 8 In the case of semileptonic decays, only the WA topology can contribute. 9 Sometimes, we will use the short-hand notation O q here h v denotes the HQET field defined by Eqs. (2.10), (2.11). The matrix elements of these operators in HQET are parameterised as where q, q = u, d, s,B q i denote the Bag parameters in HQET, withB q 1,2 corresponding to the coloursinglet operators, andB q 3,4 ≡˜ q 1,2 to the colour-octet ones, and F (µ) and f Dq are the HQET and QCD decay constants defined, respectively, as 11 The relation between f D and F up to α s and 1/m c corrections, is given e.g. in Refs. [76,77]. At the scale µ = m c , it reads whereΛ = m D − m c , and the parameters G 1 and G 2 characterise matrix elements of non-local operators. Note that in Eqs. (2.43), (2.44), in expressing the HQET decay constant in terms of the QCD one, we have included only the α s corrections -which become part of NLO dimension-six contribution -but not the 1/m c ones. The latter, as it will be explained in detail in Section 2.6, can be absorbed in the contribution of some of the dimension-seven operators in HQET.
In vacuum insertion approximation (VIA), the Bag parameters of the colour-singlet operators are equal to one,B q 1,2 = 1, and the Bag parameters of the colour-octet operators vanish,˜ q 1,2 = 0. Note that throughout this work we assume isospin symmetry, i.e.
The quantitiesδ q q i in Eq. (2.44) describe the so-called eye-contractions, see Fig. 6, and characterize "subleading" (compared to the large Bag parameters) effects in the non-perturbative matrix elements -in VIA all eye-contractions vanish i.e.δ q q i = 0. However, beyond VIA, the matrix elements of the four-quark operators with external q quarks differ from zero even when the spectator quark q in the D q meson does not coincide with the quark q , as reflected byδ q q i in Eq. (2.44). Note that in our notation the eye-contractions with q = q , are in fact included in the Bag parametersB q i . And again, due to isospin symmetry we will use: The Bag parametersB q i andδ qq i have been determined using HQET sum rules, specifically the Bag parametersB q i for the D +,0 mesons were calculated in Ref. [66], while corrections due to the strange quark mass as well as the contribution of the eye-contractions, see Fig. 6, have been computed for the first time in Ref. [68]. The numerical values of the Bag parameters will be briefly discussed in Section 3.4 and they are summarised in Appendix A.
By considering only the dominant CKM modes and by neglecting the effect of the eye-contractions, at LO-QCD and at dimension-six, the contributions of four-quark operators to the D-mesons decay rate reads respectively, for the WE, PI and WA topologies. Note that in the latter, the contribution due to the muon mass in the semileptonic decay c → sµ + ν µ has been neglected. Here some interesting numerical effects are arising. First, in the charm system, one expects that the contribution due to the spectator quark is of similar size compared to the leading term Γ 3 in the HQE, unless some additional c c q ′ q ′ q Figure 6: Diagram describing the eye-contractions.
cancellations are present. Using the pole mass m Pole c = 1.48 GeV and Lattice QCD values for the decay constants [78] we roughly obtain that This result has led the authors of Ref. [79] to propose a different ordering for the HQE series in the charm sector. However, to investigate further the size of four-quark contributions, we consider the combinations of Wilson coefficients that appear in Eqs. (2.49) -(2.51), i.e.
where the superscript S and O refers to coefficient in front of the colour-singlet and colour-octet Bag parameters, respectively. A comparison of these combinations for different values of the renormalisation scale µ 1 is shown in Table 8.
As one can see, the combination of Wilson coefficients multiplying the colour-singlet Bag parameters of WE are strongly suppressed. Note that, depending on whether we disregard α 2 s corrections in these combinations of ∆C = 1 Wilson coefficients -as we do -or not, we can get even different signs for C S WE at NLO. Moreover, in Eq. (2.49) the Bag parameters of the colour singlet operators exactly cancel in VIA at leading order in 1/m c . The coefficient of the colour-octet operator is on the other hand not suppressed for weak exchange, indicating that both singlet and octet operators might be equally important in this case. For Pauli interference, the combinations of Wilson coefficients multiplying the colour-singlet operators are significantly enhanced compared to those in WE, the same holds for the colour-octet operators. Note that C O PI and C S PI get large modifications (and even a flip of sign) compared to the case C 1 = 1 and C 2 = 0 revealing the importance of gluon radiative corrections. Moreover C O PI is enhanced compared to C S PI , indicating that both singlet and octet operators might be equally important for Pauli interference. For weak annihilation, the corresponding combination in front of the colour-singlet operators is large. On the other hand, the Bag parameters of the colour singlet operators exactly cancel in VIA at leading order in 1/m c .  The above arguments show that, by neglecting the effect of the colour-octet operators in VIA, one might be led to misleading conclusions, and therefore an accurate determination of the deviation of the Bag parameters from their VIA values, using non-perturbative methods like HQET sum rules or lattice simulations, is necessary.
Finally, by including all CKM modes as well as NLO-QCD corrections, the contribution of fourquark operators to the total decay width at order 1/m 3 c schematically reads where the matrix elements of the four-quark operators are given in Eqs. (2.43), (2.44), and the shortdistance coefficients for the WE, PI and WA topologies, c.f. Fig. 5 are denoted by been computed for HQET operators in Ref. [55]. The corresponding results for A WA i,q 1 q 2 can be obtained by Fierz transforming the ∆C = 1 operators given in Eqs. (2.2), (2.3). Since the Fierz symmetry is respected also at oneloop level, the functions A WA i,q 1 q 2 are derived from A WE i,q 1 q 2 by replacing C 1 ↔ C 2 . For the semileptonic modes, the coefficients A WA i,q 1 have been determined in Ref. [56]. Note that in our analysis, we treat the contribution of theδ q q i parameters as a subleading "NLO" effect, therefore their coefficients are included only at LO-QCD. To demonstrate the importance of the NLO-QCD corrections to the spectator effects, we show in Table 9 the dimension-six contributions to the D-meson decay widths (see Eq. (2.57)) splitting the LO and NLO parts, both in VIA and using HQET SR results for the Bag parameters. NLO-QCD corrections turn out to have an essential numerical effect for the four-quark contributions. In the case of the D 0 and D + s mesons these corrections lift the helicity suppression of weak exchange and weak annihilation being present in LO-QCD when using VIA. For the D + s meson -in addition to the CKM dominant WA contribution -there is a correction due to CKM suppressed, but nevertheless large PI topology. In the case of the D + meson the overall contribution from Pauli interference turns out to be huge, of the order of −2.5 ps −1 . In addition, the NLO correction to Pauli interference turn also out to be very large, 50% − 100% of the LO term depending on the mass scheme. Already in the B system this NLO-QCD corrections were found to be of the order of 30% for the ratio τ (B + )/τ (B d ), see e.g. Ref. [54] in the Pole scheme. Thus, neglecting these contributions for charm lifetime studies, as done in Ref. [80], is clearly not justified and a knowledge of NNLO-QCD corrections to the four-quark contributions would be highly desirable.

Dimension-seven Four-Quark Operator Contribution
The dimension-six four-quark operator contribution discussed in the previous section, is obtained by neglecting in the expression of the incoming momentum p µ = p µ c + p µ q the effect due to the small momentum of the light spectator quark p q ∼ Λ QCD . Including also corrections linear in the quantity p q /m c , leads to the contribution of order 1/m 4 c to Γ(D), which can be described in terms of the following basis of dimension-seven operators, defined in full QCD, i.e. 12 together with the corresponding colour-octet operators S q 1 , S q 2 , S q 3 , containing the generators T A , and again a summation over colour indices is implied. Due to the presence in Eqs. (2.59), (2.60) of a covariant derivative acting on the charm quark field, which scales as m c at this order, there is no immediate power counting for these operators, cf. the HQET operators in Eqs. (2.62), (2.63). Moreover, note that this basis differs from the one used in Ref. [58] for the computation of dimensionseven and dimension-eight contributions.
In order to evaluate the matrix element of the dimension-seven four-quark operators using the framework of the HQET, one has to further expand the charm quark momentum, according to p µ = m c v µ + k µ + p µ q , see Eq. (2.9), as well as to include 1/m c corrections to the effective heavy quark field and to the HQET Lagrangian, retaining only terms linear in k/m c and p q /m c . The small residual momentum of the charm quark k µ will result in a covariant derivative acting on h v and the small momentum of the spectator quark p µ q will result in a covariant derivative acting on the light quark 12 Note that e.g. in Ref. [56] it is used a redundant basis, containing also the additional operator, denoted by P q 2 , which however is related to P q 1 by hermitean conjugation, namely P q 2 = mq (c(1 + γ5)q)(q(1 + γ5)c) = (P q 1 ) † . Since it leads to the same matrix element, we do not include this operator in our basis.  Table 9: Dimension-six contributions to D-meson decay widths (see Eq. (2.57)) (in ps −1 ) and split up into LO-QCD and NLO-QCD corrections within different mass schemes and both in VIA and using the HQET SR for Bag parameters.
field q. In this case, one obtains the following basis, which includes the local operators supplemented by the corresponding colour-octet operatorsS q 1,2,3 andŨ q 1,2 , and the non-local operators also supplemented by the corresponding colour-octet operators 13 . We see that, compared to the QCD basis, there are in addition the two local operatorsR q 1 andR q 2 (and also the corresponding colour-octet ones), which emerge from the expansion in Eq. (2.11), and the four non-local operatorsM q 1,π ,M q 2,π , M q 1,G andM q 2,G (and the corresponding colour-octet ones) which are obtained by taking the timeordered product of the dimension-six operators with the 1/m c correction to the HQET Lagrangian, see e.g. Ref. [71] for details.
We parametrise the matrix elements of the operators in Eqs. (2.61) -(2.69) using VIA and account for deviations from it by including the corresponding Bag parameters, as it is explicitly shown in Appendix C. However, since for these matrix elements there is no non-perturbative evaluation available yet, in our analysis we have to rely only on VIA. It follows that, at LO-QCD the matrix element of the dimension-seven operators listed above, can be expressed in terms of the HQET nonperturbative parameters F (µ), G 1 (µ), G 2 (µ), andΛ, so far determined only with large uncertainties. For this reason, we prefer to use as an input the QCD decay constant f D , which is computed very precisely using Lattice QCD [78]. In doing so, we obtain that in VIA and at the matching scale µ = m c , the contribution of the local operatorsR q 1,2 as well as that of the non-local onesM q 1,π , M q 2,π ,M q 1,G andM q 2,G can be entirely absorbed in the QCD decay constant f D , cf. Eq. (2.47) (more precisely, in the matrix element of the dimension-six QCD operators in Eqs. (2.35), (2.36), which are proportional to f D ), so that we are left only with the 1/m c contribution due to the operatorsP q 1,2,3 , analogously to the QCD case 14 .
To make this point more clear, we consider as an example the contribution due to Pauli interference at LO-QCD and up to order 1/m 4 c , in the case of c → sdu transition, which constitutes the dominant correction to Γ(D + ), with C S PI defined in Eq. (2.55). By evaluating the matrix element of ImT PI in VIA, the contribution due to the colour-octet operators vanishes. Moreover, using the parametrisation for the matrix 13 Operators which vanish due to the equation of motion (iv · D)hv = 0 are not shown. 14 In the matrix element ofP q 1,2,3 one can replace the HQET decay constant with the QCD one, up to higher order corrections.
elements of the four-quark operators given in Eq. (2.43) and in Appendix C, we obtain in VIA and setting µ = m c , that where in the second line we have used the conversion between the QCD and HQET decay constants given in Eq. (2.47), showing that the contribution of the local operatorsR q i and non-local operators M q i,π andM q i,G is entirely absorbed in the QCD decay constant. Note that, by neglecting the effect due to the strange quark mass and using VIA we reproduce the approximate result of Eq. (19) in Ref. [79].
The same argument applies also to the remaining topologies i.e. WE and WA. However, it is worth mentioning that in VIA and neglecting the strange quark mass, the contribution of WE and WA exactly vanishes at LO-QCD, due to the helicity suppression. This suppression is lifted once the s-quark mass or perturbative gluon corrections are included, and in this case it becomes again manifest that the contributions ofR q i ,M q i,π andM q i,G in HQET can be completely absorbed in f D by evaluating the matrix elements in VIA 15 . We note that a detailed analysis of the dimension-seven contributions within the HQET has been performed in Ref. [77] for the case of B −B-mixing. Specifically, it was found that in VIA, subleading power corrections due to non-local operators can be entirely absorbed in the definition of the QCD decay constant, and that the residual 1/m b corrections, due to the running of the local dimension-seven operators from the scale m b to µ ∼ 1 GeV, is numerically small (∼ 5% for Ref. [77]). 16 Finally, by summing over all the CKM modes, at LO-QCD, the dimension-seven contribution can therefore be presented as (with q = u, d, s) where the matrix elements of the dimension-seven operators are presented in Appendix C. We confirm the results for the short-distance coefficients G WE i,q 1 q 2 , G PI i,q 1 q 2 and G WA i,q 1 q 2 , G WA i,q 1 presented in Ref. [56]. Note that, due to the current accuracy of the analysis, at dimension-seven we include only the contribution of the valence-quark, therefore e.g. D 0 |P s i |D 0 = 0. Numerical values of the dimensionseven contributions to the decay rates and the ratios will be presented in Section 4. In Table 10 we show the central values of the dimension-seven contributions in ps −1 in the kinetic mass scheme and 15 Note, that for the operator O q 2 the contribution of R q 2 is absorbed by the combination (mD fD/mc) 2 ≈ (1 + 2Λ/mc) f 2 D . 16 By neglecting the effect of running down to a lower scale, from Ref. [77] one can see that in VIA the QCD decay constant entirely absorbs all the 1/m b contributions.
we find for the D + meson a correction that is almost as large as the leading dimension three term, see Table 5.

Determination of the Non-perturbative Parameters
In the present section, we discuss the numerical determination for the matrix elements of the operators introduced in Sections 2.2 -2.6. We start with the operators of the lowest mass dimension.

Parameters of the Chromomagnetic Operator
For the B system many of non-perturbative parameters have been determined by performing fits to the experimental data for inclusive semileptonic decays [81]. In the case of the chromomagnetic operator, one finds Assuming heavy quark symmetry we expect the corresponding parameter in the D system to have a similar size. Another way of estimating the value of µ 2 G is to use the well-known spectroscopy relation [82]  Thus, from Eq. (2.28), we expect corrections to the total decay rate due to the chromomagnetic operator, c G µ 2 G /(c 3 m 2 c ) ranging between −6% and +8% with respect to the leading free-quark decay contribution. A large part of the sizable uncertainty derives from the cancellations in the coefficient c G , shown in Table 6 and Fig. 3, which could be reduced with a complete determination of the NLO-QCD corrections to c G . For semileptonic rates the contribution of the chromomagnetic operator can be even of the order of 20%, see Section 4.3.
An experimental determination of µ 2 G (D) from inclusive semileptonic D-meson decays could further reduce the uncertainties and could in particular give some insight into the numerical size of SU (3) F breaking.

Parameters of the Kinetic Operator
For the matrix element of the kinetic operator no precise determination is available so far in the charm sector. Several predictions of µ 2 π available in the literature for the B-meson cover a large range of values, see Table 11. Assuming heavy quark symmetry one can use the value obtained from the fit   In the above, we have again added an conservative uncertainty of 40% to account for the breaking of the heavy quark symmetry. This value clearly fulfills the theoretical bound µ 2 π ≥ µ 2 G , see e.g. the review [85]. Thus we expect from Eq. (2.28) corrections due to the kinetic operator of the order of −10%, which is also found in Section 4.3 -both for the total decay rate and the semileptonic one.
The SU (3) F breaking effects for the kinetic operator have been estimated in Refs. [56,86] leading to the following estimate we use for the D s meson: Again a more precise experimental determination of µ 2 π from fits to semileptonic D + , D 0 and D + s meson decays -as it was done for the B + and B 0 decays -would be very desirable.

Parameters of the Darwin Operator
For the matrix element of the Darwin operator no theoretical determination for the charm sector is available. We again could assume heavy quark symmetry and use the corresponding value in the B-system, obtained from fits of the semileptonic decays [81]: and add quadratically an uncertainty of 40% for the transition from the B to the D system, leading to a first estimate of Alternatively the Darwin parameter can be related to the matrix elements of the dimension-six fourquark operators through the equation of motion for the gluon field. At leading order in 1/m Q one obtains:  Table 17 in Appendix C. The strong coupling g s has its origin in the non-perturbative regimes -e.g. Ref. [87] suggests to set α s = 1.
Using the input from the Appendix A and applying Eq. (3.13) we derive estimates of ρ 3 D for Band D-mesons both in VIA and using the HQET SR results for the Bag parameters. The values are summarised in Table 12 for the three different choices, α s (µ = 1.5 GeV), α s (µ = 1 GeV) and α s = 1. Setting α s = 1 in Eq. (3.13), yields values for ρ 3 D that are close to the one determined from the fit of semileptonic B meson decays, Eq. (3.11), indicating 1/m b -corrections in Eq. (3.13) of the order of +30%. Moreover, we find that VIA gives in Eq. (3.13) values, which are very close to the HQET sum rule ones. We emphasise that due to the sizeable SU (3) F breaking in the decay constants, Eq. (3.13) leads also to a sizable SU (3) F breaking for the non-perturbative parameters ρ 3 D (D), ρ 3 D (D + s ). Taking the values corresponding to α s = 1 and using HQET SR results for the bag parameters we get the second estimate (last column in Table 12) where we have again added a 40% uncertainty. Finally, another possibility to extract ρ 3 D (D) is to substitute in Eq. (3.13) the values of the Bag parameters in VIA, which gives Assuming a similar size for the strong coupling in both the B-and D-meson matrix elements, from Eq. (3.15) one obtains: Using the most precise determination of the decay constants from Lattice QCD [78], and of the meson masses from PDG [1] and taking into account the value of ρ 3 D (B) in Eq. (3.11), leads to the following estimates: where we again assign in addition a conservative 40% uncertainty due to missing power corrections. These values are consistent with the numbers shown in Table 12 for α s = 1. Contrary to the case of the dimension-five non-perturbative parameters, in Eq. Again, here a more precise experimental determination of ρ 3 D from fits to semileptonic D + , D 0 and D + s meson decays -as it was done for the B + and B 0 decays -would be very desirable and could have a significant effect on the phenomenology of inclusive charm decays.

Bag parameters of Dimension-six and Dimension-seven
The dimension-six Bag parameters of the D + and D 0 mesons have been determined using HQET Sum Rules [66]; strange quark mass corrections, relevant for the Bag parameter of the D + s meson, as well as eye-contractions have been computed for the first time in Ref. [68]. The results are collected in Table 17 and the HQET sum rules suggest values for the Bag parameter that are very close to VIA.
For the dimension-seven Bag parameters (defined in HQET), we apply VIA. As one can see from Appendix C, the matrix elements of dimension-seven operators in HQET depend also on the parametersΛ (s) = m D (s) − m c , for which we use the following ranges [68] Λ = (0.5 ± 0.1) GeV, Λ s = (0.6 ± 0.1) GeV.

Numerical Results
In this section, using all the ingredients described above, we present the theoretical prediction for the total and semileptonic decay rates, and for their ratios. All the input used in our numerical analysis are collected in Appendix A. For each observable, we investigate several quark mass schemes (with the kinetic scheme as default) and compare the corresponding results using both VIA and HQET SR values for the Bag parameters. The uncertainties quoted below are obtained by varying all the input parameters within their intervals. For the renormalisation scales, we fix the central values to µ 1 = µ 0 = 1.5 GeV and vary both of them independently between 1 and 3 GeV. Moreover we add an estimated uncertainty due to missing higher power corrections. The results are discussed in the following subsections and they are summarised in Tables 13, 14, 15 and in Fig. 7.

The Total Decay Rates
We start by investigating the theory prediction of the total decay rates, which are expected to have sizable uncertainties due to the dependence of the free quark decay on the fifth power of the charm quark mass and due to large perturbative and power corrections.  Table 14: Central values of the charm observables in different quark mass schemes using HQET sum rule results [66,68] for the matrix elements of the 4-quark operators compared to the corresponding experimental values (last column).
prediction of the decay widths in several mass schemes are shown in the three first rows of Table 13, using VIA for the Bag parameters and of Table 14 using the HQET sum rules results. In Table 15 we show the theoretical prediction including the corresponding uncertainties within the kinetic mass scheme and using the HQET SR values for the dimension-six Bag parameters -the same result is visualised in Fig. 7. In each table, the corresponding experimental determinations are listed in the last column. For the D + s meson an additional subtlety is arising due to the large branching fraction of the leptonic decay D + s → τ + ν τ , which is not included in the HQE, as the tau lepton is more massive than the charm quark. Using the experimental value of the leptonic branching ratio [1] (online update) we therefore define a reduced decay rateΓ(D + s ): leading also to a reduced lifetime ratioτ  Table 15: HQE predictions for all the ten observables in the kinetic scheme (second column), using HQET SR results for the Bag parameters. The first uncertainty is parametric one, second and third uncertainties are due to µ 1 -and µ 0 -scales variation, respectively. The results are compared with the corresponding experimental measurements (third column).
The first and main result we deduce from Table 15 and Fig. 7, is that the HQE gives values of Γ(D 0 ), Γ(D + ) and Γ(D + s ) which lie in the ballpark of the experimental numbers. Looking closer we find that our prediction for Γ(D s ) is in good agreement with experiment (within large uncertainties), while the total decay rates of the D 0 and D + mesons are underestimated. As a reason for that we suspect missing NNLO-QCD corrections to the free charm quark decay. Second, using different mass schemes yields similar results, and further higher order correction will reduce the differences between these schemes. Due to the fact that the values of the HQET Bag parameters [66,68] listed in Table 17 are close to the corresponding ones in VIA, the predictions shown in Table 13 and in  Table 14 do not differ much. A peculiar role is played by the D + meson, where we get a huge theoretical uncertainty stemming from the large negative value of the Pauli interference contribution at dimension-six. This term actually dominates the total decay rate. Moreover the large negative value is further enhanced by NLO-QCD corrections, but partly compensated by the dimension-seven contribution. Here further studies of the Bag parameters, e.g. via an independent confirmation of the HQET sum rule results with lattice QCD, as well as calculation of higher order QCD corrections to dimension-six and dimension-seven might yield deeper insights.
In order to further analyse the size of the individual contributions to the total decay rate, we show below the numerical coefficients of each non-perturbative parameter in the HQE, using the central values for the input in Appendix A and (as an example) the kinetic scheme for the charm mass with µ cut = 0.5 GeV, namely 17 to indicate deviations from VIA and we conservatively normalise δB q i to 0.02. The matrix elements of the colour-octet operators are normalised to −0.04 -here using the central value of the HQET determination for˜ q 2 might underestimate its effect due to the quoted HQET uncertainties. Furthermore, we introduce also the ratios r qq i ≡δ qq i / δ qq i , with δ qq i being the central values listed in Table 17. For the neutral D meson we find a convergent series, with the largest correction due to the QCD corrections to the free quark decay and the contribution of the Darwin operator. Here a calculation of the NNLO-QCD corrections to the free-quark decay would be very desirable, as well as a more profound determination of the value of the matrix element of the Darwin operator. Note that since we take as a central value µ 1 = 1.5 GeV, the coefficient of the chromomagnetic operator in Eq. (4.4) turns out accidentally to be very small, see Fig. 3. In fact, varying the renormalisation scale µ 1 between 1 and 3 GeV one finds quite sizable contribution of ∼ 5 − 10% due to µ 2 G (D). Because of the helicity suppression, we get only small contributions from the weak exchange diagrams. In LO-QCD and VIA these corrections actually vanish, the small value ≈ −0.01 stems from NLO-QCD corrections, which break the helicity suppression. Nevertheless, depending on the size of the˜ q i , the colour-octet operator could give contributions of a similar size as the kinetic operator. Finally, according to the HQET SR determination, the numerical effect of the eye-contractions does not seem to be pronounced. 17 Here and hereafter, in the Bag parameters we use the same label q both for u or d-quarks, reflecting the isospin Similarly, we get for the D + -meson decay width: where we observe huge negative corrections due to Pauli interference. In VIA we get from dimensionsix (summing LO and NLO-QCD) a ≈ −270% correction to the LO-free-quark decay. Dimensionseven yields a large positive correction of +110%. Because of the almost perfect cancellation between the three dominant terms, 16π 2 Γ (0) 6 + α s /πΓ VIA /m 4 c , the HQE series for Γ(D + ) becomes very sensitive to sub-dominant terms, e.g. higher order QCD corrections toΓ 6 ,Γ 7 , Γ 3 , Γ 5 and Γ 6 , and to deviations of the Bag parameter from VIA. In this case it might also be interesting to further study estimates of higher orders in the HQE, see e.g. Refs. [57,58]. Else, we get for the two-quark ∆C = 0 contributions the same (due to isospin) size of corrections as in the D 0 case and we find, based on the HQET sum rules estimates, again that the eye-contractions give only tiny corrections. where we find again a converging series with the dominant contribution coming from the NLO-QCD corrections to the free quark decay and the Darwin term. For the latter a more reliable determination of the corresponding non-perturbative matrix elements would be highly desirable. In VIA, the fourquark operators show again a pronounced cancellation between dimension-six and dimension-seven contributions.

The Lifetime Ratios
In order to eliminate the contribution of the free-quark decay, we calculate the lifetime ratios as where Γ HQE (D 0 ) and Γ HQE (D + (s) ) are given in Eqs. (4.4) and (4.6), (4.7), respectively. In these ratios, Γ 3 cancels exactly and Γ 5 and Γ 6 cancel up to isospin or SU (3) F breaking corrections in the corresponding non-perturbative matrix elements. The lifetime ratios should then be dominated by the contribution of four-quark operators.
The central values for the HQE prediction of the lifetime ratios in several mass schemes are shown in the fourth and fifth rows of Table 13, Table 14, Table 15 and in Fig. 7 and it turns out that the large lifetime ratio τ (D + )/τ (D 0 ) is well reproduced in all schemes, while in the case of τ (D + s )/τ (D 0 ) the HQE predictions lie closer to one compared to the experimental values. The latter theory result is dominated by SU (3) F breaking differences of the non-perturbative matrix elements µ 2 π , µ 2 G and ρ 3 D , which are only very roughly known, see Section 3. With future, more precise determinations of these parameters our conclusion might significantly change for this lifetime ratio.
The large lifetime ratio τ (D + )/τ (D 0 ) can be expressed as In VIA, we predict a lifetime ratio of 2.5, which is already quite close to the experimental value. Again, we observe here a sizable cancellation between dimension-six and dimension-seven contributions. In order to improve the theoretical prediction, a more precise determination of the Bag parameters of the colour-octet operators is mandatory, as well as of the perturbative higher order QCD corrections inΓ 6 andΓ 7 . And finally we get for the lifetime ratio τ (D + s )/τ (D 0 ): With the estimates of µ 2 π , µ 2 G and ρ 3 D from Section 3 we find that the largest individual SU (3) F breaking effect (≈ −7%) comes from the Darwin term. Using VIA we obtain a correction of +5% due to the four-quark contributions of dimension-six and dimension-seven -finite values of the matrix elements of the colour-octet operators as well as of δB s 1,2 might lead to numerically similar effects. Else we have a large number of smaller SU (3) F breaking effects, which can be both positive and negative.

The Semileptonic Decay Widths and Their Ratios
For discussing the inclusive semileptonic decays of D mesons, we introduce the short-hand notations Γ D sl ≡ Γ(D → Xe + ν e ) and B D sl ≡ Br(D → Xe + ν e ). We determine the theory value of the semileptonic branching ratio as (4.11) The central values for the HQE prediction of the lifetime ratios in several mass schemes are shown in the sixth, seventh and eighth row of Table 13, Table 14 and Table 15 and in Fig. 7.
The semileptonic decay rate of the D 0 meson can be written (in the kinetic scheme) as where as for the total D 0 -meson decay width we find a converging series, with the largest correction due to the dimension-five operators, followed by the Darwin operator contribution and the NLO-QCD corrections to the free quark decay. Note that only the non-valence four-quark operator contributions (eye-contractions) are present here.
For the semileptonic D + -meson decay we obtain where we have a larger contributions due to CKM dominant weak annihilation as well as SU (3) F breaking corrections. Again, in VIA the four-quark contributions vanish both at dimension-six and dimension-seven, but now deviations from VIA might give sizable corrections.
Using the experimental values for the D 0 lifetime and semileptonic branching fraction, we determine the semileptonic ratios in the following way The HQE values of these ratios are shown in the ninth and tenth rows of Tables 13, 14 and 15 and in Fig. 7. In agreement with experiment HQE predicts values for Γ D + sl /Γ D 0 sl very close to one. Using the inputs from Appendix A the HQE prefers also for Γ D + s sl /Γ D 0 sl values close to one, while experiment find a value as low as 0.79 -again a more profound determination of µ 2 G , µ 2 π and ρ 3 D as well as an inclusion of dimension-seven contributions with two-quarks operators for D mesons might change this conclusion.
We Due to isospin symmetry, in Eq. (4.17) the contributions of the kinetic, chromomagnetic and the Darwin operators vanish. Moreover, in VIA there is also no correction due to the spectator quark effects. Thus this ratio, within the framework of the HQE, is predicted to be very close to one.
Finally, we obtain for the ratio which is dominated by SU (3) F -symmetry breaking corrections. The Darwin operator gives a sizable positive contribution to the ratio, which is partly compensated by the kinetic and the chromomagnetic terms. The spectator effects give in VIA a vanishing contribution, but deviations from VIA could sizably affect the ratio and also eye-contractions could yield a visible effect -here again a more precise determination of the non-perturbative parameters is necessary in order to make more profound statements.

Conclusions and Outlook
We have performed a comprehensive study of charmed mesons lifetimes, of their ratios and of the inclusive semileptonic decay rates. Compared to previous studies we have included for the first time the sizeable contribution due to the Darwin term in the charm sector (with new expressions shown in Appendix B), non-perturbative estimates of the eye-contractions [68] and strange quark mass corrections to the Bag parameters of the D + s meson [68]. Moreover we have studied different mass schemes for the charm quark.
In particular our new study supersedes the one done by some of us in Ref. [66] and we could clarify in the present work that the dimension-seven operatorsR q 1,2 (introduced in Ref. [66] as P q 5,6 ) can be absorbed in the definition of the QCD decay constant. In contrast to the present work, Ref. [66] could describe the experimental number for τ (D + s )/τ (D 0 ) by fitting the Bag parameters in order to accommodate the experimental value of Γ D + s sl /Γ D 0 sl -this can be achieved by demanding e.g. for the differenceB s 1 −B s 2 ≈ 0.032, which is in slight tension with the HQET sum rule result B s 1 −B s 2 = 0.004 +0.019 −0.012 we are using here. Ref. [80] also studies charm mesons, albeit restricting exclusively to LO-QCD expressions. Different quark mass schemes can only be differentiated starting from NLO-QCD onwards -working at leading order in QCD only, the different quark mass schemes used in our work would induce a relative uncertainty of the free-quark decay of (1.48/1.27) 5 ≈ 2.15, which is clearly not acceptable. Moreover, as can be nicely read from Table 9, NLO-QCD corrections to the four-quark operators can dominate over the LO contribution. Using exclusively LO-QCD expressions is thus a far too crude and unnecessary assumption in the charm sector. Finally, Ref. [80] considers only the MS scheme for the charm quark mass and obviously the recently determined Darwin term and the eye-contractions could not have been included, since they were not known at that point of time.
Finally there is also some overlap with two recent studies of the B c lifetime [88,89]. The first paper [88] considers also the free charm quark decay Γ 3 and the second one [89] the total D meson decay rate without the free charm quark decay, i.e. Γ(D)−Γ 3 . For Γ 3 the authors of Ref. [88] consider three quark mass schemes: MS, 1S and the meson mass scheme. They find in Table 3 and 4 of their paper values in the MS/1S scheme which are slightly smaller/slightly larger than our values in Table  5: 1.0 ps −1 vs 1.3 ps −1 and 1.7 ps −1 vs 1.5 ps −1 . Since they in principle use the same NLO-QCD expressions as we do, we expect the slight difference to root in a different treatment of higher orders in α s and some differences in the values of the input parameters. As in our study, they also find a relatively small effect due to a non-vanishing strange quark mass. In Ref. [89] the authors determine the D decay rate without the free charm quark decay. In that respect they consider all the corrections we also take into account, except contributions of dimension-seven and eye-contractions. In the end, when considering the D + meson they obtain values for the B c -meson decay rate of around 3.3 ps −1 (see Table III of Ref. [89]), compared to the experimental value of 1.961(35) ps −1 . We naively estimate that an inclusion of the dimension-seven contribution to the D + meson decay rate would decrease their result by about 1.1 ps −1 , see Table 10, to bring it in nice agreement with the measurement. On the other hand, these missing dimension-seven contributions might be partially compensated by the corresponding contributions to the B c -meson decay rate. Here a further investigation might be necessary to clarify this point.
Our main numerical results are presented in Tables 13, 14 and 15 and in Fig. 7. At a first glance all considered observables lie in the ballpark of the experimental results. In particular, we find good agreement with experiment for the ratio τ (D + )/τ (D 0 ), for the total D + s -meson decay rate, for the semileptonic rates of all three mesons D 0 , D + and D + s , and for the semileptonic ratio Γ D + sl /Γ D 0 sl . The values obtained with different mass schemes for the charm quark overlap and the exclusive use of only one scheme might underestimate the uncertainties. Including higher orders in the perturbative QCD expansion will further alleviate the differences among the mass schemes. Looking at the structure of the contributions to the total decay rates and neglecting spectator effects for a start, we find that the NLO-QCD corrections to the free quark decay give the dominant correction (of the order of 50% of LO-QCD free quark decay), followed by the Darwin term (of the order of 30% of LO-QCD free quark decay). In the case of semileptonic decay rates the chromomagnetic term provides the dominant contribution (of the order of 30%), followed by the Darwin term and NLO-QCD corrections to the free quark decay. Turning now to the spectator effects, we find them to be tiny for Γ(D 0 ), Γ D 0 sl and Γ D + sl , but they provide visible corrections to Γ D + s sl and Γ(D + s ) -in the latter case we find also sizable cancellations between dimension-six and dimension-seven contributions. For the D + meson we find, however, a huge negative Pauli interference contribution -with a substantial part stemming from the NLO-QCD corrections. Moreover, one observes here a significant cancellation between dimension-six and dimension-seven terms related to Pauli interference. The values of the HQET Bag parameters entering the spectator effects are close to the VIA values, deviations from the latter can, however, lead to sizable effects in Γ(D + ) and to visible effects in Γ(D 0 ), Γ(D + s ) and Γ D + s sl . Based on the HQET sum rule results [68] we find that eye-contractions constitute only subleading corrections, they might, however, turn out to be relevant for Γ D + s sl /Γ D 0 sl and τ (D + s )/τ (D 0 ), when more precise non-perturbative estimates will become available. In the end, the total decay rates of the D 0 and D + mesons stay underestimated in our HQE approach and we suspect that this is due to missing higher-order QCD corrections to the free charm quark decay and the Pauli interference contribution. For the SU (3) F breaking ratios τ (D + s )/τ (D 0 ) and Γ D + s sl /Γ D 0 sl our predictions lie closer to one than experiment. This might originate from the poor knowledge of the non-perturbative parameters µ 2 G , µ 2 π and ρ 3 D in the D 0 and D + s systems, as discussed in Section 3. Our numerical analysis shows that there are many possibilities for future improvements of the HQE predictions in the charm sector: • Γ 3 : NNLO-QCD [27][28][29][30][31][32][33][34][35][36][37][38] contributions to the semileptonic decays have been found to be large and NLO-QCD corrections to the non-leptonic decay rates represent one of the dominant corrections. Moreover we observe that at NLO-QCD there is pronounced cancellation -see Eq. (2.17) and Eq. (2.18) -which might not be necessarily present at NNLO-QCD. Thus a first determination of the NNLO-QCD corrections to the non-leptonic decays might have some sizable impact on the numerical studies of the total decay rates.
• Γ (1) 5 : Cancellations in the coefficient c G for the total decay rate, shown in Eq. (6) and Fig. 3 lead to large uncertainties, even the sign of these corrections is ambiguous. Here a determination of the QCD-corrections to the coefficient c G for the non-leptonic case might considerably improve the situation.
• Γ (1) 6 : The Wilson oefficients of the Darwin operator are large, therefore QCD corrections for the non-leptonic case might be important.
• Γ (0) 7,8 : Since the dimension-six contribution is sizable, the LO-QCD determination of the dimensionseven and dimension-eight contributions with two-quark operators for the non-leptonic case might bring some additional insights on the convergence of the HQE in the charm sector. •Γ 6 andΓ (0) 7 are known and their numerical values were found to be huge, see e.g. Table 9. Thus further QCD corrections will turn out to be very important.
•Γ (0) 8 : since the four-quark dimension-six contribution can dominate the total decay rate andΓ (0) 7 is also very sizable, a further study of the dimension-eight contributions might bring further insights on the convergence of the HQE in the charm sector, see Refs. [57,58].
• More precise determinations for the parameters µ 2 G , µ 2 π and ρ 3 D -both for the D 0 and the D + s mesons: the Darwin term and the chromomagnetic term provide large corrections to the decay rates and they are poorly known -in particular the size of SU (3) F breaking effects is largely unknown. An experimental determination of µ 2 G , µ 2 π and ρ 3 D from fits to semileptonic D + -, D 0 -and D + s -meson decays -as done in the B system, see e.g. Ref.
[81] -would be very desirable. This might be doable at BESIII, Belle II and a future tau-charm factory. Moreover, new theoretical determinations, e.g. via lattice simulations or sum rules could be undertaken.
• Independent lattice determination of the matrix elements of the four-quark operators of dimensionsix: here we have currently only HQET sum rule determinations [66,68] or outdated lattice results [90,91].
• A first non-perturbative determination of the matrix elements of the dimension-seven four-quark operators in order to test the validity of VIA. A similar endeavour has already been performed for B s mixing [92].
Overall, we find that the HQE can describe inclusive charm observables, in which no pronounced GIM cancellation arises 19 , albeit with very large uncertainties. We therefore do not observe a clear signal for a breakdown of the HQE in the charm sector or of violations of quark hadron duality, see e.g. Ref. [94] and we presented a long list of potential theoretical improvements, which might shed further light into the convergence properties of the HQE in the charm sector.

A Numerical Input
We use five-loop running for α s (µ) [7] with four active flavours at the scale µ ∼ m c , and the most recent value [1] α s (M Z ) = 0.1179 ± 0.0010. (A.1) For the CKM matrix elements we apply the standard parametrisation in terms of θ 12 , θ 13 , θ 22 , δ and use as an input [95] (online update) The values of the non-perturbative parameters used in the analysis are shown in Tables 16 and 17.

B Expressions for the Darwin Coefficients
The coefficients C The numerical values of the above coefficients for ρ = 0.006 are shown in Table 18.
The expressions in Eqs. (C.3) -(C.7) can be obtained using a general parametrisation of matrix elements of the HQET quark currents with a heavy pseudo-scalar meson M (see e.q. Ref. [71]): and for the non-local operators: where Γ is a generic Dirac structure, and Since we are limited to LO-QCD for the dimension-seven contribution, we can just replace the HQET decay constant F (µ) by the full QCD one f D , using F (µ) = f D √ m D .