Alternative Treatment of the Quark Mass in the Heavy Quark Expansion

The treatment of the quark mass plays an important role when it comes to increasing the precision of the predictions of the heavy quark expansion for inclusive heavy hadron decays. Various short-distance mass schemes have been invented to minimize the uncertainties induced by the quark mass, which needs to be extracted from other, independent observables. We suggest to replace the quark mass directly by an observable such as e.g. the inverse moments of the cross section for $e^+ e^- \to $ hadrons. We investigate this alternative strategy and study its impact on the perturbative series.


Introduction
Calculations in perturbative QCD have reached impressively high orders in the expansion in the strong coupling α s , opening new perspectives for the precision of theoretical predictions.However, since quarks and gluons are not the asymptotic states in QCD, one necessarily has to deal with the non-perturbative aspects of QCD to make predictions for observable quantities.Currently, in many cases, the non-perturbative part is the main factor limiting the precision of QCD predictions.
In processes where a large scale Q is present (such as a large momentum transfer or a large mass), many observables can be defined in terms of an Operator Product Expansion (OPE), which allows a factorization of short-from long-distance contributions.While the former can be computed in perturbation theory as a series in α s (Q 2 ), the latter are parametrized in terms of hadronic matrix elements of operators expressed in terms of quark and gluon fields.These matrix elements have to be fixed by independent input, either from experiment or using non-perturbative methods such as lattice QCD.
In many cases the leading term of the OPE is just the partonic, i.e. the perturbative contribution.It has been noticed long ago that the perturbative expansion of the shortdistance contributions is not a convergent series in α s , it can be at best an asymptotic series.This means that, starting at some order in the α s expansion, the coefficients of the perturbative series start to grow, leading eventually to a divergent behaviour, although the first few terms look like a convergent series.In fact, certain divergent contributions have been identified by summing the leading terms of a (formal) large n f , the number of active quark flavours, expansion, which are diagrammatically presented by "bubble chains".These diagrams lead to a factorial divergence which is still Borel summable, but the Borel transform exhibits poles on the positive real axis, the so called infrared renormalons (for a review on this see [1]).
These infrared renormalons lead to ambiguities in the expressions for the observables, which relate to the power suppressed terms in the OPE and also (in the cases we shall discuss) to the definition of the quark mass.In turn, neither the quark mass nor matrix elements appearing in the power-suppressed terms of the OPE are physical quantities, since they need to be defined.
In heavy-quark physics, the large scale is set by the mass of the heavy quark, consequently the OPE is a series in inverse powers of the heavy-quark mass.The starting point of any perturbative calculation is usually the pole mass for the quarks, which, however, is a purely perturbative concept.Its advantage is that it is gauge and scale independent.However, its disadvantage is that it suffers from infrared renormalons [2,3,4], leading to ambiguities of order Λ QCD in the definition of the heavy quark mass.Strictly speaking, this would make a systematic heavy-quark expansion in this parameter impossible.
To this end, many different mass definitions have been invented, tailored to the specific application.At high energies, the MS mass is often used, but this is restricted to scales µ ≥ m [5].In heavy-quark physics, a mass definition that can be used at scales µ ≤ m is needed, such as the kinetic mass [6,7,8] or the 1S mass [9,10].These mass definitions solve the problem of the renormalon ambiguity, and can be determined quite precisely from other independent sources.
Although both mass schemes are well established in B-meson decays, transferring similar mass definitions to the lighter D-meson decays is more challenging [11,12].In the kinetic mass scheme, a hard cut-off scale µ is introduced, which has to be perturbative, i.e. α s (µ) < 1.However, it should also satisfy µ/m Q < 1.While this can be achieved for the b quark by choosing µ ∼ 1 GeV, there is no window of this kind for the charm quark.Likewise, in the 1S scheme, one assumes that the 1S state of a Q Q quarkonium can be treated as a Coulombic system, i.e. can be treated perturbatively.While this seems to be valid for the bottomonium system, this is clearly not true for the corresponding 1S charmonium.
The usual strategy for practical calculations is to define the quark masses as well as the matrix elements appearing as power corrections in the OPEs of observables within a specific scheme and to obtain values for these parameters from fits to data.However, these parameters do not have any physical meaning, so one may also take the point of view that these parameters only transport the information obtained form one observable another observable.This suggests a slightly different strategy for perturbative QCD calculations in the framework of OPE.
It is generally assumed that the ambiguities induced by infrared renormalons cancel between the perturbative series and properly defined non-perturbative quantities, including also the quark mass, such that observables are free of such ambiguities.In fact, this has been shown explicitly in the bubble chain approximation for both the HQE and the e + e − moments [3,4].In order to fix these ambiguities, one has to chose for each parameter an additional observable, which is also computed in terms of an OPE up to the desired order in the α s and the power expansion.
Usually these parameters are extracted explicitly and used as general parameters.However, one may as well proceed in an alternative way and solve the expression obtained from the OPEs for the parameters and eliminate the parameters from the expressions for the observables.
In this way, one constructs perturbative relations between observables without any reference to quark masses or the parameters appearing in the power corrections of the OPE.We note that the OPEs for the observables has to be truncated at some order of the power expansion, which implies that the perturbative relations between the observables will still suffer from divergences, which are due to renormalon ambiguities induced by higher orders in the OPE [13].However, one would expect that the divergent behaviour of the perturbative series is shifted to higher and higher orders in α s , the more power corrections are included and fixed by new observables.
In the present paper, we shall discuss this new treatment quantitatively for the Heavy Quark Expansion (HQE) and the inverse moments of the e + e − → (heavy) hadron cross section focused on eliminating the heavy quark mass.In the next section, we will give more details on the general strategy and set the stage for the discussion of the role of the heavy quark mass in the HQE.In Sec. 3, we first discuss the moments of the cross section for e + e − → heavy hadrons, construct the perturbative relations between different inverse moments and study the resulting series in α s .We then apply the same strategy to the inclusive B → X u ℓν decay rate, relating it to the inverse moments of the cross section for e + e − → heavy hadrons and discuss the resulting perturbative series.

Infrared Renormalons in a Nutshell
In this section, we outline the strategy we pursue for the calculations.We start from an observable M which has an OPE in inverse powers of the heavy quark mass where n are operators of mass-dimension n and ... denotes a forward matrix element with appropriate states.The coefficients C (i) n (Q) can be computed in perturbation theory as a series in α s (Q), while the matrix elements QCD encode the non-perturbative input.
The coefficients n (m Q ) in the OPE can be expanded in powers of α s (m Q ), and the running of α s (µ) is governed by the β function, which can be chosen such that it depends on α s only: where n f is the number of active quark flavours and C F = 4/3, C A = 3 and T f = 1/2 for SU (3).The solution of this equation can be written as In the following we will concentrate on the leading term in the β function, which implies Furthermore, we define a scale parameter Λ QCD by For the cases we shall consider the first term of the OPE observable M is given by the perturbative expression where a n are the coefficients of the perturbative expansion and d is the mass dimension of the observable.Note that the dimensionality of M pert has to come from the quark mass, because there is no other scale involved.However, it is well known and established, that the series in α s (m Q ) is at best an asymptotic series.This conjecture is supported by calculations in the large-β 0 limit.The perturbative series exhibits contributions with factorially growing coefficients of the form with b k ∼ 1, indicating a problem with the perturbative series.
The divergent behaviour of the perturbative series is closely related to the OPE for the observable.Within the OPE, we obtain power-suppressed contributions of the from (Λ QCD /Q) n .Re-expressing this expression in terms of α s , we get which clearly does not have a Taylor series in α s (m Q ).
To clarify the relation of the divergent behaviour of the perturbative series and the power like contribution obtained form the OPE, we look at the Borel transform S(u) of a function F (x) defined by its (formal) Taylor series In case the taylor series for C(x) is convergent for positive values of x, we can reconstruct F (x) from S(u) by computing the integral However, the presence of the factorially growing contributions prevent us from computing the integral.In fact, assuming a k = k!, we get This pole occurring in S(u) at positive u prevents us from performing the t-integral in (10).Thus one needs to give a prescription of how to deal with the pole, which introduces an ambiguity in the inverse transformation (10).One way to proceed is to deform the integral in (10) into a contour integral in the complex t plane, and the ambiguity can be defined as the difference between circling the pole by moving into the upper t-half plane and by moving to the lower t-half plane.This leads to a "localized" ambiguity of the form which translate to a corresponding ambiguity in C(x) We may apply this to the perturbative series discussed above.To make the argument clear, we use (7) and set b k = η (i.e.b k independent of k).Arguing along the lines described above, this leads to an ambiguity in the observable of the form encoded in a power suppressed term.
In the general case, multiple poles and even other types of singularities appear, hindering us to compute the integral (10) without additional prescriptions how to deal with the singularities.
In addition to this, it is well known that also the quark mass can introduce a renormalon problem.Since the quark mass is not a physical parameter, one needs to chose a mass scheme for its definition.The staring point of perturbative calculations is usually the pole mass m pole Q , defined by the position of the pole in a perturbative calculation.This prescription provides a recipe to remove the ultraviolet divergence from the quark self energy, and any renormalized mass m ren Q defined in any other scheme can be related to the pole mass by a perturbative calculation with finite coefficients r i This perturbative relation suffers from renormalon ambiguities as soon as m ren Q is a so called short-distance mass.To be explicit, we quote the relation between the mass m MS Q defined in the MS scheme and the pole mass, calculated in the "bubble chain" approximation.The Borel transform of the perturbative series in (15) exhibits poles [4] inducing in general ambiguities of the form where η is the residue of the pole at u n = 1/2, 3/2, 2, .. in the relation of the pole mass to some short-distance mass.Note that the term of order (Λ QCD /m Q ) 2 is absent in the relation between the pole mass and the MS mass in this bubble-chain approximation.
It is generally assumed and explicitly demonstrated in the bubble-chain approximation that the renormalon ambiguities have to cancel in physical observables [3,4,14].However, the perturbative series remains asymptotic, even if some of the renormalon ambiguities are resoved [14], so there is no possibility to fix the perturbative series without resumming all power corrections.However, the onset of the asymptotic behaviour can be delayed by removing renormalon ambiguities.
In particular, the renormalon in the pole mass cancels against (some of) the divergent behaviour in the perturbative expansions of the Wilson coefficients in the OPE.Therefore, we expect that the choice of a renormalon-free short distance mass will shift the onset of the divergent behaviour of the perturbative series to higher orders.Likewise, a proper definition of the matrix elements O (i) n will remove further renormalon ambiguities and will further delay the onset of the divergent behaviour of perturbation theory.
In practice, the quark mass (in any scheme) as well as the matrix elements O (i) n are obtained from other, independent observables, which we assume also to have an OPE.To this end, for a calculation up to some power (Λ QCD /m Q ) k in the OPE we will have -aside from the quark mass -a finite number j of matrix elements O (i) n , since n ≤ k.These can be determined from j + 1 observables that have an OPE like (1), which is expanded to the same order k.In this way, all unknown parameters, including the quark mass, are fixed in terms of observables.In the following we pursue this strategy, namely removing the quark mass(es) and the hadronic matrix elements by inserting observables.
Doing so, generates a relation between observables expressed in terms of a perturbative expansion.As stated above, this perturbative expansion is still plagued by divergences, which are related to renormalon ambiguities induced by terms of higher order (Λ QCD /m Q ) k+1 in the OPEs of the observables.In terms of the Borel transform of the perturbative contribution this means that ambiguities related to the first k poles have been removed, leaving poles located at u ≥ k+1 2 .The lowest pole at u = k+1 2 contributes to the factorial divergences as showing the suppression factor 1/(k + 1) n+1 which eventually leads to a later onset of the asymptotic behaviour of perturbation theory.
In the following we test this conjecture by explicit calculation, using the inverse moments of the e + e − → hadrons cross section and the heavy quark expansion for semileptonic decays as an example.
3 Replacing the Quark Mass through Observables 3.1 e + e − Moments Historically, the cross section for e + e − → hadrons has been the prime quantity to discuss the notion of quark-hadron duality.The underlying elementary process is e + e − → quarks and gluons, and the early understanding of duality was that the partonic cross section e + e − → quarks and gluons should be identical to the e + e − → hadrons, once an appropriate "smearing" is applied.This idea has been quantified by considering moments of the cross section, and the statement of duality turns into the statement that the moments of the cross section e + e − → hadrons should be the same within the precision of the calculation of e + e − → quarks and gluons.
In a more modern language, the notion of duality is linked to the validity of an OPE for the relevant observable.For the case at hand, we exploit the relation between the vacuum polarization and the ratio The dispersion relation between of the vacuum polarization contains the R ratio where the vacuum polarization function is where J µ (0) = f Q f q f γ µ q f is the electromagnetic current, Q f is the charge of the quark f and q is the momentum transfer with s = q 2 .For sufficiently large s the quantity Π(s) has an OPE of the form where O (n) i are a set of local operators (labelled by i) of dimension n and a (n) i (s) are coefficients calculable in perturbation theory.As discussed previously, the leading term in (22), for n = 0, is just the perturbative contribution.The first non-vanishing nonperturbative contributions appear at dimension four and involve the quark and gluon condensates.
To relate hadronic and partonic quantities, we define the moments R(s) according to Note that R(s) is dimensionless and approaches a constant as s → ∞, so only moments with n ≥ 1 can be computed.
Inserting the dispersion relation (20) we can relate these moments to Π(s) and, combining this with ( 22), shows that one obtains an OPE for M n as well.This observation is the exact formulation of the relation between "smeared" e + e − cross sections and the formulation of duality in terms of the OPE.We are especially interested in the contribution R Q from the heavy charm and bottom quarks.In this case, the partonic cross section vanishes below the threshold value s th = 4m 2 Q , i.e.R Q (s) = 0 for s ≤ s th .In [15] the leading term, i.e. the partonic part has been computed in a perturbative series in α s .In particular, the Taylor series of Π(s) reads with z = q 2 /(4m 2 Q ) and Q Q the charge of the heavy quark.The coefficients C n are computed perturbatively using the pole mass to order k = 2 in [15,16].The four-loop contributions of C 0 and C 1 were computed in [17,18].To be concrete, the coefficients to each order contain logarithms of the form ln(µ 2 /m 2 Q ), and the series reads where C n ≡ C (1,0) n . Note that we write explicitly the µ dependence, which is spurious, since the C n overall do not depend on µ.In other words, the explicit µ dependence from the logarithms has to cancel the one of α s order by order.This means that, for the cases at hand, we have such that the µ dependence is in fact O(α 3 s ).According to (24) the inverse moments are given in terms of the coefficients of the Taylor series of Π(q 2 ) as which is used e.g. to extract the quark masses from the inverse moments of the e + e − → hadrons cross sections [19].
Power-suppressed contributions to Π appear at (relative) order (Λ QCD /m Q ) 4 and are given in terms of quark-and gluon condensates [20].For heavy quarks, the quark condensate is absent, therefore only the gluon condensate contributes.For the moments, this gives [20,21] where the coefficients a n and b n can be found in [22].The value of the gluon condensate is not very precisely known, but it is small and even compatible with zero [22,23].For this reason it is often neglected, at least for moments with small n.This means that for all practical calculations the perturbative contribution, i.e. the leading term of the OPE is sufficient, and consequently we drop the contributions from the gluon condensate.
Following the strategy outlined in the last section, we consider at the Borel transform of the perturbative series of the leading term.Schematically, we expect it to have the form where the ellipses denote terms related to renormalon singularities at even higher values of u.Note that we have dropped the first power-suppressed terms, which are related to poles in (31) at u ≥ 2 so the renormalon poles shown explicitly in (31) are related to the pole mass.
The usual way to deal with this is to define a renormlized short distance mass m ren Q defined in (15).In fact, we may insert the expression (15) of the pole mass in terms of a short distance mass into (29) and re-expand into the perturbative series1 Assuming the cancellation of the ambiguities related to the quark mass, the Borel transform of the resulting perturbative series does not have poles at u = 1/2, 1 and 3/2, since power corrections, i.e. the gluon condensate, appear only at 1/m 4 , related to poles at u = 2 and higher.We note in passing that the coefficients of the resulting series depend explicitly on n.This indicates a problem since large values of n may jeopardize the perturbative series.We shall return to this point below.
According to the strategy outlined above, we may as well replace the pole mass by an observable, e.g. by another inverse moment.Solving one of the inverse moments for the pole mass  we obtain an expression that can be inserted into other moments.By the same argument as above the resulting perturbative series for the relation between two inverse moments will not have renormalon ambiguities for u = 1/2, 1 and 3/2.Therefore, we expect that this results in a better behaviour of the perturbative series.
Eliminating the pole mass using (33) from the expression for the moment M k in terms of the moment M n , we get Re-expanding in terms of α s (m Q ) gives Note that the logarithmic term ensures that this expression is in fact independent of µ to the order we calculate.The explicit expressions for the coefficients a k,n are given by: We list the numerical values of the C i coefficients for µ = m Q in Appendix A. Using those, we obtain the numerical values for the a k,n coefficients in Table 1.For completeness, we give the a (0) k,n coefficients in Appendix A. We use n l = 4 which applies to the B meson.However, we note that also for the charm-quark, with n l = 4, we observe a similar behavior.
The values of a k,n strongly depend on the ration k/n.We consider inverse moments up to n = 7, which means that 1/7 ≤ k/n ≤ 7. The re-expansion of the expression (34) in powers of α s thus exhibits this strong k/n dependence which jeopardizes the convergence of the resulting perturbative series.More generally, the expression (34) can be written as and the re-expansion of M k involves the Taylor series of the function Φ, and small values of k/n correspond to small values of the derivatives of Φ.
In turn, this means that moments with k/n ≤ 1 should yield the most reliable results, and so we focus on the upper right triangle of tTable 1.Although we work only to α 2 s , Table 1 shows that the convergence of the perturbative series relating M k with M n works best for k ∼ n, since a k,n is of the same order as a (2) k,n .This can also be seen by computing the "typical scale" for which the O(α 2 s ) contribution vanishes.These scales are shown in Table 2 in units of m Q .For the upper-right triangle and for k ∼ n we find values of the order m Q , which may be taken as an indication of convergence of the series expressed in terms of α s (m Q ).
In fact, as pointed out above, we do not expect a convergent series here, since the power corrections will eventually induce factorial divergences.However, in the case of the inverse moments of the e + e − → hadrons cross section, the power corrections seem to be really small, such that the asymptotic behaviour induced by the power corrections is not visible in the perturbative series up to α 2 s .

Inclusive semileptonic decays of bottom hadrons
The heavy quark expansion expresses inclusive B decay rates in terms of a systematic expansion in inverse powers of the quark mass m b .Like the inverse moments of the cross section for e + e − → hadrons it is based on an OPE where the leading term is given by the perturbative (i.e.partonic) result.In particular, the perturbative result will depend on the b quark mass.
In full QCD, the heavy quark Q has the equation of motion where D µ is the usual QCD covariant derivative including the gluon fields and m pole is the pole mass of the heavy quark, which we use as a starting point.It is in general defined as the pole of the perturbatively calculated quark propagator, but -since the quarks are not the asymptotic states of QCD -it cannot be assigned a physical meaning such as for the electron mass in QED.To this end, the pole mass remains a perturbative concept.
In order to set up the HQE, we re-define the heavy quark field in (40) with some mass parameter m which we will specify below.Inserting this, we find The expansion in inverse powers of the quark mass is set up by assuming that iD µ ∼ Λ QCD ≪ m pole .In the following, we specify the mass parameter m such that the "residual mass term" δm is also counted like δm ∼ Λ QCD .Introducing a new covariant derivative by we can write which means that the derivation of the heavy mass limit an the HQE proceeds in the usual way, except that all covariant derivatives are replaced by D → ∇.This leads to the quadratic form of the equation of motion: Turning to inclusive decays, the tree level contribution for inclusive semileptonic and radiative decays can be constructed from the external field propagator where q is the momentum transferred to leptons of photons, m is the chosen mass parameter of the field redefinition in (41) and m q is the mass of the final state quark.Expanding the propagator according to [24,25] gives We note that the relevant matrix element is constrained by the equation of motion, i.e.
where α, β are spinor indices.Inserting this yields which means that the pole mass gets re-installed in the denominator, since In general, the mass which appears in the denominator is the one which appears in the Lagrangain, i.e. in the equation of motion (40).
We consider now the leading term of the HQE, which is simply the partonic result.Computing QCD corrections in the pole scheme yields a perturbative series suffering from factorial divergences.It has been shown long ago [4,26,3] that these divergences cancel against the ones induced by the pole mass, such that the renormalon at u = 1/2 is cancelled, at least when using the bubble chain approximation.
We assume that this remains true in full QCD and study the case where we use a renormalized short-distance mass for the field redefinition.Using (15), we find which starts at order α s .As argued above, these terms induced by δm should cancel, at least, the divergences related to the u = 1/2 renormalon of the perturbative series for the inclusive b → u rate computed in the pole scheme.Now we turn to the strategy eliminating the mass in favour of an observable.We proceed in a similar way as in the case of the inverse moments of the e + e − → hadrons cross section and replace the quark mass.This could be done on the one hand by using spectral moments of inclusive semileptonic decays, but we shall proceed by making use of the inverse moments of the e + e − → hadrons cross section.We consider this to be the more interesting case, since this involves now very different observables measured by very different experiments.
We consider a simple case, which is the total rate for the charmless inclusive semileptonic decay B → X u ℓν.The leading term of the HQE for this process is the partonic result which reads to O(α2 s ) in terms of the pole mass where [27] b  We now replace the pole mass in (53) using (33) 2 .Re-expanding in α s gives: n defined in (57) and the value µ/m Q for which the α 2 s contribution vanishes. with Note that a factor 5/(2n) arises because of the m 5 b dependence of the rate.This corresponds to the k/n dependence of the inverse moments of the e + e − → hadrons cross section discussed in the previous section.
We give the numerical values for the coefficients d 1,2 in table 3. We first note that the coefficients become smaller as n increases.Furthermore, compared to the expressions in the pole scheme, the sign of the coefficients has changed, since both b 1 and b 2 are negative.The ratio of the coefficients d 1 ranges between 6.87 and 7.03 (see Table / 3) and thus is not particularly small, indicating that the convergence of the pertrurbative series is not strongly improved, in particular once we compare to b 2 /b 1 = 8.8, the values obtained in the pole scheme.
Following the arguments given above this suggest the interpretation that removing the renormalons related to the mass does not significantly shift the onset of the asymptotic behaviour of the perturbative series relating the B-meson decay rate and the inverse moments.In order to improve this one would need to include renormalons of higher values of u, which then requires to include also the power corrections in the B decay rate.This lies beyond the scope of our present paper.
In Table 3, we also quote the scale at which the coefficient of the α 2 s corrections vanish.This turns out to be a scale of about 800 MeV and is remarkably constant for the various values of n.

Conclusion
It is known since almost thirty years that perturbative expansions in QCD cannot be disentangled from its non-perturbative features, since the quarks and gluons never appear as asymptotic states.The tool to disentangle perturbative from non-perturbative effects is the OPE, which yields, on top of the perturbative expansion, non-perturbative parameters, which are on the one hand the quark masses, and on the other hadronic matrix elements such as condensates and HQE parameters.
In heavy-quark physics, the precision of predictions heavily depends on the treatment of the heavy-quark mass.The pole mass, the usual starting point for the perturbative calculation, suffers from renormalon ambiguities that hinder a systematic expansion in this mass.Motivated by the assertion that these ambiguities cancel between the perturbative series and properly defined non-perturbative quantities, including the quark-masses, we discussed an alternative treatment of the heavy-quark mass by replacing it with physical observables.In this paper, we made a first numerical analysis of this idea.Assuming the pattern of cancellations of renormalon ambiguities as suggested by many seminal papers from the mid nineties mentioned previously, we used the known information on the perturbative series for various observables to study the behaviour of the resulting perturbative series relating these observables.
We found that, using the known perturbative results up to α 2 s , for the relation between different inverse moments of the e + e − → hadrons cross section, that the perturbative series in fact improves significantly.We suggest that this correlates with the fact that the power correction for this observable only start at 1/m 4 .In addition, the hadronic matrix element, the gluon condensate is very small.However, we also found that using the same reasoning for the relation between the decay rate for B → X u ℓν and the inverse moments of the e + e − → hadrons cross section does not significantly improve the perturbative series, which may be related to the presence of power corrections which start in this case at 1/m 2 .
In view of the fact that the perturbative series will remain asymptotic, since we truncated the OPE and took only the leading term into account, we expect that including more observables will shift the onset of the divergent behaviour to even higher orders.Therefore, to push this idea further for the B meson (or D meson) decay rate, the power corrections have to be investigated, which means to include more observables in order to fix the unknown matrix elements.Furthermore, the method to remove more and more renormalon ambiguities with higher values of u may be refined by clever choices of observables, possibly giving us more confidence in the methods used in heavy-quark physics.
Finally, this alternative strategy for the quark mass may also shed some light on the question, if the HQE can be used as a precision tool for charm decays.Removing the charm mass from the OPE expressions by inserting observables will eventually reveal, if the HQE, combined with the perturbative expansion in α s (m c ), is a valid method to deal with charm decays.

A Coefficients C i
In this appendix, we list the numerical value for the coefficients C i entering in (25).We first write N A,n + C F T f n l C (2) where C A = 3, C F = 4/3, T f = 1/2.Here n l is the number of light massless fermions, which is related to the number of fermions n f via n l = n f − 1.The different C coefficients can be found in [16], which also contain the logarithmic terms.Compared to (25), we have C (1) F,n and N A,n + C F T f n l C (2) Taking µ = m Q , the coefficients take the values given in Table 4.For the b quark, we have n f = 5 and n l = 4, which we use for the numerical results in the main text.
For completeness, we give the coefficient a (0) k,n given in (36).Their numerical values are given in Table 5.
for different values of n and k using n l = 4.

Table 1 :
Coefficients a

Table 2 :
The scale µ/m Q at which the α 2 s -contribution in (35) vanishes for different combinations of n and k assuming n l = 4.

Table 3 :
The coefficients d

Table 5 :
The a