improvements of the HQE predictions in the charm sector:

  * •

$\Gamma_{3}^{(2)}$: NNLO-QCD Czarnecki:1997hc ; Czarnecki:1998kt ;
vanRitbergen:1999gs ; Melnikov:2008qs ; Pak:2008cp ; Pak:2008qt ;
Dowling:2008ap ; Bonciani:2008wf ; Biswas:2009rb ; Brucherseifer:2013cu ;
Fael:2020tow ; Czakon:2021ybq  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. (30) and Eq. (31) - 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.

  * •

$\Gamma_{5}^{(1)}$: 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.

  * •

$\Gamma_{6}^{(1)}$: The Wilson oefficients of the Darwin operator are large,
therefore QCD corrections for the non-leptonic case might be important.

  * •

$\Gamma_{7,8}^{(0)}$: Since the dimension-six contribution is sizable, the LO-
QCD determination of the dimension-seven 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.

  * •

$\tilde{\Gamma}_{6}^{(2)}$, $\tilde{\Gamma}_{7}^{(1)}$: Pauli interference
dominates the total decay rate of the $D^{+}$ meson. Currently
$\tilde{\Gamma}_{6}^{(0)}$, $\tilde{\Gamma}_{6}^{(1)}$ and
$\tilde{\Gamma}_{7}^{(0)}$ 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.

  * •

$\tilde{\Gamma}_{8}^{(0)}$: since the four-quark dimension-six contribution
can dominate the total decay rate and $\tilde{\Gamma}_{7}^{(0)}$ 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.
Gabbiani:2003pq ; Gabbiani:2004tp .

  * •

More precise determinations for the parameters $\mu_{G}^{2}$, $\mu_{\pi}^{2}$
and $\rho_{D}^{3}$ – 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
$\mu_{G}^{2}$, $\mu_{\pi}^{2}$ and $\rho_{D}^{3}$ from fits to semileptonic
$D^{+}$-, $D^{0}$\- and $D_{s}^{+}$-meson decays - as done in the $B$ system,
see e.g. Ref. Alberti:2014yda  – 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 dimension-six: here we have currently only HQET sum rule
determinations Kirk:2017juj ; King:2020  or outdated lattice results
DiPierro:1998ty ; Becirevic:2001fy .

  * •

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 Davies:2019gnp
.

Overall, we find that the HQE can describe inclusive charm observables, in
which no pronounced GIM cancellation arises191919See e.g. Ref. Lenz:2020efu
for a recent discussion of the extreme GIM cancellations in mixing of neutral
$D$ mesons., 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. Jubb:2016mvq  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.

## Acknowledgments

The work of M.L.P. is supported by Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through grant 396021762 -TRR 257 “Particle Physics
Phenomenology after the Higgs Discovery”, the work of D.K. and C.V. was
supported by the STFC grant of the IPPP. We would like to thank T. Mannel, B.
Melic, D. Moreno, I. Nisandzic, and A. Pivovarov for insightful discussions.

##  Appendix A Numerical Input

We use five-loop running for $\alpha_{s}(\mu)$ Herren:2017osy  with four
active flavours at the scale $\mu\sim m_{c}$, and the most recent value
Zyla:2020zbs

$\displaystyle\alpha_{s}(M_{Z})$ $\displaystyle=$ $\displaystyle 0.1179\pm
0.0010.$ (122)

For the CKM matrix elements we apply the standard parametrisation in terms of
$\theta_{12},\theta_{13},\theta_{22},\delta$ and use as an input
Charles:2004jd  (online update)

$\displaystyle|V_{us}|$ $\displaystyle=$ $\displaystyle
0.224834^{+0.000252}_{-0.000059}\,,$ (123)
$\displaystyle\frac{|V_{ub}|}{|V_{cb}|}$ $\displaystyle=$ $\displaystyle
0.088496^{+0.001885}_{-0.002244}\,,$ (124) $\displaystyle|V_{cb}|$
$\displaystyle=$ $\displaystyle 0.04162^{+0.00026}_{-0.00080}\,,$ (125)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\left(65.80^{+0.94}_{-1.29}\right)^{\circ}.$ (126)

For the $c$-quark mass, we use different values depending of the scheme. In
the $\overline{\rm MS}$-scheme we take Zyla:2020zbs :

$\overline{m}_{c}(\overline{m}_{c})=(1.27\pm 0.02)\,{\rm GeV},$ (127)

in the kinetic scheme we employ (at NLO) Fael:2020njb :

$m_{c}^{\rm kin}(0.5\,{\rm GeV})=(1.306\pm 0.02)\,{\rm GeV},$ (128)

and in the $1S$-scheme (see Eq. (13)) we use $m_{c}^{1S}=1.44$ GeV
Herren:2017osy .
For the $s$-quark mass we take the value Zyla:2020zbs

$m_{s}=\left(93^{+11}_{-5}\right)\,{\rm MeV}.$ (129)

The masses of $D$-mesons are known very precisely Zyla:2020zbs :

$M_{D^{0}}=1.86493\,{\rm GeV},\qquad M_{D^{+}}=1.86965\,{\rm GeV},\qquad
M_{D_{s}^{+}}=1.96834\,{\rm GeV}.$

The values of the non-perturbative parameters used in the analysis are shown
in Tables 16 and 17.

Parameter |  $D^{+,0}$ |  Source |  $D_{s}^{+}$ |  Source
---|---|---|---|---
$f_{D}$ [GeV] |  $0.2120\pm 0.0007$ |  Lattice QCD Aoki:2019cca  |  $0.2499\pm 0.0005$ |  Lattice QCD Aoki:2019cca
$\mu_{\pi}^{2}(D)$ [GeV2] |  $0.465\pm 0.198$ |  Exp. fit Alberti:2014yda  and HQ symmetry |  $0.555\pm 0.232$ |  Exp. fit Alberti:2013kxa , $SU(3)_{f}$-breaking Bigi:2011gf  and HQ symmetry
$\mu_{G}^{2}(D)$ [GeV2] |  $0.34\pm 0.10$ |  Spectroscopy relations Uraltsev:2001ih  |  $0.36\pm 0.10$ |  Spectroscopy relations Uraltsev:2001ih
$\rho_{D}^{3}(D)$ [GeV3] |  $0.075\pm 0.034$ |  Exp. fit Alberti:2014yda  and E.O.M relation |  $0.110\pm 0.050$ |  Exp. fit Alberti:2014yda  and E.O.M relation
Table 16: Numerical values of the non-perturbative parameters used in our
analysis.

${\rm HQET},\,\mu_{0}=1.5\,{\rm GeV}$ | $\tilde{B}_{1}$ | $\tilde{B}_{2}$ | $\tilde{\epsilon}_{1}$ | $\tilde{\epsilon}_{2}$
---|---|---|---|---
$D^{+,0}$ | $\phantom{-}1.0026^{+0.0198}_{-0.0106}$ | $\phantom{-}0.9982^{+0.0052}_{-0.0066}$ | $-0.0165^{+0.0209}_{-0.0346}$ | $-0.0004^{+0.0200}_{-0.0326}$
$D_{s}^{+}$ | $\phantom{-}1.0022^{+0.0185}_{-0.0099}$ | $\phantom{-}0.9983^{+0.0052}_{-0.0067}$ | $-0.0104^{+0.0202}_{-0.0330}$ | $-0.0001^{+0.0199}_{-0.0324}$

${\rm HQET},\,\mu_{0}=1.5\,{\rm GeV}$ | $\tilde{\delta}_{1}$ | $\tilde{\delta}_{2}$ | $\tilde{\delta}_{3}$ | $\tilde{\delta}_{4}$
---|---|---|---|---
$\langle D_{q}|\tilde{O}^{q}|D_{q}\rangle$ | $\phantom{-}0.0026^{+0.0142}_{-0.0092}$ | $-0.0018^{+0.0047}_{-0.0072}$ | $-0.0004^{+0.0015}_{-0.0024}$ | $\phantom{-}0.0003^{+0.0012}_{-0.0008}$
$\langle D_{s}|\tilde{O}^{q}|D_{s}\rangle$ | $\phantom{-}0.0025^{+0.0144}_{-0.0093}$ | $-0.0018^{+0.0047}_{-0.0072}$ | $-0.0004^{+0.0015}_{-0.0024}$ | $\phantom{-}0.0003^{+0.0012}_{-0.0008}$
$\langle D_{q}|\tilde{O}^{s}|D_{q}\rangle$ | $\phantom{-}0.0023^{+0.0140}_{-0.0091}$ | $-0.0017^{+0.0046}_{-0.0070}$ | $-0.0004^{+0.0015}_{-0.0023}$ | $\phantom{-}0.0003^{+0.0012}_{-0.0008}$

Table 17: Numerical values of the HQET Bag parameters Kirk:2017juj ; King:2020
evaluated through a traditional HQET sum rule.

##  Appendix B Expressions for the Darwin Coefficients

The coefficients $C_{\rho_{D},mn}^{(q_{1}q_{2})}(\rho,\mu_{0})$ including full
$\rho=m_{s}^{2}/m_{c}^{2}$ dependence are given by the expressions:

$\displaystyle{\cal C}_{\rho_{D},11}^{(d\bar{d})}$ $\displaystyle=$
$\displaystyle 6+8\,\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right),$ (130)
$\displaystyle{\cal C}_{\rho_{D},12}^{(d\bar{d})}$ $\displaystyle=$
$\displaystyle-\frac{34}{3},$ (131) $\displaystyle{\cal
C}_{\rho_{D},22}^{(d\bar{d})}$ $\displaystyle=$ $\displaystyle
6+8\,\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right),$ (132)
$\displaystyle{\cal C}_{\rho_{D},11}^{(d\bar{s})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}(1-\rho)\biggl{[}9+11\rho-12\rho^{2}\log(\rho)-\,24\left(1-\rho^{2}\right)\log(1-\rho)-25\rho^{2}+5\rho^{3}\biggl{]}$
(133)
$\displaystyle+\,8\,(1-\rho)(1-\rho^{2})\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right),$
$\displaystyle{\cal C}_{\rho_{D},12}^{(d\bar{s})}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\biggl{[}17+12\rho\left(5+2\rho-2\rho^{2}\right)\log(\rho)+\,48(1-\rho)(1-\rho^{2})\log(1-\rho)$
(134)
$\displaystyle\quad\quad-26\rho+18\rho^{2}-38\rho^{3}+5\rho^{4}+24\,\rho\,(1+\rho-\rho^{2})\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\biggl{]},$
$\displaystyle{\cal C}_{\rho_{D},22}^{(d\bar{s})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}(1-\rho)\biggl{[}9+11\rho-12\rho^{2}\log(\rho)-\,24\left(1-\rho^{2}\right)\log(1-\rho)-25\rho^{2}+5\rho^{3}\biggr{]},$
(135)
$\displaystyle+\,8\,(1-\rho)(1-\rho^{2})\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right),$
$\displaystyle{\cal C}_{\rho_{D},11}^{(s\bar{d})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\biggl{[}9-16\rho-12\rho^{2}+16\rho^{3}-5\rho^{4}+12\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\biggr{]},$
(136) $\displaystyle{\cal C}_{\rho_{D},12}^{(s\bar{d})}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\biggl{[}17+12\,\rho^{2}\left(3-\rho\right)\log(\rho)-\,24(1-\rho)^{3}\log(1-\rho)$
(137)
$\displaystyle\quad\quad-50\rho+90\rho^{2}-54\rho^{3}+5\rho^{4}-12\rho\,(3-3\rho+\rho^{2})\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\biggr{]},$
$\displaystyle{\cal C}_{\rho_{D},22}^{(s\bar{d})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}(1-\rho)\biggl{[}9+11\rho-12\rho^{2}\log(\rho)-\,24\left(1-\rho^{2}\right)\log(1-\rho)$
(138)
$\displaystyle\quad\quad-25\rho^{2}+5\rho^{3}+12\,(1-\rho^{2})\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\biggr{]},$
$\displaystyle{\cal C}_{\rho_{D},11}^{(s\bar{s})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\Biggl{[}\sqrt{1-4\rho}\left(17+8\rho-22\rho^{2}-60\rho^{3}\right)-4\left(2-3\rho+\rho^{3}\right)+$
(139)
$\displaystyle\quad\quad-\,12\left(1-\rho-2\rho^{2}+2\rho^{3}+10\rho^{4}\right)\log\left(\frac{1+\sqrt{1-4\rho^{\phantom{\\!1}}}}{1-\sqrt{1-4\rho^{\phantom{\\!1}}}}\right)$
$\displaystyle\quad\quad-\,12\,(1-\rho)(1-\rho^{2})\left(\log(\rho)-\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\right)\Biggr{]},$
$\displaystyle{\cal C}_{\rho_{D},12}^{(s\bar{s})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\Biggl{[}\sqrt{1-4\rho}\left(-33+24\,\log(\rho)-24\,\log(1-4\rho)+46\rho-106\rho^{2}-60\rho^{3}\right)$
(140)
$\displaystyle\quad\quad+\,12\left(3-2\rho+4\rho^{2}-16\rho^{3}-10\rho^{4}\right)\log\left(\frac{1+\sqrt{1-4\rho^{\phantom{\\!1}}}}{1-\sqrt{1-4\rho^{\phantom{\\!1}}}}\right)$
$\displaystyle\quad\quad+\,4\left(1-\rho\right)^{2}\left(4+3(1-\rho)\log(\rho)-\rho\right)$
$\displaystyle\quad\quad-12\,\left(1-\sqrt{1-4\rho}-3\rho+3\rho^{2}-\rho^{3}\right)\,\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\Biggr{]},$
$\displaystyle{\cal C}_{\rho_{D},22}^{(s\bar{s})}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\Biggl{[}\sqrt{1-4\rho}\left(9+24\,\log(\rho)-24\,\log(1-4\rho)+22\rho-34\rho^{2}-60\rho^{3}\right)$
(141)
$\displaystyle\quad\quad+\,24\left(1-2\rho-\rho^{2}-2\rho^{3}-5\rho^{4}\right)\log\left(\frac{1+\sqrt{1-4\rho^{\phantom{\\!1}}}}{1-\sqrt{1-4\rho^{\phantom{\\!1}}}}\right)$
$\displaystyle\quad\quad\,+\,12\,\sqrt{1-4\rho}\,\log\left(\frac{\mu_{0}^{2}}{m_{c}^{2}}\right)\Biggr{]}.$

The numerical values of the above coefficients for $\rho=0.006$ are shown in
Table 18.

|  $3\,C_{1}^{2}$ |  $2\,C_{1}C_{2}$ |  $3\,C_{2}^{2}$
---|---|---|---
$c\to d\bar{d}u$ |  6 |  -11.33 |  6
$c\to d\bar{s}u$ |  6.10 |  -9.81 |  6.10
$c\to s\bar{d}u$ |  5.94 |  -11.23 |  6.10
$c\to s\bar{s}u$ |  6.04 |  -9.70 |  6.21
Table 18: Numerical values of $C_{\rho_{D},nm}^{(q_{1}q_{2})}$ for
$\rho=0.006$ and $\mu_{0}=m_{c}$.

##  Appendix C Parametrisation of the Matrix Element of Four-Quark Operators

The matrix elements of the dimension-six operators in QCD are parametrised in
the following way

$\displaystyle\langle{D}_{q}|O_{i}^{q}\,|{D}_{q}\rangle$ $\displaystyle=$
$\displaystyle A_{i}\,f_{D_{q}}^{2}m_{D_{q}}^{2}\,B_{i}^{q},$ (142)
$\displaystyle\langle{D}_{q}|O_{i}^{q^{\prime}}|{D}_{q}\rangle$
$\displaystyle=$ $\displaystyle
A_{i}\,f_{D_{q}}^{2}m_{D_{q}}^{2}\,\delta^{qq^{\prime}}_{i},\qquad
q\not=q^{\prime}\,,$ (143)

where

$A_{1}^{q}=A_{3}^{q}=1,\qquad
A_{2}^{q}=A_{4}^{q}=\frac{m_{D}^{2}}{(m_{c}+m_{q})^{2}}.$

In VIA the Bag parameters reduce to $B_{1}^{q}=B_{2}^{q}=1$ and
$B_{3}^{q}=\epsilon_{1}^{q}=0$, $B_{4}^{q}=\epsilon_{2}^{q}=0$ and all
$\delta^{qq^{\prime}}_{i}=0$.

The matrix elements of the dimension-seven four-quark operators in Eqs. (74) -
(82) in HQET are parametrised in the following way:

$\displaystyle\langle D_{q}|{\tilde{P}}_{1}^{q}|D_{q}\rangle$ $\displaystyle=$
$\displaystyle-m_{q}F^{2}(\mu_{0})\,m_{D}\,\tilde{B}_{P,1}^{q}\,,$ (144)
$\displaystyle\langle D_{q}|{\tilde{P}}_{2}^{q}|D_{q}\rangle$ $\displaystyle=$
$\displaystyle-F^{2}(\mu_{0})\,m_{D}\,\bar{\Lambda}\,\tilde{B}_{P,2}^{q}\,,$
(145) $\displaystyle\langle D_{q}|{\tilde{P}}_{3}^{q}|D_{q}\rangle$
$\displaystyle=$
$\displaystyle-F^{2}(\mu_{0})\,m_{D}\,\bar{\Lambda}\,\tilde{B}_{P,3}^{q}\,,$
(146) $\displaystyle\langle D_{q}|{\tilde{R}}_{1}^{q}|D_{q}\rangle$
$\displaystyle=$
$\displaystyle-F^{2}(\mu_{0})\,m_{D}\,(\bar{\Lambda}-m_{q})\,\tilde{B}_{R,1}^{q}\,,$
(147) $\displaystyle\langle D_{q}|{\tilde{R}}_{2}^{q}|D_{q}\rangle$
$\displaystyle=$ $\displaystyle
F^{2}(\mu_{0})\,m_{D}\,(\bar{\Lambda}-m_{q})\,\tilde{B}_{R,1}^{q}\,,$ (148)

with $\bar{\Lambda}=m_{D}-m_{c}$, and

$\displaystyle\langle D_{q}|{\tilde{M}}_{1,\pi}^{q}|D_{q}\rangle$
$\displaystyle=$ $\displaystyle
2\,F^{2}(\mu_{0})\,m_{D}\,G_{1}(\mu_{0})\,\tilde{L}_{1,\pi}^{q}\,,$ (149)
$\displaystyle\langle D_{q}|{\tilde{M}}_{2,\pi}^{q}|D_{q}\rangle$
$\displaystyle=$ $\displaystyle
2\,F^{2}(\mu_{0})\,m_{D}\,G_{1}(\mu_{0})\,\tilde{L}_{2,\pi}^{q}\,,$ (150)
$\displaystyle\langle D_{q}|{\tilde{M}}_{1,G}^{q}|D_{q}\rangle$
$\displaystyle=$ $\displaystyle
12\,F^{2}(\mu_{0})\,m_{D}\,G_{2}(\mu_{0})\,\tilde{L}_{1,G}^{q}\,,$ (151)
$\displaystyle\langle D_{q}|{\tilde{M}}_{2,G}^{q}|D_{q}\rangle$
$\displaystyle=$ $\displaystyle
12\,F^{2}(\mu_{0})\,m_{D}\,G_{2}(\mu_{0})\,\tilde{L}_{2,G}^{q}\,,$ (152)

and similar expressions for the colour-octet operators. Again, in VIA, the
dimension-seven Bag parameters are $\tilde{B}_{P,i}^{q}=1$,
$\tilde{B}_{R,i}^{q}=1$, and $\tilde{L}_{1,\pi}^{q}=1$,
$\tilde{L}_{1,G}^{q}=1$ and the corresponding colour-octet Bag parameters
vanish.

The expressions in Eqs. (144) - (148) can be obtained using a general
parametrisation of matrix elements of the HQET quark currents with a heavy
pseudo-scalar meson ${\cal M}$ (see e.q. Ref. Neubert:1993mb ):

$\displaystyle\langle 0|\bar{q}\,\Gamma\,h_{v}|M(v)\rangle$ $\displaystyle=$
$\displaystyle\frac{i}{2}F(\mu)\,{\rm Tr}[\Gamma{\cal M}(v)]\,,$ (153)
$\displaystyle\langle 0|\bar{q}\,\Gamma\,iD_{\alpha}h_{v}|M(v)\rangle$
$\displaystyle=$ $\displaystyle-\frac{i}{6}(\bar{\Lambda}-m_{q})F(\mu)\,{\rm
Tr}[(v_{\alpha}+\gamma_{\alpha})\Gamma{\cal M}(v)]\,,$ (154)
$\displaystyle\langle
0|\bar{q}(-i\overset{\leftarrow}{D}_{\alpha})\,\Gamma\,h_{v}|M(v)\rangle$
$\displaystyle=$ $\displaystyle-\frac{i}{6}F(\mu)\,{\rm
Tr}[((4\bar{\Lambda}-m_{q})v_{\alpha}+(\bar{\Lambda}-m_{q})\gamma_{\alpha})\Gamma{\cal
M}(v)],$ (155)

and for the non-local operators:

$\displaystyle\langle 0|i\int
d^{4}y\,T\left[(\bar{q}\,\Gamma\,h_{v})(0),(\bar{h}_{v}(iD)^{2}h_{v})(y)\right]|{\cal
M}(v)\rangle$ $\displaystyle=$ $\displaystyle F(\mu)\,G_{1}(\mu)\,{\rm
Tr}[\Gamma{\cal M}(v)],$ (156) $\displaystyle\langle 0|i\int
d^{4}y\,T\left[(\bar{q}\,\Gamma\,h_{v})(0),\frac{1}{2}g_{s}\left(\bar{h}_{v}\sigma_{\alpha\beta}G^{\alpha\beta}h_{v}\right)(y)\right]|{\cal
M}(v)\rangle$ $\displaystyle=$ $\displaystyle 6\,F(\mu)\,G_{2}(\mu)\,{\rm
Tr}[\Gamma{\cal M}(v)],$ (157)

where $\Gamma$ is a generic Dirac structure, and

$\displaystyle{\cal M}(v)$ $\displaystyle=$
$\displaystyle-\sqrt{m_{D}}\,\frac{(1+\not{v})}{2}\gamma_{5}\,.$ (158)

Since we are limited to LO-QCD for the dimension-seven contribution, we can
just replace the HQET decay constant $F(\mu)$ by the full QCD one $f_{D}$,
using $F(\mu)=f_{D}\,\sqrt{m_{D}}$.

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