between previous HQE determinations of $D$-mixing with data. The content of
this chapter closely follows the one of Ref. [140] and Ref. [235].

### 4.1 Theoretical study of the total decay width of charmed mesons

In the present section, we analyse the structure of the HQE in the charm
sector, to try to shed further light into the question, whether the expansion
parameters $\alpha_{s}(m_{c})$, and $\Lambda_{\rm QCD}/m_{c}$, are small
enough in order to ensure meaningful theoretical predictions for the
observables listed in Table 4.1. The Particle Data Group [191] quotes,
respectively for the pole and the $\overline{\rm MS}$ mass of the charm quark,
the values

$m_{c}^{\rm Pole}=(1.67\pm 0.07)\;{\rm GeV}\;,\hskip
28.45274pt\overline{m}_{c}(\overline{m}_{c})=(1.27\pm 0.02)\;{\rm GeV}\,,$
(4.8)

while the dependence of $\alpha_{s}$ on both the charm scale and the loop
order, obtained using the RunDec package [236], is shown in Table 4.2. We
emphasise that in our numerical analysis we use the five-loop running result.

$\alpha_{s}(m_{c})$ | $m_{c}=1.67\,{\rm GeV}$ | $m_{c}=1.48\,{\rm GeV}$ | $m_{c}=1.27\,{\rm GeV}$
---|---|---|---
two-loop | $0.322$ | $0.346$ | $0.373$
five-loop | $0.329$ | $0.356$ | $0.387$
Table 4.2:  Numerical values of the strong coupling $\alpha_{s}$ evaluated at
different scales and loop order, obtained using the RunDec package [236].

The relation between the pole and $\overline{\rm MS}$ mass schemes, up to
third order in the strong coupling, reads [237, 238, 239]

$\displaystyle m_{c}^{\rm Pole}$
$\displaystyle=\overline{m}_{c}(\overline{m}_{c})\left[1+\frac{4}{3}\frac{\alpha_{s}(\overline{m}_{c})}{\pi}+10.43\left(\frac{\alpha_{s}(\overline{m}_{c})}{\pi}\right)^{2}+116.5\left(\frac{\alpha_{s}(\overline{m}_{c})}{\pi}\right)^{3}\right]$
$\displaystyle=\overline{m}_{c}(\overline{m}_{c})\left[1+0.1642+0.1582+0.2176\right]\,,$
(4.12)

where we have used the five-loop value of $\alpha_{s}$, at the scale 1.27 GeV.
The strong dependence of $\Gamma_{3}$ on the charm pole mass, cf. Eq. (3.24),
leads to different results according to how higher orders in Eq. (4.12) are
treated. Specifically, by truncating the expansion in Eq. (4.12) at first
order in $\alpha_{s}$, and using $\overline{m}_{c}(\overline{m}_{c})=1.27$
GeV, we obtain for the pole mass the value $m_{c}^{\rm Pole}=1.479$ GeV,
which, respectively yields

$\left(m_{c}^{\rm
Pole}\right)^{5}=\overline{m}_{c}(\overline{m}_{c})^{5}\left[1+0.1642\right]^{5}=2.14\,\overline{m}_{c}(\overline{m}_{c})^{5},$
(4.13)

by computing the fifth power of $m_{c}^{\rm Pole}$, and

$\left(m_{c}^{\rm
Pole}\right)^{5}\approx\overline{m}_{c}(\overline{m}_{c})^{5}\left[1+5\cdot
0.1642\right]=1.82\,\overline{m}_{c}(\overline{m}_{c})^{5}\,,$ (4.14)

by further expanding the fifth power up to the first order in $\alpha_{s}$.
Note that the result in Eq. (4.14) is about 15 $\%$ smaller than the one in
Eq. (4.13). Conversely, by including also all the higher order terms shown in
Eq. (4.12), gives

$\left(m_{c}^{\rm
Pole}\right)^{5}=\overline{m}_{c}(\overline{m}_{c})^{5}\left[1+0.1642+0.1582+0.2176\right]^{5}=8.66\,\overline{m}_{c}(\overline{m}_{c})^{5}\,,$
(4.15)

which is roughly four times larger than the result in Eq. (4.13). In order to
deal with this numerical instability, in the following we investigate
different scenarios:

  1. 1.

Use Eq. (4.12) to first order in $\alpha_{s}$, since this is the order at
which most of the Wilson coefficients are known. In this case we fix
$m_{c}^{\rm Pole}=1.48$ GeV and $\alpha_{s}=0.356$. A further possibility
would be to use as input the pole mass value from the PDG, i.e. $m_{c}^{\rm
Pole}=1.67$ GeV. However, in this case, our numerical analysis gives results
for the decay rates which are roughly $30\%$ larger than the ones obtained in
the $1S$ scheme, discussed below. Since we expect this enhancement to be
compensated by missing NNLO corrections to the non-leptonic decay rates, we do
not present explicit results for $m_{c}^{\rm Pole}=1.67$ GeV.

  2. 2.

Express $m_{c}^{\rm Pole}$ in terms of the $\overline{\rm MS}$ mass

$m_{c}^{\rm
Pole}=\overline{m}_{c}(\overline{m}_{c})\left[1+\frac{4}{3}\frac{\alpha_{s}(\overline{m}_{c})}{\pi}\right]\,,$
(4.16)

using $\overline{m}_{c}(\overline{m}_{c})=1.27$ GeV [191], and expand
consistently up to order $\alpha_{s}$. Because of the dependence on the fifth
power of the charm quark mass, in this case, $\Gamma_{3}$ receives large
corrections $\sim 5\times(4/3)(\alpha_{s}/\pi)$.

* 3.

Express $m_{c}^{\rm Pole}$ in terms of the kinetic mass [104, 240]. 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 which is still valid at small scales $\mu\sim 1$ GeV.
The relation between the kinetic scheme and the $\overline{\rm MS}$ and pole
schemes can be found, up to N3LO corrections, in Ref. [241]. At order
$\alpha_{s}$ we have

$m_{c}^{\rm Pole}=m_{c}^{\rm
Kin}\left[1+\frac{4\alpha_{s}}{3\pi}\left(\frac{4}{3}\frac{\mu^{\rm
cut}}{m_{c}^{\rm Kin}}+\frac{1}{2}\left(\frac{\mu^{\rm cut}}{m_{c}^{\rm
Kin}}\right)^{2}\right)\right]\,,$ (4.17)

where $\mu^{\rm cut}$ is the Wilsonian cutoff separating the perturbative and
non perturbative regimes. In our numerical analysis we set $\mu^{\rm cut}=0.5$
GeV, which gives, at NLO-QCD and using as an input $\overline{m}_{c}(3\,{\rm
GeV})$ [241]

$m_{c}^{\rm kin}(0.5\,{\rm GeV})=1.363\,{\rm GeV}\,.$ (4.18)

* 4.

Express $m_{c}^{\rm Pole}$ in the $1S$-mass scheme, defined as [242, 243]

$m_{c}^{\rm
Pole}=\frac{m_{J/\psi}}{2}\left(1+\frac{(\alpha_{s}\,C_{F})^{2}}{8}\right)\,,$
(4.19)

where $C_{F}=4/3$, and we use $m_{J/\psi}=3.0969$ GeV [191], so that
$m_{J/\psi}/2\approx 1.55$ GeV. Note that in Eq. (4.19), the NLO correction
actually starts at order $\alpha_{s}^{2}$, see Ref. [242].

#### 4.1.1 Description of the computation

The non-leptonic decay of a charm quark $c\to q_{1}\bar{q}_{2}u$, with
$q_{1},q_{2}=\\{u,d,s\\}$, is described by the following effective
Hamiltonian, cf. Section 1.1, i.e.

$\displaystyle{\cal H}_{\rm eff}^{\rm NL}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\left[\sum\limits_{q_{1,2}=d,s}\lambda_{q_{1}q_{2}}\Big{(}C_{1}\,Q_{1}^{q_{1}q_{2}}+C_{2}\,Q_{2}^{q_{1}q_{2}}\Big{)}-\lambda_{b}\sum\limits_{j=3}^{6}C_{j}Q_{j}\right]+{\rm
h.c.},$ (4.20)

where we have defined the CKM factors respectively as
$\lambda_{q_{1}q_{2}}=V_{cq_{1}}^{*}V_{uq_{2}}$ and
$\lambda_{b}=V_{cb}^{*}V_{ub}$, and introduced the following notation for the
tree-level $\Delta C=1$ operators:

$\displaystyle Q_{1}^{q_{1}q_{2}}$
$\displaystyle=\left(\bar{q}_{1}^{i}\gamma_{\rho}(1-\gamma_{5})c^{i}\right)\,\left(\bar{u}^{j}\gamma^{\rho}(1-\gamma_{5})q_{2}^{j}\right)\,,$
(4.21) $\displaystyle Q_{2}^{q_{1}q_{2}}$
$\displaystyle=\left(\bar{q}_{1}^{i}\gamma_{\rho}(1-\gamma_{5})c^{j}\right)\,\left(\bar{u}^{j}\gamma^{\rho}(1-\gamma_{5})q_{2}^{i}\right)\,,$
(4.22)

while $Q_{j},$ with $j=3,\ldots,6$, refer to the penguin operators, which can
only arise in the singly Cabibbo suppressed decays $c\to s\bar{s}u$ and $c\to
d\bar{d}u$. In Eq. (4.20), $C_{i}(\mu_{1})$, with $i=1,\ldots,6$, denote the
corresponding Wilson coefficients evaluated at the renormalisation scale
$\mu_{1}\sim m_{c}$. A comparison of their values respectively at NLO-(LO-)QCD
and for different choices of $m_{c}$, is shown in Table 4.3.

$\mu_{1}[{\rm GeV}]$ |  $1$ |  $1.27$ |  $1.36$ |  $1.48$ |  $1.55$ |  $3$
---|---|---|---|---|---|---
$C_{1}(\mu_{1})$ |  $1.25$ $(1.34)$ |  $1.20$ $(1.27)$ |  $1.19$ $(1.26)$ |  $1.18$ $(1.24)$ |  $1.17$ $(1.23)$ |  $1.10$ $(1.15)$
$C_{2}(\mu_{1})$ |  $-0.48$ $(-0.62)$ |  $-0.39$ $(-0.50)$ |  $-0.40$ $(-0.53)$ |  $-0.37$ $(-0.48)$ |  $-0.36$ $(-0.47)$ |  $-0.24$ $(-0.32)$
$C_{3}(\mu_{1})$ |  $0.03$ $(0.02)$ |  $0.02$ $(0.01)$ |  $0.02$ $(0.01)$ |  $0.01$ $(0.01)$ |  $0.01$ $(0.01)$ |  $0.00$ $(0.00)$
$C_{4}(\mu_{1})$ |  $-0.06$ $(-0.04)$ |  $-0.05$ $(-0.03)$ |  $-0.04$ $(-0.03)$ |  $-0.04$ $(-0.02)$ |  $-0.04$ $(-0.02)$ |  $-0.01$ $(-0.01)$
$C_{5}(\mu_{1})$ |  $0.01$ $(0.01)$ |  $0.01$ $(0.01)$ |  $0.01$ $(0.01)$ |  $0.01$ $(0.01)$ |  $0.01$ $(0.01)$ |  $0.00$ $(0.00)$
$C_{6}(\mu_{1})$ |  $-0.08$ $(-0.05)$ |  $-0.05$ $(-0.03)$ |  $-0.05$ $(-0.03)$ |  $-0.04$ $(-0.03)$ |  $-0.04$ $(-0.02)$ |  $-0.01$ $(-0.01)$
Table 4.3: Comparison of the Wilson coefficients at NLO-QCD (LO-QCD), for
different values of $\mu_{1}=m_{c}$.

We see that the Wilson coefficients $C_{j}$, with $j=3,\ldots,6$, are very
small and additionally strongly CKM suppressed because of the factor
$\lambda_{b}\ll\lambda_{q_{1}q_{2}}$. For these reasons, in the following, the
contribution due to the penguin operators in Eq. (4.20) is neglected. However,
the most general effective Hamiltonian describing all possible $c$-quark
decays is a sum of non-leptonic, semileptonic as well as radiative
contributions, namely

$\displaystyle{\cal H}_{\rm eff}$ $\displaystyle={\cal H}_{\rm eff}^{\rm
NL}+{\cal H}_{\rm eff}^{\rm SL}+{\cal H}_{\rm eff}^{\rm rare}\,,$ (4.30)

here, ${\cal H_{\rm eff}^{\rm NL}}$ is given in Eq. (4.20),

${\cal H}_{\rm eff}^{\rm
SL}=\frac{G_{F}}{\sqrt{2}}\sum_{q=d,s}\sum_{\ell=e,\mu}V_{cq}^{*}\,Q^{q\ell}+{\rm
h.c.}\,,$ (4.31)

where we have introduced the semileptonic operator

$Q^{q\ell}=\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})c\right)\left(\bar{\nu}_{\ell}\gamma_{\mu}(1-\gamma_{5})\ell\right)\,,\qquad\ell=e,\mu\,,$
(4.32)

while ${\cal H}_{\rm eff}^{\rm rare}$ describes decays like
$D\to\pi\ell^{+}\ell^{-}$, whose branching fraction is much smaller than those
corresponding to the tree-level transitions. Hence, in the following, we also
neglect the presence of rare decays and omit to specify further the form of
${\cal H}_{\rm eff}^{\rm rare}$. Starting from the expression of the effective
Hamiltonian in Eq. (4.30), the total decay width of the heavy charm mesons
$D^{0},D^{+},D_{s}^{+}$, can be computed according to Eq. (1.144), where now
we need to set $Q=c$. The structure of the HQE is schematically given in Eq.
(1.164), and its diagrammatic representation can be visualised as in Figure
1.7. The lowest dimensional contributions, namely those due to two-quark
operators up dimension-six, and to four-quark operators up dimension-seven,
have been discussed in detail in Chapter 2 and Chapter 3. Despite having
considered explicitly the case of the $B$ meson, almost all the expressions
obtained, can be used also in the charm sector, taking into account the proper
replacements i.e. $b\to c$, $c\to s$, etc.. However, we cannot directly use
the coefficients of the Darwin operator listed in Eqs. (3.85)-(3.94), since by
setting $\rho=m_{s}^{2}/m_{c}^{2}$, it is straightforward to see that some of
the functions diverge in the limit $m_{s}\to 0$. The presence of IR
divergences reflects the fact that now there are further contributions due to
mixing of four-quark operators with external $s$-quarks and the Darwin
operator, that must be included, whereas for the $b$-system, the corresponding
operators with external $c$-quarks did not. This point will be discussed
further later on, and we refer to Ref. [169] for more details.

Following Ref. [140], we try to analyse each of the contributions that enter
the HQE of a $D$ meson, in order to identify the presence of possible
cancellations which might affect the charm system. We start from the leading
order term $\Gamma_{3}$, cf. Eq. (1.164), which, including also NLO-QCD
corrections to the short distance coefficients, can be schematically written
as

$\Gamma_{3}=\Gamma_{0}\,c_{3}=\Gamma_{0}\left[3\,C_{1}^{2}\,{\cal
C}_{3,11}+2\,C_{1}C_{2}\,{\cal C}_{3,12}+3\,C_{2}^{2}\,{\cal C}_{3,22}+{\cal
C}_{3,\rm SL}\right]\,,$ (4.33)

where a summation over all modes is implied and we stress that now
$\Gamma_{0}$ is defined slightly differently compared to Eq. (3.24), i.e.

$\Gamma_{0}=\frac{G_{F}^{2}m_{c}^{5}}{192\pi^{3}}|V_{cs}|^{2}\,.$ (4.34)

At LO-QCD, the three non-leptonic coefficients ${\cal C}_{3,11}$, ${\cal
C}_{3,12}$, and ${\cal C}_{3,22}$, for each of the $c\to q_{1}\bar{q}_{2}u$
modes, reduce to ${\cal C}^{(q_{1}\bar{q}_{2}u)}_{0}$, computed in Chapter 3,
cf. Eq. (3.23) 444Up to the CKM factor
$\lambda_{q_{1}q_{2}}^{2}/|V_{cs}|^{2}$.. The expressions for the QCD
corrections to the non-leptonic coefficients ${\cal C}_{3,11}$ and ${\cal
C}_{3,22}$, as well as to ${\cal C}_{3,\rm SL}$, are obtained from Ref. [111].
In the latter, the computation has been performed for three arbitrary massive
final states of the decaying quark, hence their results can be easily applied
to all $c$-quark decay modes, by taking the appropriate mass limits. For the
coefficient ${\cal C}_{3,12}$, we use the results of Ref. [114], respectively
for the $c\to s\bar{d}u$, $c\to d\bar{s}u$ and $c\to d\bar{d}u$ decay
channels, and those of Ref. [118] in the case of two massive final states,
e.g. $c\to s\bar{s}u$. To compare the size of $c_{3}$ between the $b$\- and
$c$-system, it is interesting to consider the effect of the NLO corrections in
the case of dominant CKM non-leptonic and semileptonic modes 555For example
$b\to c\bar{u}d$, $b\to c\ell\nu_{\ell}$ and $c\to s\bar{d}u$, $c\to
s\ell\nu_{\ell}$ transitions., neglecting for simplicity the mass of the final
state particles. The result was determined in 1991 in Ref. [112], and reads
666Note that the factor $|V_{ud}|^{2}\approx 1$ has been omitted for
simplicity.

$\displaystyle c_{3}^{\rm NLO}-c_{3}^{\rm LO}$
$\displaystyle=8\frac{\alpha_{s}}{4\pi}\left[\underbrace{\left(\frac{25}{4}-\pi^{2}\right)}_{<0}\underbrace{+(C_{1}^{2}+C_{2}^{2})\left(\frac{31}{4}-\pi^{2}\right)}_{<0}\underbrace{-\frac{2}{3}C_{1}C_{2}\left(\frac{7}{4}+\pi^{2}\right)}_{\geq
0}\right]\,.$ (4.35)

The first term on the r.h.s. of Eq. (4.35), corresponds to the semileptonic
mode while the remaining two terms to the non-leptonic one. For the $b$-quark
decay, the NLO corrections are negative, while for the charm system, the third
term can dominate over the second one and lead to a positive correction to
$c_{3}$. Moreover, there is a sizeable enhancement of the QCD corrections for
non-leptonic $b$-quark decays due to finite charm quark mass effects [113,
114, 115, 118], whereas the corresponding increase for charm is much less
pronounced since $m_{c}^{2}/m_{b}^{2}\approx 0.1\gg m_{s}^{2}/m_{c}^{2}\approx
0.005$. A comparison of $\Gamma_{3}$, both at LO- and NLO-QCD, for different
$c$-quark mass schemes is shown in Table 4.4.

|  $\Gamma_{3}^{\rm LO}$ [ps-1] |  $\Gamma_{3}^{\rm NLO}$ [ps-1]
---|---|---
$m_{c}^{\rm Pole}=1.48$ GeV | $1.45_{-0.14}^{+0.17}$ | $1.52_{-0.16}^{+0.20}$
$\overline{m}_{c}(\overline{m}_{c})=1.27$ GeV | $0.69_{-0.09}^{+0.06}$ | $1.32_{-0.03}^{+0.06}$
$m_{c}^{\rm kin}(0.5\,{\rm GeV})=1.363$ GeV | $0.97_{-0.11}^{+0.10}$ | $1.47_{-0.30}^{+0.27}$
$m_{c}^{1S}=1.548$ GeV | $1.80_{-0.16}^{+0.24}$ | $2.12_{-0.30}^{+0.51}$
Table 4.4:  Numerical values of $\Gamma_{3}$ and LO- and NLO-QCD, using
different schemes for the $c$-quark mass. The uncertainties are obtained by
varying the renormalisation scale $\mu_{1}$ between 1 GeV and 3 GeV.

The range of values between $1.3\,{\rm ps}^{-1}$ to $2.7\,{\rm ps}^{-1}$ for
the free charm-quark decay at NLO-QCD, is in good agreement with the
experimental determinations in Table 4.1, and we find that the effect of a
non-vanishing strange quark mass leads to small corrections ($<5\%$).
Interestingly, the NLO-QCD result is affected by strong cancellations. We in
fact observe a suppression of the non-leptonic contribution because of the
opposite sign between the NLO corrections to the $\Delta C=1$ operators and to
their Wilson coefficients. Furthermore, a cancellation is present between the
semileptonic and the non-leptonic modes. In the ${\rm\overline{MS}}$ scheme an
additional NLO contribution arises from the conversion factor of $m_{c}^{5}$,
which is the origin of the large shift between the LO and the NLO value. This
is explicitly indicated in the following two equations

$\displaystyle{\rm(Pole)}\qquad\Gamma_{3}$ $\displaystyle=\Gamma_{3}^{\rm
LO}\,\left[1+\left(\overbrace{\underbrace{1.84}_{\rm
oper.}-\underbrace{0.74}_{\rm WC}}^{\rm NL}-\overbrace{0.67}^{\rm
SL}\right)\,\frac{\alpha_{s}}{\pi}+{\cal
O}\left(\frac{\alpha_{s}}{\pi}\right)^{2}\right]\,,$ (4.41)
$\displaystyle{\rm(\overline{MS})}\qquad\Gamma_{3}$
$\displaystyle=\Gamma_{3}^{\rm
LO}\,\left[1+\left(\overbrace{\underbrace{2.10}_{\rm
oper.}-\underbrace{0.70}_{\rm WC}}^{\rm NL}-\overbrace{0.71}^{\rm
SL}+\overbrace{6.66}^{\rm conv.fac.}\right)\,\frac{\alpha_{s}}{\pi}+{\cal
O}\left(\frac{\alpha_{s}}{\pi}\right)^{2}\right]\,.$ (4.42)

To obtain a first indication of the behaviour of the QCD series for
$\Gamma_{3}$ at higher orders, the authors of Ref. [140] have compared the
results for the NNLO- [127] and NNNLO- [26] QCD corrections to the semi
leptonic $b$-quark decay and the preliminary NNLO-QCD corrections for the non-
leptonic $b$-quark decay [129], concluding that higher order corrections seem
to be crucial for a reliable determination of $\Gamma_{3}$ 777Note that the
results presented in Ref. [129] are not complete and hence cannot be used for
phenomenological applications..

At order $1/m_{c}^{2}$ in the HQE, Eq. (1.164), we find the contribution of
the kinetic and the chromo-magnetic operators, defined in Eqs. (1.154),
(1.155), and respectively parametrised by the non perturbative input
$\mu_{\pi}^{2}$ and $\mu_{G}^{2}$, see Eq. (1.159). At this order and at LO-
QCD, we can schematically write

$\Gamma_{5}\frac{\langle{\cal
O}_{5}\rangle}{m_{c}^{2}}=\Gamma_{0}\left[c_{\mu_{\pi}}\frac{\mu_{\pi}^{2}}{m_{c}^{2}}+c_{G}\,\frac{\mu_{G}^{2}}{m_{c}^{2}}\right]\,,$
(4.43)

where now, compared to the corresponding ones introduced in Eq. (3.23), the
short distance coefficients $c_{\mu_{\pi}}$, $c_{G}$, contain also the
contribution due to the semileptonic modes as well as the dependence on the
CKM factor $\lambda_{q_{1}q_{2}}/|V_{cs}|^{2}$, due to the different
definition of $\Gamma_{0}$. Their expressions can be then obtained from those
of ${\cal C}_{G,mn}^{(q_{1}\bar{q}_{2}u)}$ and ${\cal
C}_{0}^{(q_{1}\bar{q}_{2}u)}$ listed in Appendix D. Note that again a
summation over all modes is implied. Specifically, we can decompose $c_{G}$ as

$c_{G}=3\,C_{1}^{2}\,c_{G,11}+2\,C_{1}C_{2}\,c_{G,12}+3\,C_{2}^{2}\,c_{G,22}+c_{G,SL}\,,$
(4.44)

which leads to the following expression if we neglect the strange and muon
masses and consider only the dominant CKM modes, i.e.

$c_{G}\approx-|V_{ud}|^{2}\left[\frac{9}{2}\Bigl{(}C_{1}^{2}+C_{2}^{2}\Bigr{)}+19\,C_{1}C_{2}\right]-3.$
(4.45)

Because of the large coefficient in front of $C_{1}C_{2}$ and of its negative
value, Eq. (4.45) can be affected by cancellations. This can be visualised in
Figure 4.1, in which we show the dependence of the function $c_{G}$ in Eq.
(4.44), on the renormalisation scale $\mu_{1}$, for both LO- and NLO-QCD,
$\Delta C=1$ Wilson coefficients. Note that the latter case in Figure 4.1, is
indicated in quotation marks since it does not represent the complete NLO
result, as corrections for non-leptonic modes are still missing and their
effect could significantly reduce the strong scale dependence. In particular,
from Figure 4.1, we see that a change of sign occurs in the region between $1$
and $2$ GeV, leading to a large uncertainty due to scale variation. The
numerical determinations of $\mu_{\pi}^{2}$ and $\mu_{G}^{2}$ are presented at
the end of this section.

Figure 4.1: Scale dependence of the coefficient of the chromo-magnetic
operator $c_{G}$.

We now turn to analyse the contribution of the Darwin operator, which arises
at order $1/m_{c}^{3}$ in Eq. (1.164) and which has been discussed in detail
in Chapter 3. This can be compactly written as

$\Gamma_{6}\frac{\langle{\cal
O}_{6}\rangle}{m_{c}^{3}}=\Gamma_{0}\,c_{\rho_{D}}\frac{\rho_{D}^{3}}{m_{c}^{3}}\,,$
(4.46)

where again a summation over all modes is implied and the coefficient
$c_{\rho_{D}}$ includes the effect of non-leptonic and semileptonic channels,
namely

$c_{\rho_{D}}=3\,C_{1}^{2}\,{\cal C}_{\rho_{D},11}+2\,C_{1}C_{2}\,{\cal
C}_{\rho_{D},12}+3\,C_{2}^{2}\,{\cal C}_{\rho_{D},22}+{\cal
C}_{\rho_{D},SL}\,.$ (4.47)

As already mentioned, the expressions for the non-leptonic coefficients
obtained in Eqs. (3.85)-(3.94), cannot be directly applied to the charm
sector, since by naively replacing $m_{b}\to m_{c}$ and $m_{c}\to m_{s}$, some
of the functions would develop infrared divergences in the limit $m_{s}\to 0$,
whereas in the $b$-system the corresponding coefficients were finite functions
of $\rho=m_{c}^{2}/m_{b}^{2}$. In fact, while we can assume $m_{b}\sim
m_{c}\gg\Lambda_{\rm QCD}$, and neglect the effect of four-quark operators
with external $c$-quarks in matrix elements between $B$-meson states 888We
recall that we do not consider the case of the $B_{c}$ meson., see e.g. Ref.
[169], in the charm sector, it is $m_{c}\gg m_{s}\sim\Lambda_{\rm QCD}$, and
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 $C_{1}^{2}$ and
$C_{1}C_{2}$. Using the same procedure as discussed in Ref. [137], the
coefficients of the Darwin operator required for the study of $D$-meson decays
have been computed in Ref. [140], and the analytical expressions, including
the full $s$-quark mass dependence, however finite in the limit $m_{s}\to 0$,
are listed in Appendix F, for all non-leptonic modes. The results for ${\cal
C}_{\rho_{D},SL}$, can then be obtained by setting, $N_{c}=1$, $C_{1}=1$,
$C_{2}=0$ and $m_{s}\to m_{\mu}$ in the case of $c\to s\mu^{+}\nu_{\mu}$
decay. By neglecting the strange and muon masses and by considering only the
dominant CKM modes, we have

$c_{\rho_{D}}\approx|V_{ud}|^{2}\,\left(18\,C_{1}^{2}\,-\frac{68}{3}\,C_{1}C_{2}+18\,C_{2}^{2}\right)+12\,.$
(4.48)

It is interesting to note that in this combination all terms have the same
sign and no cancellations arise. In Figure 4.2 we show the dependence on the
renormalisation scale $\mu_{1}$ of the function $c_{\rho_{D}}$ in Eq. (4.47),
where the quotation marks in the NLO-QCD result reflect again the fact that
only corrections due to the $\Delta C=1$ Wilson coefficients have been
included, since also in this case a complete determination of the NLO
corrections is still missing. Estimates for the matrix element of the Darwin
operator will be presented at the end of this section.

Figure 4.2: Scale dependence of the coefficient of the Darwin operator
$c_{\rho_{D}}$.

In order to discuss the contribution of four-quark operators at order
$1/m_{c}^{3}$ in Eq. (1.164), see Section 2.2, we introduce the following
basis, in complete analogy to Eqs. (2.99)-(2.102) 999Note however that now, we
do not use the tilde to denote the colour-octet operators. , namely

$\displaystyle O_{1}^{q}$
$\displaystyle=\big{(}\bar{c}\,\gamma_{\mu}(1-\gamma_{5})q\big{)}\,\big{(}\bar{q}\,\gamma^{\mu}(1-\gamma_{5})c\big{)}\,,$
(4.49) $\displaystyle O_{2}^{q}$
$\displaystyle=\big{(}\bar{c}(1-\gamma_{5})q\big{)}\,\big{(}\bar{q}(1+\gamma_{5})c\big{)}\,,$
(4.50) $\displaystyle O_{3}^{q}$
$\displaystyle=\big{(}\bar{c}\,\gamma_{\mu}(1-\gamma_{5})t^{a}q\big{)}\,\big{(}\bar{q}\,\gamma^{\mu}(1-\gamma_{5})t^{a}c\big{)}\,,$
(4.51) $\displaystyle O_{4}^{q}$
$\displaystyle=\big{(}\bar{c}(1-\gamma_{5})t^{a}q\big{)}\,\big{(}\bar{q}(1+\gamma_{5})t^{a}c\big{)}\,,$
(4.52)

where a summation over colour indices is implied and we have replaced $c_{v}$
with $c$, cf. Eq. (1.145). The parametrisation of the matrix element of the
operators in Eqs. (4.49)-(4.52) is given in Appendix G. However, by evaluating
them in the framework of the HQET, the dependence on the charm quark mass can
be further extracted from the $c$-quark field and meson state cf. Section
2.2.2, and in this case, the corresponding dimension-six operators read

$\displaystyle{\cal O}_{1}^{q}$
$\displaystyle=(\bar{h}_{v}\,\gamma_{\mu}(1-\gamma_{5})q)\,(\bar{q}\,\gamma^{\mu}(1-\gamma_{5})h_{v})\,,$
(4.53) $\displaystyle{\cal O}_{2}^{q}$
$\displaystyle=(\bar{h}_{v}(1-\gamma_{5})q)\,(\bar{q}(1+\gamma_{5})h_{v})\,,$
(4.54) $\displaystyle{\cal O}_{3}^{q}$
$\displaystyle=(\bar{h}_{v}\,\gamma_{\mu}(1-\gamma_{5})t^{a}q)\,(\bar{q}\,\gamma^{\mu}(1-\gamma_{5})t^{a}h_{v})\,,$
(4.55) $\displaystyle{\cal O}_{4}^{q}$
$\displaystyle=(\bar{h}_{v}(1-\gamma_{5})t^{a}q)\,(\bar{q}(1+\gamma_{5})t^{a}h_{v})\,,$
(4.56)

where $h_{v}$ denotes the effective heavy quark field, see Section 1.2. The
parametrisation in HQET of the matrix element of the operators in Eqs.
(4.53)-(4.56), see also Appendix G, can be written as

$\displaystyle\langle{D}_{q}|{\cal O}_{i}^{q}\,|{D}_{q}\rangle$
$\displaystyle=F^{2}(m_{c})\,m_{D_{q}}\left({\tilde{B}}_{i}^{q}+\tilde{\delta}^{qq}_{i}\right)=f_{D_{q}}^{2}m_{D_{q}}^{2}\left(1+\frac{4}{3}\frac{\alpha_{s}(m_{c})}{\pi}\right)\left({\tilde{B}}_{i}^{q}+\tilde{\delta}^{qq}_{i}\right)\,,$
(4.57) $\displaystyle\langle{D}_{q}|{\cal O}_{i}^{q^{\prime}}|{D}_{q}\rangle$
$\displaystyle=F^{2}(m_{c})\,m_{D_{q}}\tilde{\delta}^{qq^{\prime}}_{i}=f_{D_{q}}^{2}m_{D_{q}}^{2}\left(1+\frac{4}{3}\frac{\alpha_{s}(m_{c})}{\pi}\right)\tilde{\delta}^{qq^{\prime}}_{i},\quad
q\neq q^{\prime}\,,$ (4.58)

where $q,q^{\prime}=u,d,s$, ${\tilde{B}}_{i}^{q}$ are the Bag parameters
computed in HQET, while $F(\mu)$ and $f_{D_{q}}$ correspond respectively to
HQET and QCD decay constants, defined, by 101010The subscript ‘QCD’ or ‘HQET’
on the states is usually omitted, however for clarity it is specified in the
definition of the decay constant.

$\langle 0|\bar{q}\gamma^{\mu}\gamma_{5}c|{D_{q}(v)}\rangle_{\rm
QCD}=if_{D_{q}}\,p^{\mu},$ (4.59)

with $p^{\mu}=m_{D}v^{\mu}$, and

$\langle 0|\bar{q}\gamma^{\mu}\gamma_{5}h_{v}|{D}_{q}(v)\rangle_{\rm
HQET}=i\,F(\mu)\,\sqrt{m_{D_{q}}}\,v^{\mu}.$ (4.60)

The relation between $f_{D}$ and $F(\mu)$ up to QCD and power corrections, can
be found e.g. in Refs. [244, 245]. At the scale $\mu=m_{c}$, it reads

$\displaystyle
f_{D}=\frac{F(m_{c})}{\sqrt{m_{D}}}\left(1-\frac{2}{3}\frac{\alpha_{s}(m_{c})}{\pi}+\frac{G_{1}(m_{c})}{m_{c}}+6\,\frac{G_{2}(m_{c})}{m_{c}}-\frac{1}{2}\frac{\bar{\Lambda}}{m_{c}}\right),$
(4.61)

where $\bar{\Lambda}=m_{D}-m_{c}$, and the parameters $G_{1}$ and $G_{2}$
characterise matrix elements of non-local operators. Note that in Eqs. (4.57),
(4.58), by expressing the HQET decay constant in terms of the one defined in
QCD, we have included only corrections due to $\alpha_{s}$, which become part
of the NLO-QCD contribution at dimension-six. In fact, as we will discuss, the
power corrections can be absorbed in the contribution of some of the
dimension-seven operators appearing in HQET.

In vacuum insertion approximation (VIA), the Bag parameters of the colour-
singlet operators are equal to one, i.e. ${\tilde{B}}_{1,2}^{q}=1$, and those
of the colour-octet operators vanish, i.e. $\tilde{B}^{q}_{3,4}=0$. Note that
we assume isospin symmetry, so that ${\tilde{B}}_{i}^{u}={\tilde{B}}_{i}^{d}$.
The parameters $\tilde{\delta}^{qq}_{i}$, $\tilde{\delta}^{qq^{\prime}}_{i}$,
in (4.58), describe subleading effects in the non perturbative matrix
elements, compared to the dominant $\tilde{B}_{i}$, and correspond to the so
called eye-contractions, shown in Figure 4.3. In VIA, the contribution of all
eye-contractions vanish i.e.
$\tilde{\delta}^{qq}_{i}=\tilde{\delta}^{qq^{\prime}}_{i}=0$. However, beyond
vacuum insertion approximation, the matrix element of four-quark operators
with external $q^{\prime}$ quarks, differ from zero even when the spectator
quark $q$ in the $D_{q}$ meson does not coincide with $q^{\prime}$, as it is
indicated in Eq. (4.58) and in Figure 4.3. Again, due to isospin symmetry, we
assume $\tilde{\delta}^{uq^{\prime}}_{i}=\tilde{\delta}^{dq^{\prime}}_{i}$ and
$\tilde{\delta}^{qu}_{i}=\tilde{\delta}^{qd}_{i}$.

Figure 4.3: Diagram describing the eye-contractions.

The Bag parameters ${\tilde{B}}_{i}$ and $\tilde{\delta}_{i}^{qq^{\prime}}$
have been computed using HQET sum rules, specifically, the formed were
obtained for the $D^{+,0}$ mesons in Ref. [246], while corrections due to the
strange quark mass as well as the contribution of the eye-contractions, have
been determined for the first time in Ref. [247]. The numerical values of the
HQET Bag parameters are listed in Table 4.10.

By considering only the dominant CKM modes and by neglecting the effect of the
eye-contractions, at LO-QCD and at dimension-six, the contribution of four-
quark operators to the decay rate of the $D^{0},D^{+}$ and $D_{s}^{+}$ mesons,
cf. Eq. (1.164), reads

$\displaystyle
16\pi^{2}\tilde{\Gamma}^{D^{0}}_{6}\frac{\langle{\tilde{O}}_{6}\rangle^{D^{0}}}{m_{c}^{3}}$
$\displaystyle=\Gamma_{0}|V_{ud}^{*}|^{2}\,16\pi^{2}\frac{M_{{D^{0}}}f_{D^{0}}^{2}}{m_{c}^{3}}(1-x_{s})^{2}\Bigg{\\{}C_{\rm
WE}^{S}\left[({\tilde{B}}_{2}^{u}-{\tilde{B}}_{1}^{u})\right]$
$\displaystyle+x_{s}\,\left(2{\tilde{B}}_{2}^{u}-\frac{{\tilde{B}}_{1}^{u}}{2}\right)+C_{\rm
WE}^{O}\Big{[}({\tilde{B}}_{4}^{u}-\tilde{B}_{3}^{u})+x_{s}\,(2\,{\tilde{B}}_{4}^{u}-\frac{\tilde{B}_{3}^{u}}{2})\Big{]}\Bigg{\\}}\,,$
(4.62)

which corresponds to the WE topology,

$\displaystyle
16\pi^{2}\tilde{\Gamma}^{D^{+}}_{6}\frac{\langle{\tilde{O}}_{6}\rangle^{D^{+}}}{m_{c}^{3}}=\Gamma_{0}|V_{ud}^{*}|^{2}16\pi^{2}\,\frac{M_{{D^{+}}}f_{D^{+}}^{2}}{m_{c}^{3}}\,(1-x_{s})^{2}\,\Big{\\{}C_{\rm
PI}^{S}\,{\tilde{B}}_{1}^{d}+C_{\rm PI}^{O}\,\tilde{B}_{3}^{d}\Big{\\}}\,,$
(4.63)

describing the PI contribution, and

$\displaystyle
16\pi^{2}\tilde{\Gamma}^{D_{s}^{+}}_{6}\frac{\langle{\tilde{O}}_{6}\rangle^{D_{s}^{+}}}{m_{c}^{3}}$
$\displaystyle=\Gamma_{0}|V_{ud}^{*}|^{2}16\pi^{2}\frac{M_{{D_{s}^{+}}}f_{D_{s}^{+}}^{2}}{m_{c}^{3}}\Bigg{\\{}\left(C_{\rm
WA}^{S}+\frac{2}{|V_{ud}^{*}|^{2}}\right)\left({\tilde{B}}_{2}^{s}-{\tilde{B}}_{1}^{s}\right)$
$\displaystyle+C_{\rm
WA}^{O}\left(\tilde{B}_{4}^{s}-\tilde{B}_{3}^{s}\right)\Bigg{\\}}\,,$ (4.64)

due to WA. Here $x_{s}=m_{s}^{2}/m_{c}^{2}$, and we have introduced the
following notation for the combinations of Wilson coefficients, cf. Eqs.
(2.80), (2.88), and (2.94), namely

$\displaystyle C_{\rm
WE}^{S}=\frac{1}{3}C_{1}^{2}+2\,C_{1}C_{2}+3\,C_{2}^{2},$ $\displaystyle
C_{\rm WE}^{O}=2\,C_{1}^{2},$ (4.65) $\displaystyle C_{\rm
PI}^{S}=C_{1}^{2}+6\,C_{1}C_{2}+C_{2}^{2},$ $\displaystyle C_{\rm
PI}^{O}=6\,(C_{1}^{2}+C_{2}^{2})\,,$ (4.66) $\displaystyle C_{\rm
WA}^{S}=3\,C_{1}^{2}+2\,C_{1}C_{2}+\frac{1}{3}C_{2}^{2},$ $\displaystyle
C_{\rm WA}^{O}=2\,C_{2}^{2}\,,$ (4.67)

where the superscript “S” and “O” refers to coefficient in front of the
colour-singlet and of colour-octet Bag parameters, respectively. Note that in
Eq. (4.64), the contribution due to the muon mass in the semileptonic decay
$c\to s\mu^{+}\nu_{\mu}$, has been neglected. The expressions in Eqs.
(4.62)-(4.64) lead to some interesting numerical effects. First, in the charm
system, one expects that the contribution due to the spectator quark is of
similar size compared to the leading term $\Gamma_{3}$ in the HQE, unless some
additional cancellations are present. Using the pole mass $m_{c}^{\rm
Pole}=1.48$ GeV and Lattice QCD values for the decay constants [248], roughly
yields

$\displaystyle
16\pi^{2}\frac{M_{{D^{0}}}f_{D^{0}}^{2}}{m_{c}^{3}}={4.1}\approx{\cal
O}(c_{3})\,,\qquad
16\pi^{2}\frac{M_{{D_{s}^{+}}}f_{D_{s}^{+}}^{2}}{m_{c}^{3}}={6.0}\approx{\cal
O}(c_{3})\,.$ (4.68)

This result has led the authors of Ref. [249] to propose a different way to
rearrange the HQE series in the charm sector. However, to investigate further
the size of four-quark contributions at dimension-six, we consider the
combinations of Wilson coefficients that appear in Eqs. (4.62) - (4.64). A
comparison of these coefficients at LO- and NLO-QCD, for different values of
the renormalisation scale $\mu_{1}$ is shown in Table 4.5.

$\mu_{1}$ [GeV] |  1 |  1.206 |  1.27 |  1.48 |  1.67 |  3
---|---|---|---|---|---|---
$C_{\rm WE}^{\rm S}(\rm LO)$ |  $\phantom{-}0.09$ |  $\phantom{-}0.04$ |  $\phantom{-}0.03$ |  $\phantom{-}0.01$ |  $\phantom{-}0.00$ |  $\phantom{-}0.01$
$C_{\rm WE}^{\rm S}(\rm NLO)$ |  $-0.03$ |  $-0.03$ |  $-0.03$ |  $-0.02$ |  $-0.02$ |  $\phantom{-}0.04$
$C_{\rm WE}^{\rm O}(\rm LO)$ |  $\phantom{-}3.57$ |  $\phantom{-}3.30$ |  $\phantom{-}3.24$ |  $\phantom{-}3.08$ |  $\phantom{-}2.98$ |  $\phantom{-}2.63$
$C_{\rm WE}^{\rm O}(\rm NLO)$ |  $\phantom{-}3.11$ |  $\phantom{-}2.93$ |  $\phantom{-}2.89$ |  $\phantom{-}2.77$ |  $\phantom{-}2.70$ |  $\phantom{-}2.44$
$C_{\rm PI}^{\rm S}(\rm LO)$ |  $-2.80$ |  $-2.25$ |  $-2.12$ |  $-1.79$ |  $-1.57$ |  $-0.79$
$C_{\rm PI}^{\rm S}(\rm NLO)$ |  $-1.74$ |  $-1.36$ |  $-1.28$ |  $-1.04$ |  $-0.88$ |  $-0.27$
$C_{\rm PI}^{\rm O}(\rm LO)$ |  $\phantom{-}13.0$ |  $\phantom{-}11.7$ |  $\phantom{-}11.4$ |  $\phantom{-}10.6$ |  $\phantom{-}10.1$ |  $\phantom{-}8.50$
$C_{\rm PI}^{\rm O}(\rm NLO)$ |  $\phantom{-}10.6$ |  $\phantom{-}9.73$ |  $\phantom{-}9.55$ |  $\phantom{-}9.05$ |  $\phantom{-}8.72$ |  $\phantom{-}7.60$
$C_{\rm WA}^{\rm S}(\rm LO)$ |  $\phantom{-}3.82$ |  $\phantom{-}3.65$ |  $\phantom{-}3.61$ |  $\phantom{-}3.51$ |  $\phantom{-}3.45$ |  $\phantom{-}3.24$
$C_{\rm WA}^{\rm S}(\rm NLO)$ |  $\phantom{-}3.57$ |  $\phantom{-}3.45$ |  $\phantom{-}3.42$ |  $\phantom{-}3.35$ |  $\phantom{-}3.31$ |  $\phantom{-}3.16$
$C_{\rm WA}^{\rm O}(\rm LO)$ |  $\phantom{-}0.77$ |  $\phantom{-}0.59$ |  $\phantom{-}0.55$ |  $\phantom{-}0.46$ |  $\phantom{-}0.40$ |  $\phantom{-}0.21$
$C_{\rm WA}^{\rm O}(\rm NLO)$ |  $\phantom{-}0.41$ |  $\phantom{-}0.32$ |  $\phantom{-}0.30$ |  $\phantom{-}0.24$ |  $\phantom{-}0.21$ |  $\phantom{-}0.10$
Table 4.5: Comparison of the combinations $C_{\rm WE,PI,WA}^{\rm S,O}$,
respectively at LO- and NLO-QCD, for different values of the renormalisation
scale $\mu_{1}$.

The first observation is that $C^{\rm S}_{\rm WE}$ is strongly suppressed.
Moreover, in Eq. (4.65) the Bag parameters of the colour-singlet operators
exactly cancel in VIA. On the other side, the combination of Wilson
coefficients in front of the colour-octet operators is not suppressed for weak
exchange, indicating that both colour structures 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
$C_{\rm PI}^{O}$ and $C_{\rm PI}^{S}$ get large modifications, and even a
change of sign, compared to the case $C_{1}=1$ and $C_{2}=0$ revealing the
importance of gluon radiative corrections. Moreover $C_{\rm PI}^{O}$ is
enhanced compared to $C_{\rm PI}^{S}$, again indicating that both colour
structures might be equally important for PI. In the case of weak
annihilation, $C_{\rm WA}^{S}$ is large. On the other hand, the Bag parameters
of the colour-singlet operators exactly cancel in VIA. 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 four-quark operators to the total decay width at order
$1/m_{c}^{3}$, schematically reads

scheme | $D^{0}$ | $D^{+}$ | $D^{+}_{s}$
---|---|---|---
VIA
Pole | $\underbrace{-0.06}_{\rm NLO}=\underbrace{0.00}_{\rm LO}\underbrace{-0.06}_{\rm\Delta NLO}$ | $\underbrace{-11.3}_{\rm NLO}=\underbrace{-7.18}_{\rm LO}\underbrace{-4.13}_{\rm\Delta NLO}$ | $\underbrace{-0.85}_{\rm NLO}=\underbrace{-0.51}_{\rm LO}\underbrace{-0.34}_{\rm\Delta NLO}$
$\overline{\rm MS}$ | $\underbrace{-0.09}_{\rm NLO}=\underbrace{0.00}_{\rm LO}\underbrace{-0.09}_{\rm\Delta NLO}$ | $\underbrace{-22.9}_{\rm NLO}=\underbrace{-11.3}_{\rm LO}\underbrace{-11.5}_{\rm\Delta NLO}$ | $\underbrace{-1.66}_{\rm NLO}=\underbrace{-0.77}_{\rm LO}\underbrace{-0.89}_{\rm\Delta NLO}$
Kinetic | $\underbrace{-0.08}_{\rm NLO}=\underbrace{0.00}_{\rm LO}\underbrace{-0.08}_{\rm\Delta NLO}$ | $\underbrace{-16.3}_{\rm NLO}=\underbrace{-9.18}_{\rm LO}\underbrace{-7.14}_{\rm\Delta NLO}$ | $\underbrace{-1.21}_{\rm NLO}=\underbrace{-0.64}_{\rm LO}\underbrace{-0.57}_{\rm\Delta NLO}$
$1S$ | $\underbrace{-0.05}_{\rm NLO}=\underbrace{0.00}_{\rm LO}\underbrace{-0.05}_{\rm\Delta NLO}$ | $\underbrace{-10.1}_{\rm NLO}=\underbrace{-6.27}_{\rm LO}\underbrace{-3.82}_{\rm\Delta NLO}$ | $\underbrace{-0.76}_{\rm NLO}=\underbrace{-0.45}_{\rm LO}\underbrace{-0.31}_{\rm\Delta NLO}$
HQET SR
Pole | $\underbrace{0.06}_{\rm NLO}=\underbrace{0.10}_{\rm LO}\underbrace{-0.04}_{\rm\Delta NLO}$ | $\underbrace{-12.3}_{\rm NLO}=\underbrace{-7.97}_{\rm LO}\underbrace{-4.37}_{\rm\Delta NLO}$ | $\underbrace{-0.93}_{\rm NLO}=\underbrace{-0.69}_{\rm LO}\underbrace{-0.24}_{\rm\Delta NLO}$
$\overline{\rm MS}$ | $\underbrace{0.23}_{\rm NLO}=\underbrace{0.17}_{\rm LO}\underbrace{+0.06}_{\rm\Delta NLO}$ | $\underbrace{-24.9}_{\rm NLO}=\underbrace{-12.6}_{\rm LO}\underbrace{-12.3}_{\rm\Delta NLO}$ | $\underbrace{-1.78}_{\rm NLO}=\underbrace{-1.06}_{\rm LO}\underbrace{-0.72}_{\rm\Delta NLO}$
Kinetic | $\underbrace{0.12}_{\rm NLO}=\underbrace{0.13}_{\rm LO}\underbrace{-0.01}_{\rm\Delta NLO}$ | $\underbrace{-17.8}_{\rm NLO}=\underbrace{-10.2}_{\rm LO}\underbrace{-7.61}_{\rm\Delta NLO}$ | $\underbrace{-1.31}_{\rm NLO}=\underbrace{-0.87}_{\rm LO}\underbrace{-0.43}_{\rm\Delta NLO}$
$1S$ | $\underbrace{0.06}_{\rm NLO}=\underbrace{0.09}_{\rm LO}\underbrace{-0.03}_{\rm\Delta NLO}$ | $\underbrace{-11.0}_{\rm NLO}=\underbrace{-6.97}_{\rm LO}\underbrace{-4.05}_{\rm\Delta NLO}$ | $\underbrace{-0.83}_{\rm NLO}=\underbrace{-0.60}_{\rm LO}\underbrace{-0.23}_{\rm\Delta NLO}$
Table 4.6: Dimension-six contributions to the $D$-mesons decay width
normalised by $\Gamma_{0}$ and split up into LO-QCD and NLO-QCD corrections
within different mass schemes and using both VIA and HQET SR values for the
Bag parameters.

$\displaystyle\tilde{\Gamma}^{D_{q}}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle^{D_{q}}}{m_{c}^{3}}$
$\displaystyle=\frac{\Gamma_{0}}{{|V_{cs}|^{2}}}\\!\\!\\!\sum_{q_{1},q_{2}=d,s}\\!\\!\\!\left|\lambda_{q_{1}q_{2}}\right|^{2}\sum_{i=1}^{4}\Bigg{[}A_{i,q_{1}q_{2}}^{\rm
WE}\frac{\langle D_{q}|{\cal
O}_{i}^{u}|D_{q}\rangle}{m_{c}^{3}}+A_{i,q_{1}q_{2}}^{\rm PI}\frac{\langle
D_{q}|{\cal O}_{i}^{q_{2}}|D_{q}\rangle}{m_{c}^{3}}$
$\displaystyle+A_{i,q_{1}q_{2}}^{\rm WA}\frac{\langle D_{q}|{\cal
O}_{i}^{q_{1}}|D_{q}\rangle}{m_{c}^{3}}\Bigg{]}+\frac{\Gamma_{0}}{{|V_{cs}|^{2}}}\sum_{q_{1}=d,s}|V_{cq_{1}}|^{2}\sum_{\ell=e,\mu}\sum_{i=1}^{4}\left[A_{i,q_{1}\ell}^{\rm
WA}\frac{\langle D_{q}|{\cal
O}_{i}^{q_{1}}|D_{q}\rangle}{m_{c}^{3}}\right]\,,$ (4.93)

where the matrix elements of the four-quark operators are given in Eqs.
(4.57), (4.58), and the short-distance coefficients for the WE, PI and WA
topologies are denoted by $A_{i,q_{1}q_{2}}^{\rm WE}$, $A_{i,q_{1}q_{2}}^{\rm
PI}$ and $A_{i,q_{1}q_{2}}^{\rm WA}$, $A_{i,q_{1}\ell}^{\rm WA}$,
respectively. Their expressions at LO-QCD have been derived in Section 2.2.1,
while NLO corrections to $A_{i,q_{1}q_{2}}^{\rm WE}$ and
$A_{i,q_{1}q_{2}}^{\rm PI}$ have been computed in Ref. [142]. The
corresponding results for $A_{i,q_{1}q_{2}}^{\rm WA}$ can be obtained by using
Eq. (2.92), since the Fierz symmetry is respected also at one-loop. For the
semileptonic modes, the coefficients $A_{i,q_{1}\ell}^{\rm WA}$ have been
determined in Ref. [143]. Note that in our analysis, we treat the contribution
of the $\tilde{\delta}_{i}^{qq^{\prime}}$ parameters effectively as a NLO
effect, therefore their coefficients are included only at LO-QCD. Finally, in
Table 4.6, we compare the size of the LO- and NLO-QCD corrections in Eq.
(4.93), normalised by $\Gamma_{0}$, both in VIA and using HQET SR results for
the Bag parameters. The NLO-QCD corrections turn out to have an essential
numerical effect for the contribution of four-quark at order $1/m_{c}^{3}$. In
particular, in the case of the $D^{0}$ and $D_{s}^{+}$ mesons, they lift the
helicity suppression present in the weak exchange and weak annihilation
topologies at LO-QCD and in VIA. Note that for the $D^{+}_{s}$ meson, in
addition to the CKM dominant WA contribution, there is a correction due to the
CKM suppressed, but nevertheless large PI topology. In the case of the $D^{+}$
meson the NLO corrections to Pauli interference are very large, $50\%-100\%$
depending on the mass scheme. Already in the $B$ system they were found to be
of the order of $30\%$ for the ratio $\tau(B^{+})/\tau(B_{d})$, in the pole
scheme, see e.g. Ref. [141]. We conclude that, neglecting these contributions
in the study of charm lifetime, as it has been previously done in Ref. [250],
is clearly not justified and the determination of higher order corrections
would be highly desirable.

We now consider the contribution of four-quark operators at order
$1/m_{c}^{4}$ to the HQE in Eq. (1.164), which have been discussed in detail
in Section 2.2.2. By expanding $p^{\mu}=p_{c}^{\mu}\pm p_{q}^{\mu}$, only in
the small momentum of the light spectator quark $p_{q}\sim\Lambda_{QCD}$,
leads to the following basis for the dimension-seven operators 111111 Note
that the basis used e.g. in Ref. [143], is redundant, since it contains also
the additional operator denoted by $P_{2}^{q}$, related to $P_{1}^{q}$ by
hermitean conjugation, namely
$P_{2}^{q}=m_{q}\,(\bar{c}(1+\gamma_{5})q)(\bar{q}(1+\gamma_{5})c)=(P^{q}_{1})^{\dagger}$.
Leading to the same matrix element, we do not include this operator in our
basis.

$\displaystyle P_{1}^{q}$
$\displaystyle=m_{q}\,(\bar{c}(1-\gamma_{5})q)(\bar{q}(1-\gamma_{5})c)\,,$
(4.94) $\displaystyle P_{2}^{q}$
$\displaystyle=\frac{1}{m_{c}}(\bar{c}\overset{\leftarrow}{D_{\nu}}\gamma_{\mu}(1-\gamma_{5})D^{\nu}q)(\bar{q}\gamma^{\mu}(1-\gamma_{5})c)\,,$
(4.95) $\displaystyle P_{3}^{q}$
$\displaystyle=\frac{1}{m_{c}}(\bar{c}\overset{\leftarrow}{D_{\nu}}(1-\gamma_{5})D^{\nu}q)(\bar{q}(1+\gamma_{5})c)\,,$
(4.96)

together with the corresponding
$\tilde{P}_{1}^{q},\tilde{P}_{2}^{q},\tilde{P}_{3}^{q}$, containing the
generators $t^{a}$. Due to the presence in Eqs. (4.95), (4.96), 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, contrary to
those defined in HQET, cf. Eqs. (4.98), (4.99). Moreover, note that this basis
differs from the one used in Ref. [144] for the computation of the dimension-
seven and dimension-eight contributions. By evaluating the matrix elements of
the dimension-seven four-quark operators using the framework of the HQET, we
have to further expand the charm quark momentum, according to
$p_{c}^{\mu}=m_{c}v^{\mu}+k^{\mu}$, see Section 1.2, as well as to include
$1/m_{c}$ corrections to the effective heavy quark field and to the HQET
Lagrangian, cf. Section 2.2.2. In this case, we obtain the following basis, in
complete analogy to Eqs. (2.112)-(2.120), namely

$\displaystyle{\cal P}_{1}^{q}$
$\displaystyle=m_{q}\,(\bar{h}_{v}(1-\gamma_{5})q)(\bar{q}(1-\gamma_{5})h_{v})\,,$
(4.97) $\displaystyle{\cal P}_{2}^{q}$
$\displaystyle=(\bar{h}_{v}\gamma_{\mu}(1-\gamma_{5})(iv\cdot
D)q)(\bar{q}\gamma^{\mu}(1-\gamma_{5})h_{v})\,,$ (4.98) $\displaystyle{\cal
P}_{3}^{q}$ $\displaystyle=(\bar{h}_{v}(1-\gamma_{5})(iv\cdot
D)q)(\bar{q}(1+\gamma_{5})h_{v})\,,$ (4.99)

due to the contribution of the light spectator quark momentum,

$\displaystyle{\cal R}_{1}^{q}$
$\displaystyle=(\bar{h}_{v}\gamma_{\mu}(1-\gamma_{5})q)(\bar{q}\gamma^{\mu}(1-\gamma_{5})(i\not{D})h_{v})\,,$
(4.100) $\displaystyle{\cal R}_{2}^{q}$
$\displaystyle=(\bar{h}_{v}(1-\gamma_{5})q)(\bar{q}(1+\gamma_{5})(i\not{D})h_{v})\,,$
(4.101)

due to $1/m_{c}$ corrections to the effective heavy quark field $h_{v}$, and

$\displaystyle{\cal M}_{1}^{q}$ $\displaystyle=i\int d^{4}y\,{\rm
T}\left[{\cal O}_{1}^{q}(0),(\bar{h}_{v}(iD)^{2}h_{v})(y)\right],$ (4.102)
$\displaystyle{\cal M}_{2}^{q}$ $\displaystyle=i\int d^{4}y\,{\rm
T}\left[{\cal
O}_{1}^{q}(0),\frac{1}{2}g_{s}\left(\bar{h}_{v}\sigma_{\alpha\beta}G^{\alpha\beta}h_{v}\right)(y)\right],$
(4.103) $\displaystyle{\cal M}_{3}^{q}$ $\displaystyle=i\int d^{4}y\,{\rm
T}\left[{\cal O}_{2}^{q}(0),(\bar{h}_{v}(iD)^{2}h_{v})(y)\right],$ (4.104)
$\displaystyle{\cal M}_{4}^{q}$ $\displaystyle=i\int d^{4}y\,{\rm
T}\left[{\cal
O}_{2}^{q}(0),\frac{1}{2}g_{s}\left(\bar{h}_{v}\sigma_{\alpha\beta}G^{\alpha\beta}h_{v}\right)(y)\right],$
(4.105)

due to $1/m_{c}$ corrections to the HQET Lagrangian, which we have explicitly
indicated, cf. Eq. (1.69), (1.70). Moreover, the set of operators in Eqs.
(4.97)-(4.105), are supplemented by the corresponding colour octet ones. To
parametrise the matrix element of the dimension-seven operators in HQET, we
use VIA and account for deviations from it by including the corresponding Bag
parameters, as it is explicitly shown in Appendix G. However, since for these
matrix elements, does not exist a 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 non perturbative parameters $F(\mu)$, $G_{1}(\mu)$,
$G_{2}(\mu)$, and $\bar{\Lambda}$, 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 [248]. In
doing so, we obtain that in VIA and at the matching scale $\mu=m_{c}$, the
contribution of the local operators ${\cal R}_{1,2}^{q}$, as well as that of
the non-local ${\cal M}_{1}^{q}$, ${\cal M}_{2}^{q}$, ${\cal M}_{3}^{q}$ and
${\cal M}_{4}^{q}$, can be entirely absorbed in the QCD decay constant
$f_{D}$, cf. Eq. (4.61), more precisely, in the QCD matrix element of the
dimension-six operators in Eqs. (4.49), (4.52), which are proportional to
$f_{D}$, and the only remaining $1/m_{c}$ contribution is due to the operators
${\cal P}_{1,2,3}^{q}$, analogously to the QCD case 121212In the matrix
element of $\tilde{P}_{1,2,3}^{q}$ one can replace the HQET decay constant
with the QCD one, up to higher order corrections.. To make this point more
clear, we consider as an example the contribution due to Pauli interference at
LO-QCD and up to order $1/m_{c}^{4}$, in the case of $c\to s\bar{d}u$
transition, which constitutes the dominant correction to $\Gamma(D^{+})$,
namely

$\displaystyle{\rm Im}\,{\cal T}^{\rm PI}$
$\displaystyle=\Gamma_{0}\,|V_{ud}^{*}|^{2}\,\frac{32\pi^{2}}{m_{c}^{3}}(1-x_{s})^{2}\Biggl{[}C_{\rm
PI}^{S}\left({\cal O}_{1}^{d}+\frac{{\cal R}_{1}^{d}}{m_{c}}+\frac{{\cal
M}_{1}^{d}}{m_{c}}+\frac{{\cal
M}_{2}^{d}}{m_{c}}+2\frac{1+x_{s}}{1-x_{s}}\frac{{\cal
P}_{3}^{q}}{m_{c}}\right)$ $\displaystyle+({\rm singlet}\to{\rm
octet})\Biggr{]}\,.$ (4.106)

By evaluating the matrix element of ${\rm Im}{\cal T}^{\rm PI}$ in VIA, the
contribution due to the colour-octet operators vanishes. Moreover, using the
parametrisation for the matrix elements of the four-quark operators given in
Eq. (4.57) and in Appendix G, we obtain in VIA and setting $\mu=m_{c}$, that

$\displaystyle\langle{\cal O}_{1}^{d}+\frac{{\cal
R}_{1}^{d}}{m_{c}}+\frac{{\cal M}_{1}^{d}}{m_{c}}+\frac{{\cal
M}_{2}^{d}}{m_{c}}\rangle_{\rm HQET}$
$\displaystyle=F^{2}(m_{c})\,m_{D^{+}}\left[1-\frac{\bar{\Lambda}}{m_{c}}+\frac{2\,G_{1}(m_{c})}{m_{c}}+\frac{12\,G_{2}(m_{c})}{m_{c}}\right]$
$\displaystyle=f_{D}^{2}\,m_{D^{+}}^{2}=\langle O_{1}^{d}\rangle_{\rm QCD},$
(4.107)

where in the second line we have used the conversion between the QCD and HQET
decay constants given in Eq. (4.61). From Eq. (4.107) we see that the
contribution of the local operators ${\cal R}_{1}^{q}$ and non-local operators
${\cal M}^{q}_{1}$ and ${\cal M}^{q}_{2}$, is entirely absorbed by using the
QCD decay constant. Note that, by neglecting the effect due to the strange
quark mass and using VIA we reproduce the result in Eq. (19) of Ref. [249].
The same argument applies also to the remaining topologies i.e. WE and WA.
However, it is worth remarking 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 contribution of ${\cal R}_{i}^{q}$, and ${\cal M}^{q}_{i}$
in HQET, can be completely absorbed in $f_{D}$ by evaluating the matrix
elements in VIA. 131313Note, that for the operator ${O}_{2}^{q}$ the
contribution of ${\cal R}_{2}^{q}$ is absorbed by the combination
$(m_{D}\,f_{D}/m_{c})^{2}\approx(1+2\,\bar{\Lambda}/m_{c})\,f_{D}^{2}$.. We
stress that in our numerical analysis, we employ this argument also when using
the results of the Bag parameters determined from HQET SR, by neglecting the
small deviation from their corresponding VIA values. Note that a detailed
analysis of the dimension-seven contributions within the HQET has been
performed in Ref. [245] for the case of $B-\bar{B}$-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 $\mu\sim$ 1 GeV, is numerically
small ($\sim 5\%$ for Ref. [245]).141414By neglecting the effect of running
down to a lower scale, from Ref. [245] one can see that in VIA the QCD decay
constant entirely absorbs all the $1/m_{b}$ contributions. Finally, by summing
over all the CKM modes, at LO-QCD, the dimension-seven contribution can be
presented as

$\displaystyle 16\pi^{2}\tilde{\Gamma}^{D_{q}}_{7}\frac{\langle\tilde{\cal
O}_{7}\rangle^{D_{q}}}{m_{c}^{4}}$
$\displaystyle=\frac{\Gamma_{0}}{{|V_{cs}|^{2}}}\\!\\!\\!\sum_{q_{1},q_{2}=d,s}\Bigg{\\{}\left|\lambda_{q_{1}q_{2}}\right|^{2}\\!\sum_{i=1}^{3}\Big{[}G_{i,q_{1}q_{2}}^{\rm
WE}\frac{\langle D_{q}|{\cal
P}_{i}^{u}|D_{q}\rangle}{m_{c}^{4}}+G_{i,q_{1}q_{2}}^{\rm PI}\frac{\langle
D_{q}|{\cal P}_{i}^{q_{2}}|D_{q}\rangle}{m_{c}^{4}}$
$\displaystyle+G_{i,q_{1}q_{2}}^{\rm WA}\frac{\langle D_{q}|{\cal
P}_{i}^{q_{1}}|D_{q}\rangle}{m_{c}^{4}}\Big{]}+|V_{cq_{1}}|^{2}\sum_{\ell=e,\mu}\sum_{i=1}^{3}\left[G_{i,q_{1}\ell}^{\rm
WA}\frac{\langle D_{q}|{\cal P}_{i}^{q_{1}}|D_{q}\rangle}{m_{c}^{4}}\right]$
$\displaystyle+(\mbox{colour-octet part})\Bigg{\\}}\,,$ (4.108)

and we confirm the results for the short-distance coefficients
$G_{i,q_{1}q_{2}}^{\rm WE}$, $G_{i,q_{1}q_{2}}^{\rm PI}$ and
$G_{i,q_{1}q_{2}}^{\rm WA}$, $G_{i,q_{1}\ell}^{\rm WA}$ presented in Ref.
[143]. 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.
$\langle D^{0}|{\cal P}_{i}^{s}|D^{0}\rangle=0$.

Having presented each of the contributions that enter the HQE of $D$-mesons,
we now turn to discuss the numerical evaluation of the corresponding matrix
elements. For most of the non perturbative parameters in the charm-sector
there is no determination available yet, contrary to the $b$-system, where
e.g. the value of $\mu_{G}^{2}$, $\mu_{\pi}^{2}$, and $\rho_{D}^{3}$, has been
extracted performing fits to experimental data for $B^{0}$ and $B^{+}$
semileptonic decays [251]. Specifically, for $\mu_{G}^{2}$ they find [251]

$\mu_{G}^{2}(B)=(0.332\pm 0.062)\,{\rm GeV}^{2}\,.$ (4.109)

By using the heavy quark symmetry, we could expect the corresponding parameter
in the $D$ system to have a similar size. The value of $\mu_{G}^{2}$ can also
be obtained taking into account the spectroscopy relation [252]

$\mu_{G}^{2}(D_{(s)})=\frac{3}{2}m_{c}\,\left(M_{D_{(s)}^{*}}-M_{D_{(s)}}\right)\,,$
(4.110)

which holds up to power corrections. Using the value for the meson masses
given in the PDG [191] and setting $m_{c}=1.27\,{\rm GeV}$, we obtain the
following estimates

$\mu_{G}^{2}(D)=(0.268\pm 0.107)\,{\rm
GeV}^{2},\qquad\mu_{G}^{2}(D_{s})=(0.274\pm 0.110)\,{\rm GeV}^{2},$ (4.111)

where we have conservatively added an uncertainty of $40\%$ due to unknown
power corrections of order $1/m_{c}$. The values in Eq. (4.111) are roughly
19$\%$ smaller than those obtained from experimental fits for semileptonic
$B$-meson decays, see Eq. (4.109). Moreover, Eq. (4.110) leads to a tiny
amount of $SU(3)_{f}$-symmetry breaking of $\approx 2\%$, which might,
however, be enhanced by the neglected power corrections. In the literature
instead of Eq. (4.110) it is often adopted the relation [70, 52]

$\mu_{G}^{2}(D_{(s)})=\frac{3}{4}\left(M_{D_{(s)}^{*}}^{2}-M_{D_{(s)}}^{2}\right)\,,$
(4.112)

which coincides with Eq. (4.110) up to corrections of order $1/m_{c}$.
Numerically we find that Eq. (4.112) yields

$\mu_{G}^{2}(D)=0.41\,{\rm GeV}^{2}\,,\qquad\mu_{G}^{2}(D_{s}^{+})=0.44\,{\rm
GeV}^{2}\,,$ (4.113)

which are roughly 23$\%$ higher than the values in Eq. (4.109). In our
numerical analysis we take the average of the two determinations in Eq.
(4.111) and Eq. (4.113). This gives

$\mu_{G}^{2}(D)=(0.34\pm 0.10)\,{\rm
GeV}^{2},\qquad\mu_{G}^{2}(D_{s}^{+})=(0.36\pm 0.10)\,{\rm GeV}^{2}\,,$
(4.114)

Source |  LQCD [253] |  LQCD [254] |  Exp. fit [251] |  QCD SR [255] |  QCD SR [256]
---|---|---|---|---|---
$\mu_{\pi}^{2}[{\rm GeV}^{2}]$ |  0.05(22) |  0.314(15) |  0.465(68) |  0.10(5) |  0.6(1)
Table 4.7: Different determinations of $\mu_{\pi}^{2}(B)$ available in the
literature.

which agrees well with the one in Eq. (4.109). From Eq. (4.43), we expect
corrections to the total decay rate due to the chromo-magnetic operator,
$c_{G}\,\mu_{G}^{2}/(c_{3}\,m_{c}^{2})$ ranging between $-6\%$ and $+8\%$ with
respect to the leading free-quark decay contribution. A large part of the
sizeable uncertainty derives from the cancellations in the coefficient
$c_{G}$, as shown in Figure 4.1, which could be reduced with a complete
determination of the NLO-QCD corrections. For semileptonic rates the
contribution of the chromomagnetic operator can be even of the order of
$20\%$. An experimental determination of $\mu_{G}^{2}(D)$ 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.

For the matrix element of the kinetic operator, there are several predictions
of $\mu_{\pi}^{2}$, available in the literature for the $B$-meson, which cover
a large range of values, see Table 4.7. Assuming heavy quark symmetry we can
again use the determination in Ref. [251]

$\mu_{\pi}^{2}(B)=(0.465\pm 0.068)\,{\rm GeV}^{2}\,,$ (4.117)

to obtain the following estimate in the case of $D$-meson

$\mu_{\pi}^{2}(D)=(0.465\pm 0.198)\,{\rm GeV}^{2},$ (4.118)

where we have added a conservative uncertainty of $40\%$ in order to account
for the breaking of the heavy quark symmetry. This value fulfils the
theoretical bound $\mu_{\pi}^{2}\geq\mu_{G}^{2}$, see e.g. the review [257].
We then expect, from Eq. (4.43), corrections due to the kinetic operator of
the order of $-10\%$. The $SU(3)_{f}$ breaking effects for the matrix element
of the kinetic operator have been estimated in Refs. [258, 143], i.e.

$\mu_{\pi}^{2}(D_{s}^{+})-\mu_{\pi}^{2}(D^{0})\approx 0.09\,{\rm GeV}^{2}\,,$
(4.119)

leading to the following estimate in the case of the $D_{s}$ meson

$\mu_{\pi}^{2}(D_{s}^{+})=(0.555\pm 0.232)\,{\rm GeV}^{2}.$ (4.120)

Again a more precise experimental determination of $\mu_{\pi}^{2}$ from fits
to semileptonic $D^{+}$, $D^{0}$ and $D_{s}^{+}$ meson decays, as it has been
done for the $B^{+}$ and $B^{0}$ decays, would be very desirable.

For the matrix element of the Darwin operator, we can again assume the
validity of the heavy quark symmetry and use the corresponding value obtained
from fits of the semileptonic $B$ decays [251], namely

$\rho_{D}^{3}(B)=(0.170\pm 0.038)\,{\rm GeV}^{3}\,,$ (4.121)

which, by adding quadratically an uncertainty of $40\%$ to account for the
breaking of the heavy-quark symmetry, leads to a first estimate of

$\displaystyle\rho_{D}^{3}(D)^{\rm I}=(0.17\pm 0.07)\,{\rm GeV}^{3}\,.$
(4.122)

Alternatively the Darwin parameter can be related to the matrix elements of
the dimension-six four-quark operators through the equation of motion for the
gluon field. At leading order in $1/m_{Q}$, we obtain

$\displaystyle\rho_{D}^{3}(H_{Q})$
$\displaystyle=\frac{g_{s}^{2}}{18}f_{H_{Q}}^{2}\,m_{H_{Q}}\Bigg{[}2\,\tilde{B}_{2}^{q}-\tilde{B}_{1}^{q}+\frac{3}{4}\tilde{B}_{3}^{q}-\frac{3}{2}\tilde{B}_{4}^{q}$
$\displaystyle+\sum_{q^{\prime}=u,d,s}\left(2\tilde{\delta}_{2}^{q^{\prime}q}-\tilde{\delta}_{1}^{q^{\prime}q}+\frac{3}{4}\tilde{\delta}_{3}^{q^{\prime}q}-\frac{3}{2}\tilde{\delta}_{4}^{q^{\prime}q}\right)\Bigg{]}\,,$
(4.123)

where $H_{Q}$ is a heavy hadron with the mass $m_{H_{Q}}$ and the decay
constant $f_{H_{Q}}$, $q=u,d,s,$ is the light valence quark in $H_{Q}$, and
the Bag parameters have bee introduced in Eqs. (4.57). The strong coupling
$g^{2}_{s}=4\pi\alpha_{s}$, should be evaluated at a non perturbative scale
and e.g. Ref. [259] suggests to set $\alpha_{s}=1$.

|  $\mu=1.5$ GeV |  $\mu=1.0$ GeV | $\alpha_{s}=1$
---|---|---|---
$\rho_{D}^{3}[{\rm GeV^{3}}]$ | VIA | HQET | VIA | HQET | VIA | HQET
$B^{+},B_{d}$ | 0.048 | 0.047 | 0.066 | 0.064 | 0.133 | 0.129
$B_{s}$ | 0.072 | 0.070 | 0.098 | 0.095 | 0.199 | 0.193
$D^{+},D^{0}$ | 0.021 | 0.020 | 0.027 | 0.026 | 0.059 | 0.056
$D_{s}^{+}$ | 0.030 | 0.029 | 0.040 | 0.038 | 0.086 | 0.082
Table 4.8: Values of $\rho_{D}^{3}(H)$ for $B$\- and $D$-mesons in VIA and
using HQET SR for the Bag parameters for three different choices of
$\alpha_{s}$ in Eq. (4.123).

From the input listed in Table 4.9 and the expression in Eq. (4.123), we can
estimate the size of $\rho_{D}^{3}$ for both the $B$\- and $D$-mesons and
using the VIA as well as the HQET SR values for the Bag parameters. The
results are summarised in Table 4.8 for the three different choices, namely
$\alpha_{s}(\mu=1.5\,{\rm GeV})$, $\alpha_{s}(\mu=1\,{\rm GeV})$ and
$\alpha_{s}=1$. By setting $\alpha_{s}=1$ in Eq. (4.123), we obtain values for
$\rho_{D}^{3}(B)$ that are close to the one in Eq. (4.121), indicating
$1/m_{b}$-corrections in Eq. (4.123) of the order of $+30\%$. Moreover, the
difference between using VIA and HQET sum rule is small. We emphasise that
because of the sizeable $SU(3)_{F}$ breaking in the decay constants, Eq.
(4.123) leads also to a sizeable $SU(3)_{F}$ breaking for the non-perturbative
parameters $\rho_{D}^{3}(D)$, $\rho_{D}^{3}(D_{s}^{+})$. By setting
$\alpha_{s}=1$ and using HQET SR results for the Bag parameters we arrive at
the second estimate, cf. last column in Table 4.8

$\displaystyle\rho_{D}^{3}(D)^{II}=(0.056\pm 0.022)\,{\rm
GeV}^{3}\,,\quad\rho_{D}^{3}(D_{s}^{+})^{II}=(0.82\pm 0.033)\,{\rm
GeV}^{3}\,,$ (4.130)

where we have again added $40\%$ uncertainty. Finally, another possibility to
extract $\rho_{D}^{3}(D)$ is to substitute in Eq. (4.123) the values of the
Bag parameters in VIA, which gives

$\rho_{D}^{3}(H_{Q})\approx\frac{g_{s}^{2}}{18}f_{H_{Q}}^{2}\,m_{H_{Q}}.$
(4.131)

Assuming the strong coupling to have a similar size for both the $B$\- and
$D$-meson matrix elements, from Eq. (4.131) we obtain

$\displaystyle\rho_{D}^{3}(D)\approx\frac{f_{D}^{2}\,m_{D}}{f_{B}^{2}\,m_{B}}\,\rho_{D}^{3}(B)\,,\quad\rho_{D}^{3}(D_{s})\approx\frac{f_{D_{s}}^{2}m_{D_{s}}}{f_{B}^{2}\,m_{B}}\,\rho_{D}^{3}(B)\,.$
(4.132)

Using the most precise determination of the decay constants from Lattice QCD
[248], and of the meson masses from PDG [191] and taking into account the
value of $\rho_{D}^{3}(B)$ in Eq. (4.121), leads to the third estimate

$\displaystyle\rho_{D}^{3}(D)^{III}=(0.075\pm 0.034)\,{\rm
GeV^{3}}\,,\quad\rho_{D}^{3}(D_{s})^{III}=(0.110\pm 0.050)\,{\rm GeV^{3}}\,,$
(4.133)

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 4.8 for $\alpha_{s}=1$. Contrary to the case of the dimension-five
matrix elements, in Eq. (4.133) we observe a large $SU(3)_{f}$ symmetry
breaking of $\approx 46\%$, and similarly of $\approx 49\%$ for the
$B_{(s)}$-mesons, as already stated above, mostly stemming from the ratios
$f_{B_{s}}/f_{B_{d}}$ and $f_{D_{s}^{+}}/f_{D^{0}}$. In our numerical analysis
we use the values shown in Eq. (4.133), which lie between the estimates
obtained in Eq. (4.122) and Eq. (4.130). Again, a more precise experimental
determination of $\rho_{D}^{3}$ from fits to semileptonic $D^{+}$, $D^{0}$ and
$D_{s}^{+}$ meson decays, as it has been 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.

Finally, the dimension-six Bag parameters of the $D^{+}$ and $D^{0}$ mesons
have been determined using HQET sum rules in Ref. [246]. Corrections due to
the inclusion of the strange quark mass, needed in the case of the $D_{s}^{+}$
meson, as well as the effect of the eye-contractions, have been computed for
the first time in Ref. [247], again using HQET sum rules. The results,
collected in Table 4.10, show only a small deviation from the corresponding
VIA values. For the dimension-seven Bag parameters we use only VIA. In HQET
the matrix elements of dimension-seven operators depend also on the parameters
$\bar{\Lambda}_{(s)}=m_{D_{(s)}}-m_{c}$, for which we use the following range
of values [247]

$\bar{\Lambda}=(0.5\pm 0.1)\,{\rm GeV},\qquad\bar{\Lambda}_{s}=(0.6\pm
0.1)\,{\rm GeV}\,.$ (4.134)

#### 4.1.2 Numerical results

Parameter |  Value |  Source
---|---|---
$\alpha_{s}(M_{Z})$ |  $0.1179\pm 0.0010$ |  PDG [191]
$|V_{us}|$ |  $0.224834^{+0.000252}_{-0.000059}$ |  CKMfitter [260]
$|V_{ub}|/|V_{cb}|$ |  $0.088496^{+0.001885}_{-0.002244}$ |
$|V_{cb}|$ |  $0.04162^{+0.00026}_{-0.00080}$ |
$\delta$ |  $\left(65.80^{+0.94}_{-1.29}\right)^{\circ}$ |
$\overline{m}_{c}(\overline{m}_{c})$ |  $(1.27\pm 0.02)\,{\rm GeV}$ |  PDG [191]
$m_{c}^{\rm kin}(0.5\,{\rm GeV})$ |  $1.306\,{\rm GeV}$ |  [241]
$m_{J/\psi}$ |  $3.0969$ GeV |  PDG [191]
$m_{s}$ |  $\left(93^{+11}_{-5}\right)\,{\rm MeV}$ |  PDG [191]
$M_{D^{0}}$ |  $1.86493\,{\rm GeV}$ |
$M_{D^{+}}$ |  $1.86965\,{\rm GeV}$ |  PDG [191]
$M_{D_{s}^{+}}$ |  $1.96834\,{\rm GeV}$ |
$f_{D}$ |  $(0.2120\pm 0.0007)$ GeV |  Lattice QCD [248]
$f_{D_{s}}$ |  $(0.2499\pm 0.0005)$ GeV |
$\mu_{\pi}^{2}(D)$ |  $(0.465\pm 0.198)$ GeV2 |  Exp. fit [251] and HQ symmetry
$\mu_{\pi}^{2}(D_{s})$ |  $(0.555\pm 0.232)$ GeV2 |  $SU(3)_{f}$-breaking [258] and HQ symmetry
$\mu_{G}^{2}(D)$ |  $(0.339\pm 0.098)$ GeV2 |  Spectroscopy relations [252, 70]
$\mu_{G}^{2}(D_{s})$ |  $(0.357\pm 0.104)$ GeV2 |
$\rho_{D}^{3}(D)$ |  $(0.075\pm 0.034)$ GeV3 |  Exp. fit [251] and E.O.M relations
$\rho_{D}^{3}(D_{s})$ |  $(0.110\pm 0.050)$ GeV3 |
Table 4.9: Numerical input used in the numerical analysis.

${\rm HQET}$ | $\tilde{B}_{1}$ | $\tilde{B}_{2}$ | $\tilde{B}_{3}$ | $\tilde{B}_{4}$
---|---|---|---|---
$D^{+,0}$ | $1.0000^{+0.0020}_{-0.0006}$ | $\phantom{-}1.0000^{+0.0007}_{-0.0000}$ | $-0.0161^{+0.0115}_{-0.0206}$ | $-0.0007^{+0.0104}_{-0.0170}$
$D_{s}^{+}$ | $1.0000^{+0.0014}_{-0.0003}$ | $\phantom{-}1.0000^{+0.0007}_{-0.0000}$ | $-0.0094^{+0.0103}_{-0.0171}$ | $-0.0001^{+0.0104}_{-0.0169}$

${\rm HQET}$ | $\tilde{\delta}_{1}$ | $\tilde{\delta}_{2}$ | $\tilde{\delta}_{3}$ | $\tilde{\delta}_{4}$
---|---|---|---|---
$\langle D_{q}|\tilde{O}^{q}|D_{q}\rangle$ | $0.0026^{+0.0004}_{-0.0009}$ | $-0.0018^{+0.0005}_{-0.0002}$ | $-0.0004^{+0.0001}_{-0.0001}$ | $\phantom{-}0.0003^{+0.0000}_{-0.0001}$
$\langle D_{s}|\tilde{O}^{q}|D_{s}\rangle$ | $0.0025^{+0.0004}_{-0.0008}$ | $-0.0018^{+0.0005}_{-0.0002}$ | $-0.0004^{+0.0001}_{-0.0001}$ | $\phantom{-}0.0003^{+0.0000}_{-0.0001}$
$\langle D_{q}|\tilde{O}^{s}|D_{q}\rangle$ | $0.0017^{+0.0005}_{-0.0009}$ | $-0.0012^{+0.0005}_{-0.0003}$ | $-0.0003^{+0.0001}_{-0.0001}$ | $\phantom{-}0.0002^{+0.0001}_{-0.0001}$
$\langle D_{s}|\tilde{O}^{s}|D_{s}\rangle$ | $0.0023^{+0.0005}_{-0.0009}$ | $-0.0017^{+0.0005}_{-0.0002}$ | $-0.0004^{+0.0002}_{-0.0001}$ | $\phantom{-}0.0003^{+0.0000}_{-0.0001}$

Table 4.10: Numerical values of the HQET Bag parameters [246, 247], at the
renormalisation scale $\mu_{0}=1.5\,{\rm GeV}$.

In this section, using all the ingredients described above, we present the
theoretical predictions for the total and semileptonic decay rates of the
$D^{0}$, $D^{+}$ and $D_{s}^{+}$ mesons, and for their ratios. All the input
included in our numerical analysis are collected in Table 4.9. For each
observable we investigate several quark mass schemes, using as default the
kinetic and the $1S$ scheme, and compare the corresponding results with both
VIA and HQET SR values for the dimension-six Bag parameters. The uncertainties
quoted are obtained by varying all the input parameters within their
intervals. For the renormalisation scales, we fix the central values to
$\mu_{1}=\mu_{0}=1.5\,{\rm GeV}$ 151515The renormalisation scale $\mu_{0}$
enters in the NLO-QCD corrections to the dimension-six coefficients as well as
in the running of the Bag parameters., and vary both of them independently
between 1 and 3 GeV. Moreover, we add an estimated uncertainty due to missing
higher power and QCD corrections.

VIA
---
Observable |  Pole |  $\overline{\rm MS}$ |  Kinetic |  $1S$ | Exp. value
$\Gamma(D^{0})[{\rm ps}^{-1}]$ |  $1.68$ |  $1.47$ |  $1.56$ |  $2.31$ | $2.44$
$\Gamma(D^{+})[{\rm ps}^{-1}]$ |  $0.19$ |  $-0.03$ |  $0.09$ |  $0.56$ | $0.96$
$\bar{\Gamma}(D_{s}^{+})[{\rm ps}^{-1}]$ |  $1.72$ |  $1.48$ |  $1.58$ |  $2.34$ | $1.88$
$\tau(D^{+})/\tau(D^{0})$ |  $2.55$ |  $2.56$ |  $2.53$ |  $2.82$ | $2.54$
$\bar{\tau}(D_{s}^{+})/\tau(D^{0})$ |  $0.99$ |  $1.00$ |  $0.99$ |  $0.99$ | $1.30$
$B_{sl}^{D^{0}}[\%]$ |  $5.31$ |  $6.46$ |  $6.03$ |  $8.48$ | $6.49$
$B_{sl}^{D^{+}}[\%]$ |  $13.5$ |  $16.4$ |  $15.3$ |  $21.5$ | $16.07$
$B_{sl}^{D^{+}_{s}}[\%]$ |  $6.88$ |  $8.24$ |  $7.74$ |  $10.8$ | $6.30$
$\Gamma_{sl}^{D^{+}}/\Gamma_{sl}^{D^{0}}$ |  $1.000$ |  $1.000$ |  $1.000$ |  $1.000$ | $0.985$
$\Gamma_{sl}^{D^{+}_{s}}/\Gamma_{sl}^{D^{0}}$ |  $1.04$ |  $1.04$ |  $1.04$ |  $1.05$ | $0.790$
Table 4.11: Central values of the charm observables in different quark mass
schemes using VIA for the matrix elements of the four-quark operators compared
to the corresponding experimental values (last column).

We start by considering the total decay rates, which are expected to have
sizeable 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. A comparison of the central values for the HQE prediction of the
decay widths in several mass schemes is shown in the three first rows of Table
4.11, using VIA for the Bag parameters and of Table 4.12 using the HQET sum
rules results. In Table 4.13 we present the complete theoretical prediction
including the corresponding uncertainties, using the $1S$ and kinetic scheme
for the quark masses and the HQET SR values for the dimension-six Bag
parameters, the same results can be visualised also in Figure 4.4. In each
table, the corresponding experimental determinations are listed in the last
column. For the $D_{s}^{+}$ meson there is an additional subtlety due to the
large branching fraction of the leptonic decay
$D_{s}^{+}\to\tau^{+}\nu_{\tau}$, which however is not included in the HQE,
since the tau lepton is more massive than the charm quark.

HQET SR
---
Observable |  Pole |  $\overline{\rm MS}$ |  Kinetic |  $1S$ | Exp. value
$\Gamma(D^{0})[{\rm ps}^{-1}]$ |  $1.71$ |  $1.50$ |  $1.59$ |  $2.34$ | $2.44$
$\Gamma(D^{+})[{\rm ps}^{-1}]$ |  $-0.05$ |  $-0.25$ |  $-0.14$ |  $0.29$ | $0.96$
$\bar{\Gamma}(D_{s}^{+})[{\rm ps}^{-1}]$ |  $1.70$ |  $1.46$ |  $1.56$ |  $2.32$ | $1.88$
$\tau(D^{+})/\tau(D^{0})$ |  $2.83$ |  $2.83$ |  $2.80$ |  $3.14$ | $2.54$
$\bar{\tau}(D_{s}^{+})/\tau(D^{0})$ |  $1.01$ |  $1.02$ |  $1.01$ |  $1.01$ | $1.30$
$B_{sl}^{D^{0}}[\%]$ |  $5.18$ |  $6.37$ |  $5.93$ |  $8.34$ | $6.49$
$B_{sl}^{D^{+}}[\%]$ |  $13.2$ |  $16.2$ |  $15.1$ |  $21.2$ | $16.07$
$B_{sl}^{D^{+}_{s}}[\%]$ |  $6.79$ |  $8.19$ |  $7.67$ |  $10.7$ | $6.30$
$\Gamma_{sl}^{D^{+}}/\Gamma_{sl}^{D^{0}}$ |  $1.002$ |  $1.001$ |  $1.001$ |  $1.002$ | $0.985$
$\Gamma_{sl}^{D^{+}_{s}}/\Gamma_{sl}^{D^{0}}$ |  $1.05$ |  $1.05$ |  $1.05$ |  $1.05$ | $0.790$
Table 4.12: Central values of the charm observables in different quark mass
schemes using HQET sum rule results [246, 247] for the matrix elements of the
four-quark operators compared to the corresponding experimental values (last
column).

Using the experimental value of the leptonic branching ratio [191] (online
update), we obtain

${\rm Br}(D_{s}^{+}\to\tau^{+}\nu_{\tau})=(5.48\pm 0.23)\%\,,$ (4.188)

accordingly, we define the reduced decay rate $\bar{\Gamma}(D_{s}^{+})$, as

$\bar{\Gamma}(D_{s}^{+})\equiv{\Gamma}(D_{s}^{+})-\Gamma(D_{s}^{+}\to\tau^{+}\nu_{\tau})=(1.88\pm
0.02)\,\mbox{ps}^{-1}\,,$ (4.189)

which leads to the reduced lifetime ratio

$\frac{\bar{\tau}(D_{s}^{+})}{\tau(D^{0})}=1.30\pm 0.01\,.$ (4.190)

The first and main result we can derive from Table 4.13 and from Figure 4.4,
is that the HQE can reproduce the experimental values of $\Gamma(D^{0})$,
$\Gamma(D^{+})$, and $\Gamma(D_{s}^{+})$, within very large uncertainties.

Observable | Kinetic scheme |  $1S$-scheme | Exp. value
---|---|---|---
$\Gamma(D^{0})[{\rm ps}^{-1}]$ | $1.590\pm 0.242^{+0.451\,+0.002}_{-0.365\,-0.002}$ | $2.348\pm 0.247^{+0.651\,+0.002}_{-0.489\,-0.001}$ | $2.44\pm 0.01$
$\Gamma(D^{+})[{\rm ps}^{-1}]$ | $-0.138\pm 0.572^{+0.581\,+0.252}_{-0.273\,-0.102}$ | $0.293\pm 0.664^{+0.939\,+0.360}_{-0.453\,-0.161}$ | $0.96\pm 0.01$
$\bar{\Gamma}(D_{s}^{+})[{\rm ps}^{-1}]$ | $1.572\pm 0.309^{+0.508\,+0.018}_{-0.399\,-0.004}$ | $2.330\pm 0.349^{+0.734\,+0.027}_{-0.540\,-0.009}$ | $1.88\pm 0.02$
$\tau(D^{+})/\tau(D^{0})$ | $2.798\pm 0.606^{+0.001\,+0.109}_{-0.135\,-0.263}$ | $3.137\pm 0.691^{+0.023\,+0.169}_{-0.299\,-0.376}$ | $2.54\pm 0.02$
$\bar{\tau}(D_{s}^{+})/\tau(D^{0})$ | $1.010\pm 0.105^{+0.018\,+0.003}_{-0.030\,-0.010}$ | $1.010\pm 0.118^{+0.027\,+0.006}_{-0.044\,-0.015}$ | $1.30\pm 0.01$
$B_{sl}^{D^{0}}[\%]$ | $5.94\pm 1.15^{+0.33}_{-0.28}$ | $8.36\pm 1.31^{+0.23}_{-0.08}$ | $6.49\pm 0.11$
$B_{sl}^{D^{+}}[\%]$ | $15.1\pm 2.91^{+0.83}_{-0.72}$ | $21.2\pm 3.32^{+0.58}_{-0.19}$ | $16.07\pm 0.30$
$B_{sl}^{D^{+}_{s}}[\%]$ | $7.73\pm 1.80^{+0.45}_{-0.40}$ | $10.76\pm 2.18^{+0.33}_{-0.04}$ | $6.30\pm 0.16$
$\Gamma_{sl}^{D^{+}}/\Gamma_{sl}^{D^{0}}$ | $1.002\pm 0.002\pm 0.001$ | $1.002\pm 0.003\pm 0.001$ | $0.985\pm 0.028$
$\Gamma_{sl}^{D^{+}_{s}}/\Gamma_{sl}^{D^{0}}$ | $1.053\pm 0.130^{+0.006}_{-0.007}$ | $1.060\pm 0.164^{+0.006}_{-0.007}$ | $0.790\pm 0.026$
Table 4.13:  HQE predictions for all the ten observables in the kinetic
(second column) and in the $1S$-schemes (third column), using HQET SR results
for the Bag parameters. The first uncertainty is parametric one, second and
third uncertainties are due to $\mu_{1}$\- and $\mu_{0}$-scales variation,
respectively. The results are compared with the corresponding experimental
measurements (fourth column).

Moreover, we find that in the 1$S$ scheme we obtain larger values for the
decay rates, while the kinetic and the $\overline{\rm MS}$ scheme typically
result in smaller values, close to the pole scheme. Within the uncertainties
the predictions in the different mass schemes are compatible with each other,
however, to a large extent. Given the current precision then, to consider only
one quark mass scheme might lead to considerably underestimate the
uncertainties. Due to the fact that the values of the HQET Bag parameters
[246, 247] are close to the corresponding ones in VIA, the predictions shown
in Table 4.11 and in Table 4.12 do not differ much. A peculiar role is played
by the $D^{+}$ meson, for which we obtain huge theoretical uncertainties
because of the large negative value of the Pauli interference contribution at
dimension-six. This term actually dominates the total decay rate. Furthermore,
the large negative value is enhanced by the NLO-QCD corrections, but partly
compensated by the dimension-seven contribution. In this respect, having an
independent determination of the HQET sum rule results, e.g. with a lattice
QCD computation, as well as higher order QCD corrections to dimension-six and
seven might could significantly bring more insights.

Figure 4.4: A comparison of the HQE prediction for the charm observables in
the kinetic scheme (magenta) and in the $1S$ scheme (blue), with the
corresponding experimental data (green). Note that all the quantities are
normalised to the corresponding experimental central values.

For the determination of the lifetime ratios, in order to eliminate the
contribution of the free-quark decay, we use

$\frac{\tau(D^{+}_{(s)})}{\tau(D^{0})}=1+\left(\Gamma^{\rm
HQE}(D^{0})-\Gamma^{\rm HQE}(D^{+}_{(s)})\right)\tau^{\rm
exp}(D^{+}_{(s)})\,.$ (4.202)

In Eq. (4.202), $\Gamma_{3}$ cancels exactly and $\Gamma_{5}$ and $\Gamma_{6}$
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 results for the HQE
prediction of the lifetime ratios, in several mass schemes, are shown in the
fourth and fifth rows of Table 4.11, Table 4.12, Table 4.13 as well as in
Figure 4.4. We observe that the large lifetime ratio $\tau(D^{+})/\tau(D^{0})$
is well reproduced in all schemes considered, while in the case of
$\tau(D_{s}^{+})/\tau(D^{0})$ the HQE result lies closer to one, compared to
the experimental value. In the latter case, the theoretical prediction is
dominated by $SU(3)_{f}$-symmetry breaking effects in the non-perturbative
parameters $\mu_{\pi}^{2}$, $\mu_{G}^{2}$ and $\rho_{D}^{3}$, which are only
very roughly known. Future and more precise determinations of their values can
considerably improve our conclusion for these lifetime ratios.

Figure 4.5: A comparison of the HQE prediction for the charm observables in
the kinetic scheme (magenta) and in the $1S$ scheme (blue) with the
corresponding experimental data (green). All the quantities are normalised to
the corresponding experimental central values. In comparison to Fig. 4.4 now
the NNLO corrections to the semileptonic branching fractions are included,
taken from the talk of Matteo Fael at the CHARM-2020 conference.

In the case of the inclusive semileptonic decays, we introduce the shorthand
notations $\Gamma_{sl}^{D}\equiv\Gamma(D\to Xe^{+}\nu_{e})$ and
$B_{sl}^{D}\equiv{\rm Br}(D\to Xe^{+}\nu_{e})$. The theoretical predictions
are then obtained as

$B_{sl}^{D,\rm HQE}=\Gamma_{sl}^{D,\rm HQE}\cdot\tau(D)^{\rm Exp.}\,.$ (4.203)

The HQE results in several mass schemes are shown in the sixth, seventh and
eighth row of Table 4.11, Table 4.12 and Table 4.13, as well as in Figure 4.4.
In the kinetic scheme all HQE predictions for the semi-leptonic branching
fractions cover the experimental values, while the results in the 1$S$ scheme
tend to be too large. It is interesting to note, that by adding NNLO-QCD
corrections to the semileptonic decays, the difference between the two quark
mass schemes is considerably reduced, see Figure 4.5. Using the experimental
values respectively for the $D^{0}$ lifetime and the semileptonic branching
fraction, we determine the semileptonic ratios in the following way

$\displaystyle\frac{\Gamma_{sl}^{D^{+}}}{\Gamma_{sl}^{D^{0}}}=1+\left[\Gamma_{sl}^{D^{+}}-\Gamma_{sl}^{D^{0}}\right]^{\rm
HQE}\left[\frac{\tau(D^{0})}{B_{sl}^{D^{0}}}\right]^{\rm exp}\,,$ (4.204)
$\displaystyle\frac{\Gamma_{sl}^{D_{s}^{+}}}{\Gamma_{sl}^{D^{0}}}=1+\left[\Gamma_{sl}^{D^{+}_{s}}-\Gamma_{sl}^{D^{0}}\right]^{\rm
HQE}\left[\frac{\tau(D^{0})}{B_{sl}^{D^{0}}}\right]^{\rm exp}\,.$ (4.205)

The HQE results for these ratios are shown in the ninth and tenth row of Table
4.11, Table 4.12 and Table 4.13 and in Figure 4.4. In agreement with the
experimental data, the HQE leads to values for
$\Gamma_{sl}^{D^{+}}/\Gamma_{sl}^{D^{0}}$ very close to one. Also for
$\Gamma_{sl}^{D_{s}^{+}}/\Gamma_{sl}^{D^{0}}$ the corresponding theoretical
prediction is close to one, however, the corresponding experimental number is
as low as 0.79, confirming the necessity of having better control over the
$SU(3)_{f}$-symmetry breaking effects in the non-perturbative parameters
$\mu_{G}^{2}$, $\mu_{\pi}^{2}$ and $\rho_{D}^{3}$ for the $D$ mesons.

### 4.2 Charm mixing

In Section 4.1 we have shown that the HQE is able to reproduce, within large
theoretical uncertainties, the experimental pattern for the lifetime of
charmed mesons. However, a naive application of the HQE yields results for the
decay rate difference of neutral $D$ mesons that are four orders of magnitude
smaller than the experimental ones. It is well known that this huge
suppression results from severe Glashow-Iliopoulos-Maiani (GIM) cancellations
[261]. Following Ref. [235], we discuss a possible explanation for the large
discrepancy between the theoretical prediction for $D$-mixing and experimental
data. We stress though, that we do not present a detailed derivation of the
fundamentals of the theory of mixing, for which, instead, we refer to the
comprehensive reviews [194, 195, 196].

#### 4.2.1 GIM cancellations in $D$-mixing

Because of the weak interaction, neutral mesons, here we consider the case of
the $D^{0}$ meson, can mix with their corresponding antiparticles through the
box diagrams shown in Figure 4.6. The process is described by a $2\times 2$
Hamiltonian matrix with non vanishing off-diagonal entries $M_{12}$ and
$\Gamma_{12}$. By diagonalising the mixing matrix of the $D^{0}$ and the
$\bar{D}^{0}$ mesons, we can obtain the two eigenstates with definite mass and
decay width. The corresponding observables $\Delta M_{D}$ and
$\Delta\Gamma_{D}$, denoting respectively the mass and decay width difference
between the two eigenstates, are functions of $\Gamma_{12}$ and $M_{12}$,.
Moreover we define

$x=\frac{\Delta M_{D}}{\Gamma_{D^{0}}}\,,\qquad
y=\frac{\Delta\Gamma_{D}}{2\Gamma_{D^{0}}}\,,$ (4.206)

and

$x_{12}=\frac{2\,|M_{12}|}{\Gamma_{D^{0}}}\,,\qquad
y_{12}=\frac{2\,|\Gamma_{12}|}{\Gamma_{D^{0}}}\,,\qquad\phi_{12}=\arg\left(\frac{M_{12}}{\Gamma_{12}}\right)\,.$
(4.207)

In the following we only discuss the computation of $\Gamma_{12}$ and, by
taking into account the bound $\Delta\Gamma_{D}\leq 2\,|\Gamma_{12}|$, see
e.g. Refs. [195, 262], we derive the theoretical prediction for
$\Delta\Gamma_{D}$. In fact, we do not consider the calculation of $M_{12}$,
and hence we can only determine one contribution to the mixing phase
$\phi_{12}$ in Eq. (4.207).

Figure 4.6: Box diagrams contributing to mixing of neutral $D$-mesons.

$\Gamma_{12}$ corresponds to the absorptive part of the mixing amplitude of
the $D^{0}$ \- $\bar{D}^{0}$ system, and it is then obtained by computing the
imaginary part of the matrix element of the effective Hamiltonian describing
the $c$-quark decay, between the $D^{0}$ and $\bar{D}^{0}$ states. Using the
formalism described in Section 1.4, $\Gamma_{12}$ can be expanded in inverse
powers of the heavy $c$-quark mass, leading to

$\Gamma_{12}=\left[\Gamma_{6}^{(0)}+\frac{\alpha_{s}}{4\pi}\,\Gamma_{6}^{(1)}+\ldots\right]\frac{\langle
Q_{6}\rangle}{m_{c}^{3}}+\ldots\,,$ (4.208)

where the ellipsis stand for terms of higher order and we have explicitly
shown the perturbative expansion of the short distance coefficient
$\Gamma_{6}$, cf. Eq. (1.165). Eq. (1.164), is diagrammatically represented in
Figure 4.7. The product of $\Delta C=1$ operators in the effective
Hamiltonian, i.e. in the “full” theory, cf. Eq. (4.20), is now matched into a
series of local $\Delta C=2$ operators $Q_{d}$ of increasing dimension $d\geq
6$, with the short distance coefficients denoted by $\Gamma_{d}$. The
expressions for $\Gamma_{6}^{(0)}$ can be easily derived in complete analogy
to what it has been done in Section 2.2, while those for $\Gamma_{6}^{(1)}$
can be obtained from the corresponding ones for $B$-mixing determined in Refs.
[263, 264, 265, 266, 267, 268] by replacing $m_{b}\to m_{c}$, $m_{c}\to
m_{s}$, etc. Furthermore, the matrix elements of the dimension-six operators
have been computed e.g. in Refs. [246, 269]. The experimental value of the
decay rate difference reads 161616Note that for consistency we present the
numbers used in the analysis of Ref. [235] based on the previous determination
$y=0.68^{+0.06}_{-0.07}\,\%$, however, using the new value quoted in Eq.
(4.7), would not lead to any significant difference, and we would have instead
$\Delta\Gamma_{D}^{\rm Exp}\geq 0.027$ ps-1.

$\Delta\Gamma_{D}^{\rm Exp}=2y/\tau(D^{0})=(0.032\pm 0.003){\,\rm ps}^{-1},$
(4.209)

which, at one standard deviation, leads to the following bound

$\Delta\Gamma_{D}^{\rm Exp}\geq 0.028\,{\rm ps}^{-1}\,.$ (4.210)

In order to compare the theoretical predictions with the experimental
determinations, we investigate the quantities

$\alpha=-\arg(\Gamma_{12})\,,\qquad\Omega=\frac{2\,|\Gamma_{12}|^{\rm
SM}}{0.028\,\mbox{ps}^{-1}}\,,$ (4.211)

where $\alpha$ contributes to CP violation in mixing and values of $\Omega$
smaller than one signal a discrepancy between the theoretical and experimental
description of $D$-mixing, within the one sigma range. A naive application of
the HQE leads to $\Omega=3.4\cdot 10^{-5}$ at LO-QCD and to $6.2\cdot 10^{-5}$
at NLO-QCD, showing that the theoretical prediction for the decay rate
difference is more than four orders of magnitude smaller than the
corresponding experimental number. Moreover, the phase $\alpha$ is very large,
i.e. $\alpha=93^{\circ}$ at LO-QCD and $\alpha=99^{\circ}$ at NLO-QCD. By
default in our numerical analysis we have used PDG [270] values for the quark,
in the $\overline{\mbox{MS}}$ scheme, and meson masses, as well as for the
strong coupling, CKM input from Ref. [260], the results of Ref. [246] for the
non-perturbative matrix elements and Ref. [248] for the $D^{0}$ decay
constant.

Figure 4.7:  Diagrams describing mixing of neutral $D$ mesons via intermediate
$s\bar{s}$, $s\bar{d}$, $d\bar{s}$, and $d\bar{d}$, states in the“full” theory
at LO-QCD (left) and NLO-QCD (center) and at NLO-QCD in the HQE (right). The
crossed circles denote the insertion of $\Delta C=1$ operators of the
effective Hamiltonian describing the charm-quark decay, while the full dot
indicates the insertion of $\Delta C=2$ operators in the HQE.

In order to analyse the peculiarities of $D$-mixing, we decompose
$\Gamma_{12}$ according to the flavour of the internal quark pair, cf. Figure
4.7. We denote the corresponding three contributions by $\Gamma_{12}^{ss}$,
$\Gamma_{12}^{dd}$, and $\Gamma_{12}^{sd}$, i.e.

$\displaystyle\Gamma_{12}$
$\displaystyle=-\Big{(}\lambda_{s}^{2}\,\Gamma^{ss}_{12}+2\,\lambda_{s}\lambda_{d}\,\Gamma^{sd}_{12}+\lambda_{d}^{2}\,\Gamma^{dd}_{12}\Big{)}$
$\displaystyle=-\,\lambda_{s}^{2}\Big{(}\Gamma^{ss}_{12}-2\Gamma^{sd}_{12}+\Gamma^{dd}_{12}\Big{)}+\,2\lambda_{s}\lambda_{b}\Big{(}\Gamma^{sd}_{12}-\Gamma^{dd}_{12}\Big{)}-\lambda_{b}^{2}\Gamma^{dd}_{12}\,,$
(4.212)

where the CKM factors are defined as $\lambda_{q}=V_{cq}V_{uq}^{*}$, and we
have used the unitarity relation $\lambda_{d}+\lambda_{s}+\lambda_{b}=0$, to
eliminate $\lambda_{d}$ in the second line of Eq. (4.212). Taking into account
the numerical value of the CKM elements, we see that Eq. (4.212) shows the
presence of a very pronounced hierarchy, namely

$\displaystyle-\lambda_{s}^{2}=-4.791\cdot 10^{-2}+3.094\cdot 10^{-6}i,$
(4.213) $\displaystyle+2\lambda_{s}\lambda_{b}=+2.751\cdot 10^{-5}+6.121\cdot
10^{-5}i,$ (4.214) $\displaystyle-\lambda_{b}^{2}=+1.560\cdot
10^{-8}-1.757\cdot 10^{-8}i.$ (4.215)

The CKM factor in the first term of Eq. (4.212) has considerably the largest
real part, whereas the second term has the largest imaginary part and it
should then be important for the determination of the potential size of CP
violation in $D$-mixing. The relative size between the imaginary and real
part, is much larger in $\lambda_{b}$ than in $\lambda_{s}$ and we therefore
suggest to include all terms in Eq. (4.212). Moreover, extreme GIM
cancellations [261] affect the short distance coefficients of the CKM elements
in Eq. (4.212). By expanding in the small mass parameter
$z=m_{s}^{2}/m_{c}^{2}$, we obtain at LO- (top line) and at NLO-QCD (lower
line), respectively

$\displaystyle{\Gamma^{ss}_{12}}$
$\displaystyle=\left\\{\begin{array}[]{ll}{1.62-2.34\,z-5.07\,z^{2}+\ldots}\,,\\\\[5.69054pt]
{1.42-4.30\,z-12.45\,z^{2}+\ldots}\,,\end{array}\right.$ (4.218)
$\displaystyle{\Gamma^{sd}_{12}-\Gamma^{dd}_{12}}$
$\displaystyle=\left\\{\begin{array}[]{ll}{-1.17\,z-2.53\,z^{2}+\ldots}\,,\\\\[5.69054pt]
{-2.15\,z-6.26\,z^{2}+\ldots}\,,\end{array}\right.$ (4.221)
$\displaystyle{\Gamma^{ss}_{12}-2\Gamma^{sd}_{12}+\Gamma^{dd}_{12}}$
$\displaystyle=\left\\{\begin{array}[]{ll}{-13.38\,z^{3}+\ldots}\,,\\\\[8.53581pt]
{0.07\,z^{2}-29.72\,z^{3}+\ldots}\,.\end{array}\right.$ (4.224)

Note that in the NLO result in Eq. (4.224), the GIM suppression is lowered by
one power of $z$, as it has been observed before [271, 272]. We conclude that
the peculiarity of Eq. (4.212) lies in the fact that the CKM dominant factor
$\lambda_{s}^{2}$ multiplies the extremely GIM suppressed term given in Eq.
(4.224), the CKM suppressed factor $\lambda_{s}\lambda_{b}$ multiplies the GIM
suppressed term given in Eq. (4.221), while the very CKM suppressed factor
$\lambda_{b}^{2}$ multiplies $\Gamma_{12}^{dd}$, obtained taking the limit
$z\to 0$ in Eq. (4.218), in which no GIM suppression is present. Therefore,
the three contributions in Eq. (4.212) have actually a similar size, in fact

$\displaystyle\Gamma_{12}$ $\displaystyle=\left(2.08\cdot 10^{-7}-1.34\cdot
10^{-11}i\right)\mbox{(1st term)}$ $\displaystyle-\left(3.74\cdot
10^{-7}+8.31\cdot 10^{-7}i\right)\mbox{(2nd term)}$
$\displaystyle+\left(2.22\cdot 10^{-8}-2.5\cdot 10^{-8}i\right)\mbox{(3rd
term)}.$ (4.225)

Because of Eq. (4.214), it also follows that a sizeable contribution to the
mixing phase can only arise if the slightly GIM suppressed term in Eq. (4.221)
is enhanced. In order to explain the mismatch between the HQE prediction and
experimental determination, in the literature different solutions have been
proposed.

  * i)

At higher order in the HQE, the GIM suppression could be less pronounced [273,
274, 275]. First estimates of the dimension-nine contribution to $D$-mixing,
performed in Ref. [276], show indeed such an enhancement, but not on a scale
sufficient to reproduce the experimental result. For a final statement about
this possibility, the complete determination of the dimension-nine and twelve
contributions, would be necessary.

  * ii)

The discrepancy is a signal of the violation of quark hadron duality. However,
while it was originally suggested that deviations of quark hadron duality
should be as large as $10^{5}$, because of $\Omega\approx 10^{-5}$, this seems
unlikely given the many successful tests of the HQE. In fact in Ref. [262] it
was shown that violations as small as 20 per cent could be sufficient to
explain the experimental data.

  * iii)

The HQE is not applicable and we have to consider different methods, like to
sum over all the exclusive decays channels contributing to the decay rate
difference, see e.g. Refs. [277, 278, 279].

#### 4.2.2 Alternative scale setting

$\Gamma_{12}$ depends on the two scales $\mu_{1}$ and $\mu_{2}$. The former
denotes the renormalisation scale of the $\Delta C=1$ operators, and it is
explicitly present in the expressions of the corresponding Wilson coefficients
in the effective Hamiltonian, the latter is the renormalisation scale of the
$\Delta C=2$ operators that arise in the HQE and it appears also in their
short distance coefficients. Up to higher order terms, the dependence on
$\mu_{1}$ and $\mu_{2}$ must cancel between the local matrix elements and the
corresponding short distance functions. Without discussing the dependence on
$\mu_{2}$, for which the cancellation is very effective, in the following we
consider only that on $\mu_{1}$. In the $B_{s}$ system the cancellation is
numerically only weakly realised when moving from LO- to NLO-QCD, see Refs.
[280, 281], indicating the importance of higher order corrections. First steps
in this direction show indeed large NNLO-QCD effects [280, 281]. In the $D$
system a reduction of the $\mu_{1}$-dependence, when moving from LO- to NLO-
QCD, is present in the individual contributions $\Gamma^{ss,sd,dd}_{12}$, but
not in $\Gamma_{12}$, see Figure 4.8, which seems to be again consequence of
the severe GIM cancellations.

Figure 4.8: Comparison of $\mu_{1}$-dependence of $|\Gamma_{12}|$ at LO-QCD
(dotted blue) and NLO-QCD (solid pink).

By explicitly showing the scale dependence in $\Gamma_{12}$, we can write

$\Gamma_{12}=\sum\limits_{q_{1}q_{2}=ss,sd,dd}\Gamma_{6}^{q_{1}q_{2}}(\mu_{1}^{q_{1}q_{2}},\mu_{2}^{q_{1}q_{2}})\langle
Q_{6}\rangle(\mu_{2}^{q_{1}q_{2}})\frac{1}{m_{c}^{3}}+\ldots\,.$ (4.226)

In general different internal quark pairs contribute to different decay
channels of the $D^{0}\,(\bar{D}^{0})$ meson e.g. $s\bar{s}$ to a $K^{+}K^{-}$
final state and $s\bar{d}$ to a $\pi^{+}K^{-}$ final state. For each of these
observables the choice of the renormalisation scales is a priori arbitrary,
nevertheless typically one fixes $\mu_{x}^{ss}=\mu_{x}^{sd}=\mu_{x}^{dd}=\mu$,
which is then chosen to be equal to the mass of the decaying heavy quark, i.e.
$\mu=m_{Q}$ for $Q$ quark decays, in order to minimise the effect of the
logarithmic terms $\alpha_{s}(\mu)\log(\mu^{2}/m_{Q}^{2})$. Uncertainties due
to unknown higher order corrections are estimated varying $\mu$ between
$m_{Q}/2$ and $2\,m_{Q}$ and in the case of the charm quark we fix the lower
bound to $1$ GeV to still ensure reliable perturbative results.

We consider two alternative possibilities to treat the renormalisation scale
$\mu_{1}^{q_{1}q_{2}}$, which both allow to reduce the discrepancy between the
theoretical and the experimental determination of $D$-mixing, while leaving
the other HQE predictions unchanged, namely

  * A)

We fix the central value of the three scales $\mu_{1}^{ss}$, $\mu_{1}^{sd}$,
and $\mu_{1}^{dd}$, to $m_{c}$, but we vary them independently between $1$ GeV
and $2m_{c}$.

  * B)

We choose different central values for the three scales $\mu_{1}^{ss}$,
$\mu_{1}^{sd}$, and $\mu_{1}^{dd}$, according to the size of the available
phase space. In particular we evaluate $\Gamma^{ss}_{6}$ at the scale
$\mu_{1}^{ss}=\mu-2\epsilon$, $\Gamma^{sd}_{6}$ at the scale
$\mu_{1}^{sd}=\mu-\epsilon$, and $\Gamma^{dd}_{6}$ at the scale
$\mu_{1}^{dd}=\mu$, where $\epsilon$ is an unknown parameter, related to the
kinematics of the decays.

If $\epsilon$ is not too large, both methods yield results for the individual
$\Gamma^{ss}_{6}$, $\Gamma^{sd}_{6}$ and $\Gamma^{dd}_{6}$, which lie within
the usually quoted theory uncertainties obtained following the prescription
stated above, but they affect in a sizeable way the severe GIM cancellations
in Eqs. (4.221) and (4.224). The first method gives a considerably enhanced
range of values for $\Omega$ in Eq. (4.210), i.e.

$\Omega\in[4.6\cdot 10^{-5},1.3]\,,$ (4.227)

which nicely covers also the experimental determination of the decay rate
difference. Scanning independently over $\mu_{1}^{ss}$, $\mu_{1}^{sd}$, and
$\mu_{1}^{dd}$, in 11 equidistant steps we find that out of the 1331 points
only 14 give a value of $\Omega<0.001$, while 984 give a value of
$\Omega>0.1$. The large discrepancy between the theoretical and the
experimental determinations, seems then to be an artefact of fixing the scales
$\mu_{1}^{ss}$, $\mu_{1}^{sd}$, and $\mu_{1}^{dd}$, to be the same. The range
of values shown in Eq. (4.227) does not change significantly if we use the
pole scheme for the quark masses, lattice results instead of the HQET sum rule
results, or a different $\Delta C=2$ operator basis. In all these cases we can
obtain $\Omega\geq 1$. For $\alpha$ in Eq. (4.211), we observe that the
results lie in the range $[-\pi,\pi]$. A closer look however, shows that for
$\Omega>0.5$ only values of $\alpha<0.1^{\circ}$ are allowed, and conversely
large values of $\alpha$ correspond to results for $\Omega$ inconsistent with
the experimental data. The second method for the scale setting requires the
introduction of a mass scale $\epsilon$. A possible estimate for the size of
this parameter could be the strange quark mass $\epsilon=m_{s}\approx 0.1$ GeV
or the phase space difference of the corresponding exclusive decay channels,
specifically, by comparing the energy release of $D^{0}\to K^{+}K^{-}$,
$M_{D^{0}}-2M_{K^{+}}=0.88$ GeV, with that of $D^{0}\to\pi^{+}\pi^{-}$,
$M_{D^{0}}-2M_{\pi^{+}}=1.59$ GeV we might expect that $\epsilon\approx 0.35$
GeV.

Figure 4.9: Comparison of the $\epsilon$ dependence of $\Omega$ at LO-QCD
(blue) and NLO-QCD (pink) for different values of $\mu$: the dashed line
corresponds to $\mu=\leavevmode\nobreak\ m_{c}$ while the two solid lines to
$\mu=1$ GeV and $\mu=2m_{c}$.

In Figure 4.9 it is shown how the HQE prediction of $\Omega$ would be affected
in this scenario. Also in this case, it appears possible that the theoretical
prediction could reach the experimental value for $\epsilon\approx 0.2$ GeV.

Finally, we have to consider how other HQE predictions would be affected by
choosing a different scale setting procedure. In the case of observables in
which GIM-like cancellations are not present, like the lifetime, both in the
charm and bottom system, and the decay rate difference $\Delta\Gamma_{s}$,
there is no significant change, but only a shift within the usually quoted
theory uncertainties. However, the semileptonic CP asymmetries are governed by
the weakly GIM suppressed contribution in $B_{s}$-mixing. The SM predictions
read

$\displaystyle{\rm Re}\left(\frac{\Gamma_{12}^{q}}{M_{12}^{q}}\right)^{\rm
SM}$ $\displaystyle=-\frac{\Delta\Gamma_{q}}{\Delta
M_{q}}=\left\\{\begin{array}[]{ll}-(49.9\pm 6.7)\cdot
10^{-4}&q=s\\\\[8.53581pt] -(49.7\pm 6.8)\cdot
10^{-4}&q=d\end{array}\right.\,,$ (4.230) $\displaystyle{\rm
Im}\left(\frac{\Gamma_{12}^{q}}{M_{12}^{q}}\right)^{\rm SM}$
$\displaystyle=a_{sl}^{q}=\left\\{\begin{array}[]{ll}(+2.2\pm 0.2)\cdot
10^{-5}&q=s\\\\[8.53581pt] (-5.0\pm 0.4)\cdot
10^{-4}&q=d\end{array}\right.\,,$ (4.233)

while in the scenario B we obtain

$\begin{array}[]{|c||c|c|}\hline\cr\epsilon\,({\rm
GeV})&\Gamma_{12}^{s}/M_{12}^{s}&\Gamma_{12}^{d}/M_{12}^{d}\\\
\hline\cr\hline\cr 0.&{\bf-0.00499+0.000022i}&{\bf-0.00497-0.00050i}\\\
\hline\cr 0.2.&{\bf-0.00494+0.000023i}&{\bf-0.00492-0.00053i}\\\ \hline\cr
0.5.&{\bf-0.00484}+0.000026i&{\bf-0.00482}-0.00059i\\\ \hline\cr
1.0&{\bf-0.00447}+0.000037i&{\bf-0.00448}-0.00084i\\\ \hline\cr
1.5.&-0.00287+0.000091i&-0.00309-0.0021i\\\ \hline\cr\end{array}$

We see that for values of $\epsilon\leq 1$ GeV, the predictions for the real
part lie within the usually quoted theory uncertainties, indicated in bold
type. However, they would be increased by almost $100\%$ in correspondence of
larger values of $\epsilon$.

Our conclusion is that, by modifying the usually adopted scale setting, the
theoretical uncertainty of $y$, within the HQE, becomes larger than previously
thought and it can cover the experimental value. However, this does not
represent a complete solution and more precise estimates of higher power
corrections to the HQE, as well as full NNLO-QCD corrections to the leading
dimension-six term, could bring further insights. The alternative scale
setting procedure shows that a small contribution to CP violation in mixing
stemming from the decay rate can be up to one per mille within in the SM,
which agrees with estimates made in Refs. [282, 283]. For a prediction of CP
violation in mixing, the contribution coming from $M_{12}$ needs to be
determined in addition. This might be done in future via the help of
dispersion relations, see e.g. Refs. [284, 278, 283]. We would like to note
that our suggested procedure still respects the GIM mechanism, because for
vanishing internal strange quark mass, also the parameter $\epsilon$ is zero.
Finally this alternative scale setting does not affect quantities like
$\tau(D^{+})/\tau(D^{0})$, $b$-hadron lifetimes and $\Delta\Gamma_{s}$ outside
the range of their quoted theoretical errors, but it affects the semileptonic
CP asymmetries and we get enhanced SM ranges

$a_{sl}^{d}\in[-9.2;-4.6]\cdot 10^{-4}\,,\qquad a_{sl}^{s}\in[2.0;4.0]\cdot
10^{-5}\,.$ (4.240)

Note that in Ref.[14], the CP violating effects responsible for creating the
baryon asymmetry stem actually from $a_{sl}^{d}$ and $a_{sl}^{s}$.

## Conclusion

As discussed at the beginning this work, indirect BSM searches with quark
flavour observables, represent a promising route to improve the current
understanding of the fundamental laws of physics. However, this strategy
strongly relies on the ability to systematically increase both the
experimental and the theoretical precision. In the present thesis we have
analysed the theoretical status for the study of the inclusive decay widths of
heavy hadrons, which define observables of primary phenomenological importance
in heavy flavour physics. In particular, we have discussed the computation of
higher power corrections to the HQE, and tested its applicability in the charm
sector, for the case of inclusive quantities like lifetimes, semileptonic
branching fractions and mixing observables. Specifically, the first part of
this thesis has been dedicated to presenting the main ingredients required for
the computation. We have started by introducing the weak effective Hamiltonian
and the heavy quark effective theory, which constitute the two effective
theories that allow to disentangle a multi-scale problem like the weak decay
of heavy hadrons, by progressively integrating out the heavier degrees of
freedom, respectively, the $W$-boson and the massive component of the heavy
quark field. We stress that our exposure, far from being exhaustive, has only
covered the aspects relevant for the subsequent discussions and has mostly
followed the excellent reviews available in the literature. We have then
performed a detailed derivation of the expansion of the quark propagator in
the external gluon field using the Fock-Schwinger gauge, which provides a
gauge covariant parametrisation of the soft interaction with the non
perturbative QCD field, for a quark propagating with large momentum inside the
hadronic state. The corresponding expressions, up to terms proportional to one
covariant derivative of the gluon field strength tensor, have been computed
both in momentum and in coordinate space. Finally, we have presented a
pedagogical introduction to the heavy quark expansion, the theoretical
framework in which all of the remaining computations and discussions are
embedded. In the second part of this work, we have shown the explicit
calculation of the lowest dimensional contributions to the HQE of a $B$ meson,
namely those of two-quark operators up to order $1/m_{b}^{2}$ and of four-
quark operators up to order $1/m_{b}^{4}$. For the former, the computation has
been performed using the coordinate representation of the quark propagator and
considering the single decay mode $b\to c\bar{u}d$. Moreover, we have
reproduced the results given in the literature and in some cases derived more
general expressions. Also in this case we have tried to provide a very
comprehensive and detailed presentation. The most important results, from a
technical point of view, have been discussed in the third part of this work.
Here, we have outlined the detailed computation of the contribution of two-
quark operators up to order $1/m_{b}^{3}$, for arbitrary non-leptonic decays
of the heavy $b$-quark and using the momentum representation of the quark
propagator in the external gluon field. Particular emphasis has been put in
describing the mixing between four-quark operators and the Darwin operator at
dimension-six, which ensures the cancellations of the infrared divergences
otherwise present in the coefficients of the Darwin operator, due to the
emission of a soft gluon from a light quark propagator. The contribution of
the Darwin operator for non-leptonic $b$-quark decays, has been only recently
determined and found to be sizeable, hence its effect, previously neglected,
might have important consequences for $b$-physics phenomenology. In the last
part of this work, we have considered two phenomenological applications of the
HQE in the charm sector, specifically the study of the inclusive decay width
of $D$-mesons and of the Glashow-Iliopoulos-Maiani (GIM) cancellations in
$D$-meson mixing. Due to the value of its mass, the charm quark sits at the
boundary between the heavy- and light-quark region, and the applicability of
the HQE in the charm sector is a priori questionable. In fact, both the
perturbative and the power corrections might not describe a well converging
series. For this reason, the charm system can be considered an important
testing ground for the theoretical framework here discussed. We have then
performed a comprehensive study of the structure of the HQE for the inclusive
decay width of charmed mesons, including for the first time the contribution
of the Darwin operator and in addition clarified some inconsistencies related
to the contribution of dimension-seven four-quark operators. Our predictions
appear to be consistent with the corresponding experimental pattern, albeit
with large theoretical uncertainties. Given the current poor knowledge of many
of the non perturbative input in the charm sector, as well as the absence of
determinations of higher order perturbative and power corrections, we conclude
that our numerical analysis does not show signals for a breakdown of the HQE
in the charm system. Finally, we have also attempted to clarify the long
standing puzzle due to the big discrepancy between the experimental
determination for neutral $D$-meson mixing and the corresponding HQE
prediction, which might naively point at a complete failure of the HQE in the
charm sector. In this respect, we have proposed a novel procedure to treat the
renormalisation scale for observables affected by GIM cancellations. Our
results show that the experimental value can be accommodated within the HQE,
albeit again with very large theoretical uncertainties.

In conclusion, we have improved the current theoretical status of the HQE, by
computing the contribution of the Darwin operator to non-leptonic decays of
heavy quarks and by testing its applicability in the charm system. The same
framework can in future be applied to the $B$ system, to improve the
theoretical prediction for lifetimes and mixing observables. A precise
determination of e.g. $\tau(B_{s})/\tau(B_{d})$, could in fact increase the
bounds on the size of potential new physics contributions in the decay $b\to
s\tau\tau$, which are predicted by some of the current BSM models that explain
the $B$-anomalies, or could further constrain the baryogengesis model
discussed in Ref. [14].

## Acknowledgements

First and outmost I thank Prof. Dr. Alexander Lenz for his precious guidance
and the inestimable support provided during the last three years. I also thank
him for being an inexhaustible source of energy and motivation and I consider
myself very lucky to have had the possibility to learn from him. A special
thanks goes to my collaborator and dear friend Dr. Aleksey Rusov, he has
deeply contributed to my technical growth and I am grateful for the chance had
to work together. Moreover, I sincerely thank him for the patience and the
constant support offered in the last few months while writing this thesis. I
warmly thank Prof. Dr. Michael Spannowsky and Prof. Dr. Frank Krauss for
giving me the opportunity to start my PhD program at the IPPP in Durham and
Dr. Daniel Maitre for the supervision during the first year there. A big
thanks to everyone met in the past four years, first in Durham and now in
Siegen, for each of them has contributed in some way to this journey. Arrived
at this point, I feel obliged to also thank all the teachers and professors I
have had the fortune to find on my lifelong learning path, many of them have
certainly inspired me and motivated me to continue studying. Last but not
least, I thank my family, my friends, and Peter, for understanding the many
times I have been absent and for taking care of me despite the distance.

##  Appendix A Expansion of operators containing the heavy quark momentum

Here we discuss a general procedure to generate operators of higher dimension,
bilinear in the heavy quark field, starting from expressions containing lower
dimensional operators with coefficients proportional to the heavy quark
momentum. For definiteness we consider the case of the $b$-quark in order to
make the connection to Chapter 2 and Chapter 3 easier. We start by recalling
that inside a heavy hadron, the momentum of the heavy quark can be
conveniently parametrised as $p_{b}^{\mu}=m_{b}v^{\mu}+k^{\mu}$.
Correspondingly the rescaled heavy quark field $b_{v}(x)$ is defined by

$b(x)=e^{-im_{b}v\cdot x}\,b_{v}(x)\,,$ (A.1)

where the phase factor removes the large ‘kinetic’ part of the heavy quark
momentum so that a derivative acting on $b_{v}(x)$ returns only the residual
component $k$. Consider now the following expression $p_{b}^{\mu}\,\bar{b}b$,
with the $b$-fields evaluated at the origin $x=0$. By taking into account Eq.
(A.1), we can write, see Refs. [130, 131]

$\displaystyle p_{b}^{\mu}\bar{b}b=\lim_{x\to
0}\bar{b}(x)i\partial^{\mu}b(x)=\lim_{x\to
0}\bar{b}_{v}(x)(m_{b}v^{\mu}+i\partial^{\mu})b_{v}(x)\,,$ (A.2)

here the partial derivative is acting on the right. Using that in the FS
gauge, see Section 1.3, the gauge field satisfies the useful property
$A^{\mu}(0)=0$, cf. Eq. (1.87), it follows that on the r.h.s. of Eq. (A.2),
the partial derivative acting on $b_{v}(x)$ can be replaced by the
corresponding covariant derivative in the limit $x\to 0$, namely

$p_{b}^{\mu}\bar{b}b=\lim_{x\to
0}\bar{b}_{v}(x)\big{(}m_{b}v^{\mu}+i\partial^{\mu}+A^{\mu}(x)\big{)}b_{v}(x)=\bar{b}_{v}(m_{b}v^{\mu}+iD^{\mu})b_{v}\,,$
(A.3)

which generates the dimension-four operator $\bar{b}_{v}iD^{\mu}b_{v}$. If the
original expression contains more than one power of the heavy quark momentum,
to keep track of the order of the covariant derivatives we can symmetrise the
action of the partial derivatives, e.g. the case of
$p_{b}^{\mu}p_{b}^{\nu}\,\bar{b}b$ gives

$\displaystyle p_{b}^{\mu}p_{b}^{\nu}\,\bar{b}b=\frac{1}{2}\lim_{x\to
0}\bar{b}(x)\big{(}i\partial^{\mu}i\partial^{\nu}+i\partial^{\nu}i\partial^{\mu}\big{)}b(x)$
$\displaystyle=\frac{1}{2}\lim_{x\to 0}\bar{b}_{v}(x)\Big{(}(m_{b}v^{\mu}$
$\displaystyle+i\partial^{\mu})(m_{b}v^{\nu}+i\partial^{\nu})+(m_{b}v^{\nu}+i\partial^{\nu})(m_{b}v^{\mu}+i\partial^{\mu})\Big{)}b_{v}(x)\,.$
(A.4)

Here again we would like to replace the action of the partial derivative with
that of the covariant derivative. Note that in this case, apart from terms
containing the gauge field on the most left, which vanish when taking the
limit $x\to 0$, we introduce also terms with derivatives of the gauge field
evaluated at the origin, that are in general non zero. From Eq. (1.87), it
follows that $\partial^{\mu}A^{\nu}(0)=(1/2)G^{\mu\nu}(0)$, however, due to
the antisymmetry of the gluon field strength tensor, these contributions
cancel in the symmetric combination in Eq. (A.4), namely

$\displaystyle\hskip
128.0374pt\frac{1}{2}\,\bar{b}_{v}\Big{(}iD^{\mu}iD^{\nu}+iD^{\nu}iD^{\mu}\Big{)}b_{v}$
$\displaystyle=\frac{1}{2}\lim_{x\to
0}\bar{b}_{v}(x)\Big{(}\big{(}i\partial^{\mu}i\partial^{\nu}+A^{\mu}(x)i\partial^{\nu}+i\partial^{\mu}A^{\nu}(x)+A^{\nu}(x)i\partial^{\mu}+A^{\mu}(x)A^{\nu}(x)\big{)}$
$\displaystyle\quad\quad+\big{(}i\partial^{\nu}i\partial^{\mu}+A^{\nu}(x)i\partial^{\mu}+i\partial^{\nu}A^{\mu}(x)+A^{\mu}(x)i\partial^{\nu}+A^{\nu}(x)A^{\mu}(x)\big{)}\Big{)}b_{v}(x)$
$\displaystyle\hskip
62.59596pt=\frac{1}{2}\,\bar{b}_{v}\big{(}i\partial^{\mu}i\partial^{\nu}+i\partial^{\nu}i\partial^{\mu}+\frac{i}{2}G^{\mu\nu}(0)+\frac{i}{2}G^{\nu\mu}(0)\big{)}b_{v}$
$\displaystyle\hskip
122.34692pt=\frac{1}{2}\bar{b}_{v}\big{(}i\partial^{\mu}i\partial^{\nu}+i\partial^{\nu}i\partial^{\mu}\big{)}b_{v}\,.$
(A.5)

We then obtain that

$p_{b}^{\mu}p_{b}^{\nu}\,\bar{b}b=\frac{1}{2}b_{v}\Big{\\{}(m_{b}v^{\mu}+iD^{\mu}),(m_{b}v^{\nu}+iD^{\nu})\Big{\\}}b_{v}\,,$
(A.6)

where the curly brackets denote the anticommutator. Notice that we generate
operators of dimension-four and dimension-five with respectively one and two
covariant derivatives. In the case of three powers of the heavy quark
momentum, $p_{b}^{\mu}p_{b}^{\nu}p_{b}^{\rho}\,\bar{b}b$, in rewriting this
expression in terms of partial derivatives we consider all the permutations in
the three Lorentz indices $\mu\nu\rho$. The replacement $i\partial^{\mu}\to
iD^{\mu}$ again follows from the symmetric combination of antisymmetric
tensors, since Eq. (1.87) now gives
$\partial^{\mu}\partial^{\nu}A^{\rho}(0)=(1/2)\partial^{\mu}G^{\nu\rho}(0)+(1/3)D^{\mu}G^{\nu\rho}(0)$.
For $n$-powers of the heavy quark momentum, we use the general expression

$\displaystyle p_{b}^{\mu_{1}}\ldots
p_{b}^{\mu_{n}}\,\bar{b}b=\frac{1}{n!}\sum_{\sigma\in
S_{n}}\bar{b}_{v}\,\big{(}m_{b}v+iD\big{)}^{\sigma(\mu_{1})}\,\ldots\,\big{(}m_{b}v+iD\big{)}^{\sigma(\mu_{n})}\,b_{v}\,,$
(A.7)

where $S_{n}$ is the group of all permutations of $n$ elements. Furthermore
consider the following function

$f(r)=a+b(r)\,r+c(r)\,r^{2}+d(r)\,r^{3}+e(r)\,r^{4}\,,$ (A.8)

with the dependence on the heavy quark momentum contained in the argument
$r=m^{2}/p_{b}^{2}$. In this case, we can use that
$p_{b}^{2}=m_{b}^{2}\,(1+X)$, where

$X=2\,\frac{v\cdot k}{m_{b}}+\frac{k^{2}}{m_{b}^{2}}\ll 1\,,$ (A.9)

and expand Eq. (A.8) in series, i.e.

$f(r)=\sum\limits_{n=0}^{\infty}\,g_{n}(\rho)\,(-X)^{n}\,,$ (A.10)

here $g_{0}(\rho)=f(\rho)$ and the dimensionless mass parameter
$\rho=m^{2}/m_{b}^{2}$. The series in Eq. (A.10) can be truncated at a certain
value of $n$, leading to corrections up to order $1/m_{b}^{2n}$. Expressing
$X$ in Eq. (A.10) back in terms of the four-momentum, we then obtain

${\\\
}f(r)=\sum_{n=0}^{\infty}\,g_{n}(\rho)\,\Bigg{(}1-\frac{p_{b}^{2}}{m_{b}^{2}}\Bigg{)}^{n}\,,$

from which it follows that $f(r)\,\bar{b}b$ can be expanded according to Eq.
(A.7). Finally, for the computation of the coefficient of the Darwin operator
discussed in Chapter 3, we need to expand also the structure
$p_{b}^{\rho}\,\bar{b}\,G^{\mu\nu}b$. In this case we write

$\displaystyle p_{b}^{\rho}\,\bar{b}\,G^{\mu\nu}b$
$\displaystyle=\frac{1}{2}\,\lim_{x\to
0}\Big{(}\bar{b}(x)(-i\overset{\leftarrow}{\partial^{\rho}})G^{\mu\nu}(x)b(x)+\bar{b}(x)G^{\mu\nu}(x)(i\overset{\rightarrow}{\partial^{\rho}})b(x)\Big{)}$
$\displaystyle=\frac{1}{2}\,\lim_{x\to
0}\Big{(}\bar{b}_{v}(x)(m_{b}v^{\rho}-i\overset{\leftarrow}{\partial^{\rho}})G^{\mu\nu}(x)b_{v}(x)+\bar{b}_{v}(x)G^{\mu\nu}(x)(m_{b}v^{\rho}+i\overset{\rightarrow}{\partial^{\rho}})b_{v}(x)\Big{)}$
$\displaystyle=\frac{1}{2}\,\lim_{x\to
0}\Big{(}\bar{b}_{v}(x)(m_{b}v^{\rho}+i\overset{\rightarrow}{\partial^{\rho}})G^{\mu\nu}(x)b_{v}(x)+\bar{b}_{v}(x)G^{\mu\nu}(x)(m_{b}v^{\rho}+i\overset{\rightarrow}{\partial^{\rho}})b_{v}(x)\Big{)}$
$\displaystyle=\frac{1}{2}\,\Big{(}\bar{b}_{v}(m_{b}v^{\rho}+iD^{\rho})G^{\mu\nu}b_{v}+\bar{b}_{v}G^{\mu\nu}(m_{b}v^{\rho}+iD^{\rho})b_{v}\Big{)}\,,$
(A.11)

where in the third step we have applied the chain rule for the derivative
operator acting on the left and dropped the corresponding term with a total
derivative since it does not contribute in forward matrix elements with zero
momentum transfer.

In general, by expressing powers of the heavy quark momentum in terms of
operators acting on the heavy quark field, we generate a set of higher
dimensional operators with multiple covariant derivatives. Specifically, in
order to compute the contributions discussed in Chapter 2 and Chapter 3, it is
sufficient to consider the expansions up to three covariant derivatives. The
operators obtained in this way, must be then evaluated between external $B$
meson states for the calculation of the total decay width. This can be easily
done by taking into account the results presented in Ref. [105], in which the
complete parametrisation of these matrix elements, up to order $1/m_{b}^{4}$
was derived.

##  Appendix B Results for one-loop integrals in dimensional regularisation

Here we list some useful results for the computation of one-loop integrals in
dimensional regularisation [47, 48, 49, 50] with $D=4-2\epsilon$. All the
expressions presented can be found in standard QFT textbooks like Refs. [81,
38, 285]. Note that in the following we use the notation

$\mu^{2\varepsilon}\int\frac{d^{D}l}{(2\pi)^{D}}\equiv\int_{l}\,.$ (B.1)

The scalar tadpole and bubble integrals are respectively given by

$\int_{l}\frac{1}{(l^{2}-m^{2}+i\varepsilon)}={\cal A}_{0}(m^{2})\,,$ (B.2)

and

$\int_{l}\frac{1}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}={\cal
B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})\,,$ (B.3)

with

$\displaystyle{\cal
A}_{0}(m^{2})=\frac{i\,m^{2}}{16\pi^{2}}\,\Bigg{(}\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)+1-\log\left(\frac{m^{2}}{\mu^{2}}\right)\Bigg{)}+{\cal
O}(\epsilon)\,,$ (B.4) $\displaystyle{\cal
B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})=\frac{i}{16\pi^{2}}\Bigg{(}\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)+f(p^{2},m_{1}^{2},m_{2}^{2})\Bigg{)}+{\cal
O}(\epsilon)\,,$ (B.5)

where $\gamma_{E}$ is the Euler-Mascheroni constant and

$f(p^{2},m_{1}^{2},m_{2}^{2})=-\int\limits_{0}^{1}dx\,\log\left(\frac{(1-x)\,m_{1}^{2}+x\,m_{2}^{2}-x\,(1-x)\,p^{2}-i\varepsilon}{\mu^{2}}\right)\,,$
(B.6)

Note that as $\varepsilon\to 0$, the function $f(p^{2},m_{1}^{2},m_{2}^{2})$
is real and analytic in $s\in{\mathbb{C}}$, with ${\rm Re}(s)=p^{2}$, only for
$p^{2}<\big{(}m_{1}+m_{2}\big{)}^{2}\vee 0\leq x\leq 1$, while it exhibits a
branch point when the argument of the logarithm in the integrand on the r.h.s
of Eq. (B.6) vanishes. This leads to a cut in the complex $s$-plane, in
correspondence of the points $p^{2}>\big{(}m_{1}+m_{2}\big{)}^{2}$ in which
the argument of logarithm becomes negative. By introducing $\varepsilon$, the
computation of the integral, performed above and below the branch cut, gives
see e.g. the lecture notes [286, 287]

${\rm
Disc}\,f(p^{2},m_{1}^{2},m_{2}^{2})=2i\pi\,\frac{\sqrt{\lambda(p^{2},m_{1}^{2},m_{2}^{2})}}{p^{2}}\,\,\theta\left(p^{2}-\big{(}m_{1}+m_{2}\big{)}^{2}\right)\,,$
(B.7)

and $\theta(x)$ is the Heaviside function. From Eqs. (B.4)-(B.7) it follows
that

${\rm Im}\,{i\cal A}_{0}(m^{2})=0\,,$ (B.8)

while

$\displaystyle{\rm Im}\,{i\cal
B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})=-\frac{1}{16\pi^{2}}\,{\rm
Im}f(p^{2},m_{1}^{2},m_{2}^{2})=-\frac{1}{16\pi}\,\frac{\sqrt{\lambda(p^{2},m_{1}^{2},m_{2}^{2})}}{p^{2}}\,,$
(B.9)

for $p^{2}\geq\big{(}m_{1}+m_{2}\big{)}^{2}$. The corresponding one- and two-
point one-loop tensor integrals of rank $r$ can be computed, using the
Passarino-Veltman reduction algorithm [155], as a linear combination of
tensors of the same rank, built from the metric $g^{\mu\nu}$ and the external
momentum $p^{\mu}$, with coefficients proportional to the scalar integrals in
Eqs. (B.2), (B.3). Let us consider explicitly the cases $r=1,2$. Rank-1
integrals can only depend on the four-vector $p^{\mu}$, namely

$\int_{l}\,\frac{l^{\rho}}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}=B_{11}\,p^{\rho}\,,$
(B.10)

where the coefficient $B_{11}$ is obtained contracting both sides of Eq.
(B.10) with $p_{\rho}$ i.e.

$B_{11}=\frac{1}{p^{2}}\,\int_{l}\,\frac{l\cdot
p}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}\,.$
(B.11)

By substituting the identity

$l\cdot
p=\frac{1}{2}\Big{(}l^{2}+p^{2}-(l-p)^{2}+m_{1}^{2}-m_{1}^{2}+m_{2}^{2}-m_{2}^{2}\Big{)}\,,$
(B.12)

into Eq. (B.11), evidently yields

$\displaystyle B_{11}$
$\displaystyle=\frac{1}{2p^{2}}\int_{l}\frac{(l^{2}-m_{1}^{2})-((l-p)^{2}-m_{2}^{2})+(p^{2}+m_{1}^{2}-m_{2}^{2})}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}$
$\displaystyle=\frac{1}{2p^{2}}\,\Bigg{(}{\cal A}_{0}(m_{2}^{2})-{\cal
A}_{0}(m_{1}^{2})+\Big{(}p^{2}+m_{1}^{2}-m_{2}^{2}\Big{)}\,{\cal
B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})\Bigg{)}\,,$ (B.13)

where we have taken into account Eqs. (B.2), (B.3). In the case of rank-2
integrals, we can build two independent tensors, $g^{\rho\sigma}$ and
$p^{\rho}p^{\sigma}$, namely

$\int_{l}\,\frac{l^{\rho}l^{\sigma}}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}=B_{21}\,p^{\rho}p^{\sigma}+B_{22}\,p^{2}g^{\rho\sigma}\,,$
(B.14)

here, the coefficients $B_{21}$, $B_{22}$, are the solutions of the system of
two equations obtained contracting both sides of Eq. (B.14) respectively with
$p_{\rho}$ and $g_{\rho\sigma}$, i.e.

$\displaystyle\int_{l}\,\frac{l^{2}}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}$
$\displaystyle=B_{21}\,p^{2}+\,p^{2}\,B_{22}\,D\,,$ (B.15)
$\displaystyle\int_{l}\,\frac{(l\cdot
p)\,l^{\sigma}}{(l^{2}-m_{1}^{2}+i\varepsilon)((l-p)^{2}-m_{2}^{2}+i\varepsilon)}$
$\displaystyle=B_{21}\,p^{2}\,p^{\sigma}+\,p^{2}\,B_{22}\,p^{\sigma}\,.$
(B.16)

Adding and subtracting $m_{1}^{2}$ in the numerator of Eq. (B.15) and
substituting Eq. (B.12) into Eq. (B.16), we obtain 111Note that integrals of
odd functions of $l^{\mu}$ vanish due to parity.

$\displaystyle\Big{(}B_{21}+D\,B_{22}\Big{)}\,p^{2}={\cal
A}_{0}(m_{2}^{2})+m_{1}^{2}\,{\cal B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})\,,$
(B.17)
$\displaystyle\Big{(}B_{21}+B_{22}\Big{)}\,p^{2}=\displaystyle{\frac{1}{2}}\Big{(}{\cal
A}_{0}(m_{2}^{2})+(p^{2}+m_{1}^{2}-m_{2}^{2})\,B_{11}\Big{)}\,,$ (B.18)

solved by

$\displaystyle B_{21}=$
$\displaystyle\displaystyle{\frac{1}{D-1}\,\frac{1}{2p^{2}}}\Bigg{(}\,{\cal
A}_{0}(m_{2}^{2})(D-2)-2m_{1}^{2}\,{\cal B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})$
$\displaystyle+\displaystyle{\frac{D\,(m_{1}^{2}-m_{2}^{2}+p^{2})\Big{(}{\cal
A}_{0}(m_{2}^{2})-{\cal A}_{0}(m_{1}^{2})+(p^{2}+m_{1}^{2}-m_{2}^{2}){\cal
B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})\Big{)}}{2p^{2}}}\Bigg{)}\,,$ (B.19)

and

$\displaystyle B_{22}=$
$\displaystyle\displaystyle{\frac{1}{D-1}\,\frac{1}{2p^{2}}}\Bigg{(}\,{\cal
A}_{0}(m_{2}^{2})+2m_{1}^{2}\,{\cal B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})$
$\displaystyle-\displaystyle{\frac{(m_{1}^{2}-m_{2}^{2}+p^{2})\Big{(}{\cal
A}_{0}(m_{2}^{2})-{\cal A}_{0}(m_{1}^{2})+(p^{2}+m_{1}^{2}-m_{2}^{2}){\cal
B}_{0}(p^{2},m_{1}^{2},m_{2}^{2})\Big{)}}{2p^{2}}}\Bigg{)}\,,$ (B.20)

where Eq. (B.13) has been used. The expressions in Eq. (B.2)-(B.20) are needed
for the computation of the four-quark operators contribution discussed in
Chapter 2.

Finally we derive some useful results used in the calculation of the one-loop
diagram in Figure 3.4, describing the mixing of four-quark operators with two-
quark operators at dimension-six, discussed in Chapter 3. Consider the
following scalar integrals

$\int_{l}\frac{1}{(l^{2}-m^{2}+i\varepsilon)^{2}}={\cal I}_{0}(m^{2})\,,$
(B.21)

and

$\int_{l}\frac{l^{2}}{(l^{2}-m^{2}+i\varepsilon)^{3}}=\frac{D}{4}\,{\cal
I}_{0}(m^{2})\,,$ (B.22)

with

${\cal
I}_{0}(m^{2})=\frac{i}{16\pi^{2}}\Bigg{(}\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)+\log\left(\frac{\mu^{2}}{m^{2}}\right)\Bigg{)}+{\cal
O}(\epsilon)\,.$ (B.23)

Rank-2 integrals, since now there is no external momentum they can depend on,
can be only parametrised by the metric tensor i.e.

$\int_{l}\frac{l^{\mu}l^{\nu}}{(l^{2}-m^{2}+i\varepsilon)^{3}}=g(m^{2})\,g^{\mu\nu}\,,$
(B.24)

where the coefficient $g(m^{2})$ is obtained by contracting both sides of Eq.
(B.24) with $g_{\mu\nu}$. This gives

$\int_{l}\frac{l^{2}}{(l^{2}-m^{2}+i\varepsilon)^{3}}=D\,g(m^{2})\,,$ (B.25)

and from Eq. (B.22) it then follows that

$g(m^{2})=\frac{1}{4}\,{\cal I}_{0}(m^{2})\,.$ (B.26)

##  Appendix C Decomposition of tensor integrals

In the following section we describe how to reduce rank-$r$ integrals to a
linear combination of tensors of rank-$r$ with coefficients given by scalar
integrals. We consider explicitly integrals of the sunset type encountered in
Chapter 3 although the same procedure can be generalised to any tensor
integral without loss of generality. We work again in dimensional
regularisation with $D=4-2\varepsilon$ and to simplify the notation we define

$\int_{l_{1}l_{2}}\equiv\int_{l_{1}}\,\int_{l_{2}}\,\frac{1}{(l_{1}^{2}-m_{1}^{2}+i\varepsilon)\,(l_{2}^{2}-m_{2}^{2}+i\varepsilon)((l_{1}+l_{2}-p)^{2}-m_{3}^{2}+i\varepsilon)}\,,$
(C.1)

with $\int_{l}$ defined as in Eq. (B.1). Rank-$1$ integrals can only be
parametrised in terms of the external momentum $p^{\mu}$, hence

$\int_{l_{1}l_{2}}l_{j}^{\mu}=a_{j}(p^{2},m_{i}^{2})\,p^{\mu}\,,$ (C.2)

where $j=1,2$ and the coefficient $a_{j}(p^{2},m_{i}^{2})$ is obtained by
contracting both sides of Eq. (C.2) with $p_{\mu}$ i.e.

$a_{j}(p^{2},m_{i}^{2})=\frac{1}{p^{2}}\int_{l_{1}l_{2}}(l_{j}\cdot p)\,.$
(C.3)

In the case of rank-$2$ integrals, we can build two independent tensors,
namely $p^{2}g^{\mu\nu}$ and $p^{\mu}p^{\nu}$, hence

$\int_{l_{1}l_{2}}l_{j}^{\mu}l_{k}^{\nu}=b_{jk}(p^{2},m_{i}^{2})\,p^{2}g^{\mu\nu}+c_{jk}(p^{2},m_{i}^{2})\,p^{\mu}p^{\nu}\,,$
(C.4)

where $j,k=1,2$ and the coefficients $b_{jk}(p^{2},m_{i}^{2})$,
$c_{jk}(p^{2},m_{i}^{2})$ are the solutions of the system of two equations
obtained by contracting both sides of Eq. (C.4) respectively with $g_{\mu\nu}$
and $p_{\mu}p_{\nu}$ i.e.

$\displaystyle b_{jk}(p^{2},m_{i}^{2})$
$\displaystyle=\frac{1}{D-1}\,\frac{1}{p^{4}}\,\int_{l_{1}l_{2}}\Big{(}(l_{j}\cdot
l_{k})p^{2}-(l_{j}\cdot p)(l_{k}\cdot p)\Big{)}\,,$ (C.5) $\displaystyle
c_{jk}(p^{2},m_{i}^{2})$
$\displaystyle=\frac{1}{D-1}\,\frac{1}{p^{4}}\,\int_{l_{1}l_{2}}\Big{(}D\,(l_{j}\cdot
p)(l_{k}\cdot p)-(l_{j}\cdot l_{k})p^{2}\Big{)}\,.$ (C.6)

In the case of rank-$3$ integrals, there are four independent tensors
$p^{2}g^{\\{\mu\nu}p^{\rho\\}}$ with the curly brackets denoting all possible
permutation of the Lorentz indices, hence

$\displaystyle\int_{l_{1}l_{2}}l_{j}^{\mu}l_{k}^{\nu}l_{m}^{\rho}$
$\displaystyle=d_{1,jkm}(p^{2},m_{i}^{2})\,p^{2}\,g^{\mu\nu}p^{\rho}+d_{2,jkm}(p^{2},m_{i}^{2})\,p^{2}\,g^{\rho\mu}p^{\nu}$
$\displaystyle+d_{3,jkm}(p^{2},m_{i}^{2})\,p^{2}\,g^{\nu\rho}p^{\mu}+e_{jkm}(p^{2},m_{i}^{2})\,p^{\mu}p^{\nu}p^{\rho}\,,$

where $j,k,m=1,2$ and the coefficients $d_{r,jkm}(p^{2},m_{i}^{2})$,
$e_{jkm}(p^{2},m_{i}^{2})$ are the solutions of the system of four equations
obtained by contracting both sides of Eq. (LABEL:eq:r3-int) respectively with
$g_{\mu\nu}p_{\rho}$, $g_{\rho\mu}p_{\nu}$, $g_{\nu\rho}p_{\mu}$ and
$p_{\mu}p_{\nu}p_{\rho}$. Defining for simplicity

$\displaystyle d_{jkm}(p^{2},m_{i}^{2})$
$\displaystyle=\frac{1}{D-1}\,\frac{1}{p^{6}}\,\int_{l_{1}l_{2}}\Big{(}(l_{j}\cdot
l_{k})(l_{m}\cdot p)p^{2}-(l_{j}\cdot p)(l_{k}\cdot p)(l_{m}\cdot
p)\Big{)}\,,$ (C.8)

leads to

$\displaystyle d_{1,jkm}(p^{2},m_{i}^{2})=d_{jkm}(p^{2},m_{i}^{2})\,,$ (C.9)
$\displaystyle d_{2,jkm}(p^{2},m_{i}^{2})=d_{kmj}(p^{2},m_{i}^{2})\,,$ (C.10)
$\displaystyle d_{3,jkm}(p^{2},m_{i}^{2})=d_{mjk}(p^{2},m_{i}^{2})\,,$ (C.11)

while

$\displaystyle e_{jkm}(p^{2},m_{i}^{2})$
$\displaystyle=\frac{1}{D-1}\,\frac{1}{p^{6}}\,\int_{l_{1}l_{2}}\Big{(}(D+2)\,(l_{j}\cdot
p)(l_{k}\cdot p)(l_{m}\cdot p)$ $\displaystyle-(l_{j}\cdot l_{k})(l_{m}\cdot
p)\,p^{2}-(l_{k}\cdot l_{m})(l_{j}\cdot p)\,p^{2}-(l_{m}\cdot
l_{j})(l_{k}\cdot p)\,p^{2}\Big{)}\,.$ (C.12)

Finally in the case of rank-$4$ integrals there are $10$ independent tensor
structures we can build, namely $p^{4}\,g^{\\{\mu\nu}g^{\rho\sigma\\}}$,
$p^{2}g^{\\{\mu\nu}p^{\rho}p^{\sigma\\}}$, and
$p^{\mu}p^{\nu}p^{\rho}p^{\sigma}$, hence

$\displaystyle\int_{l_{1}l_{2}}l_{j}^{\mu}l_{k}^{\nu}l_{m}^{\rho}l_{n}^{\sigma}$
$\displaystyle=f_{1,jkmn}(p^{2},m_{i})\,p^{4}\,g^{\mu\nu}g^{\rho\sigma}+f_{2,jkmn}(p^{2},m_{i})\,p^{4}\,g^{\mu\rho}g^{\nu\sigma}$
$\displaystyle+f_{3,jkmn}(p^{2},m_{i})\,p^{4}\,g^{\mu\sigma}g^{\nu\rho}+g_{1,jkmn}(p^{2},m_{i})\,p^{2}\,g^{\mu\nu}p^{\rho}p^{\sigma}$
$\displaystyle+g_{2,jkmn}(p^{2},m_{i})\,p^{2}\,g^{\mu\rho}p^{\nu}p^{\sigma}+g_{3,jkmn}(p^{2},m_{i})\,p^{2}\,g^{\mu\sigma}p^{\nu}p^{\rho}$
$\displaystyle+g_{4,jkmn}(p^{2},m_{i})\,p^{2}\,g^{\nu\rho}p^{\mu}p^{\sigma}+g_{5,jkmn}(p^{2},m_{i})\,p^{2}\,g^{\nu\sigma}p^{\mu}p^{\rho}$
$\displaystyle+g_{6,jkmn}(p^{2},m_{i})\,p^{2}\,g^{\rho\sigma}p^{\mu}p^{\nu}+h_{jkmn}(p^{2},m_{i})\,p^{\mu}p^{\nu}p^{\rho}p^{\sigma}\,,$
(C.13)

where the coefficients $f_{r,jmk}(p^{2},m_{i})$, $g_{s,jmk}(p^{2},m_{i})$ and
$h_{jmk}(p^{2},m_{i})$ are the solutions of the system of $10$ equations
obtained by contracting both sides of Eq. (C.13) with each of the tensor
structures. Defining

$\displaystyle f_{jkmn}(p^{2},m_{i})$
$\displaystyle=\frac{1}{(D^{2}-1)(D-2)}\,\frac{1}{p^{8}}\,\int_{l_{1}l_{2}}\,\Big{[}D\,\Big{(}(l_{j}\cdot
p)(l_{k}\cdot p)(l_{m}\cdot p)(l_{n}\cdot p)$ $\displaystyle+(l_{j}\cdot
l_{k})\Big{(}(l_{m}\cdot l_{n})-(l_{m}\cdot p)(l_{n}\cdot
p)\Big{)}-(l_{j}\cdot p)(l_{k}\cdot p)(l_{m}\cdot l_{n})\Big{)}$
$\displaystyle+(l_{j}\cdot l_{m})(l_{k}\cdot p)(l_{n}\cdot p)+(l_{j}\cdot
l_{n})(l_{k}\cdot p)(l_{m}\cdot p)$ $\displaystyle+(l_{j}\cdot p)(l_{n}\cdot
p)(l_{k}\cdot l_{m})+(l_{j}\cdot p)(l_{m}\cdot p)(l_{k}\cdot l_{n})$
$\displaystyle-\Big{(}(l_{j}\cdot l_{m})(l_{k}\cdot l_{n})+(l_{j}\cdot
l_{n})(l_{k}\cdot l_{m})+2\,(l_{j}\cdot p)(l_{k}\cdot p)(l_{m}\cdot
p)(l_{n}\cdot p)\Big{)}\Big{]}\,,$ (C.14)

and

$\displaystyle g_{jkmn}(p^{2},m_{i})$
$\displaystyle=-\frac{1}{(D^{2}-1)(D-2)}\,\frac{1}{p^{8}}\,\int_{l_{1}l_{2}}\,\Big{[}-D^{2}(l_{j}\cdot
l_{k})(l_{m}\cdot p)(l_{n}\cdot p)$ $\displaystyle+D\Big{(}D\,(l_{j}\cdot
p)(l_{k}\cdot p)(l_{m}\cdot p)(l_{n}\cdot p)+(l_{j}\cdot l_{k})(l_{m}\cdot
l_{n})-(l_{j}\cdot p)(l_{k}\cdot p)(l_{m}\cdot l_{n})\Big{)}$
$\displaystyle+2\,(l_{j}\cdot l_{k})(l_{m}\cdot p)(l_{n}\cdot p)+(l_{j}\cdot
l_{m})(l_{k}\cdot p)(l_{n}\cdot p)+(l_{j}\cdot l_{n})(l_{k}\cdot p)(l_{m}\cdot
p)$ $\displaystyle+(l_{j}\cdot p)(l_{n}\cdot p)(l_{k}\cdot l_{m})+(l_{j}\cdot
p)(l_{m}\cdot p)(l_{k}\cdot l_{n})$ $\displaystyle-\big{(}(l_{j}\cdot
l_{m})(l_{k}\cdot l_{n})+(l_{j}\cdot l_{n})(l_{k}\cdot l_{m})+4\,(l_{j}\cdot
p)(l_{k}\cdot p)(l_{m}\cdot p)(l_{n}\cdot p)\big{)}\Big{]}\,,$ (C.15)

it follows that

$\displaystyle f_{1,jkmn}(p^{2},m_{i})$
$\displaystyle=f_{jkmn}(p^{2},m_{i})\,,$ (C.16) $\displaystyle
f_{2,jkmn}(p^{2},m_{i})$ $\displaystyle=f_{jmkn}(p^{2},m_{i})\,,$ (C.17)
$\displaystyle f_{3,jkmn}(p^{2},m_{i})$
$\displaystyle=f_{jnkm}(p^{2},m_{i})\,,$ (C.18) $\displaystyle
g_{1,jkmn}(p^{2},m_{i})$ $\displaystyle=g_{jkmn}(p^{2},m_{i})\,,$ (C.19)
$\displaystyle g_{2,jkmn}(p^{2},m_{i})$
$\displaystyle=g_{jmkn}(p^{2},m_{i})\,,$ (C.20) $\displaystyle
g_{3,jkmn}(p^{2},m_{i})$ $\displaystyle=g_{jnkm}(p^{2},m_{i})\,,$ (C.21)
$\displaystyle g_{4,jkmn}(p^{2},m_{i})$
$\displaystyle=g_{kmjn}(p^{2},m_{i})\,,$ (C.22) $\displaystyle
g_{5,jkmn}(p^{2},m_{i})$ $\displaystyle=g_{knjm}(p^{2},m_{i})\,,$ (C.23)
$\displaystyle g_{6,jkmn}(p^{2},m_{i})$
$\displaystyle=g_{mnjk}(p^{2},m_{i})\,,$ (C.24)

while

$\displaystyle h_{jkmn}(p^{2},m_{i})$
$\displaystyle=-\frac{1}{D^{2}-1}\,\frac{1}{p^{8}}\,\int_{l_{1}l_{2}}\,\Bigg{[}-(D+4)(D+2)\,(l_{j}\cdot
p)(l_{k}\cdot p)(l_{m}\cdot p)(l_{n}\cdot p)$
$\displaystyle+(D+2)\Big{(}(l_{j}\cdot l_{k})(l_{m}\cdot p)(l_{n}\cdot
p)+(l_{j}\cdot l_{m})(l_{k}\cdot p)(l_{n}\cdot p)$ $\displaystyle+(l_{j}\cdot