$D^{0}\to\pi^{-}\mu^{+}\nu$, and $D^{+}\to\pi^{0}\mu^{+}\nu$). We have
experimented with different models for the missing off-diagonal blocks of the
full correlation matrix, ranging from zero correlation to 100% correlation.
Our results for $|V_{cd}|$ and $|V_{cs}|$ are extremely insensitive to the
precise treatment of these off-diagonal correlations and give statistically
indistinguishable results. We therefore report values from our preferred
analysis, which uses a simple model for the correlations in which the off-
diagonal blocks are taken to be constant, with correlation coefficient equal
to the mean of the corresponding diagonal blocks.101010We thank the BES III
collaboration for providing us with the correlations for the differential rate
$d\Gamma/dq^{2}$ for the decays $D_{s}^{+}\to K^{0}e^{+}\nu$,
$D^{0}\to\pi^{-}\mu^{+}\nu$, and $D^{+}\to\pi^{0}\mu^{+}\nu$ as well as for
information and guidance regarding the treatment of off-diagonal correlations
between different decays (Lei Li, private communication, 22 July 2022; Hailong
Ma, private communication, 11 Dec 2022). Regarding the measurements of
$D^{0}\to K^{-}e^{+}\nu$ coming from BaBar [121], our fits drop the largest
$q^{2}$ bin, since it is constructed by a normalization constraint (one minus
the sum of the other bins). Especially when fitting $d\Gamma/dq^{2}$ for
semimuonic channels, including the scalar form factor is essential to
achieving a good description of the data at low $q^{2}$. However, the higher
parameters $b_{1}$ and $b_{2}$ associated with the scalar form factor are
constrained entirely by our precise synthetic data for $f_{0}$. The influence
of neglecting $f_{0}$ is considered below.

The plots in Figs. 25 and 26 also include comparisons with the shape obtained
from our form factors, given by the parameters in Table 13 (lattice-QCD-only
results), normalized by $|V_{cx}|^{2}_{\rm joint}$. Those correspond to the
blue curves in the plots. In addition, fits are also conducted to the
experimental data alone, although we do not use these results to extract the
CKM matrix elements. All those $z$-expansion fits, as well as the joint fit,
enforce the kinematic identity $f_{+}(0)=f_{0}(0)$ by imposing $a_{0}=b_{0}$
[cf. Eq. 54 and Eq. 55]. The best-fist posterior values for those
$z$-expansion fits are also given in Sec. D.5.

Another visualization of the form factors’ shapes, which is independent of the
overall normalization, comes from comparing the ratios $r_{1}\equiv
a_{1}/a_{0}$ and $r_{2}\equiv a_{2}/a_{0}$ of the $z$-expansion coefficients
from Eq. 56 after applying the refitting procedure of Ref. [62]. These ratios
are displayed for $D\to\pi$ and $D\to K$ in Fig. 27. Our results, given by the
black ellipses, show good agreement with the experimental shapes. For $D\to
K$, we also find good agreement with the lattice QCD calculation from HPQCD
[62]. For $D\to\pi$, we find $r_{1}=-2.009(55)$ and $r_{2}=0.14(36)$, with a
correlation of $\rho_{12}=-0.58$. For $D\to K$, we find $r_{1}=-2.05(11)$ and
$-0.59(52)$, with a correlation of $\rho_{12}=-0.29$.

Figure 27:  Comparing the shapes of the vector form factor $f_{+}$ between
lattice QCD and experiment for the decays $D\to\pi$ (left) and $D\to K$
(right) via ratios of $z$-expansion coefficients from Eq. 56. Where published
correlations are available, the ellipses show the $68\%$ confidence intervals.
Systematic errors from QED and isospin breaking are not included in the
lattice QCD results.

The second method we use to extract CKM matrix elements, the _binned method_ ,
combines lattice-QCD results with experimental data for the rate
$d\Gamma/dq^{2}$ to give a binwise estimate of the CKM matrix element:

$\displaystyle\left[|V_{cx}(q_{i}^{2})|\right]_{\rm
Binned}\equiv\left[\left(\frac{d\Gamma}{dq^{2}}\right)_{\rm
Expt}\frac{24\pi^{3}}{G_{F}^{2}\eta_{\rm
EW}^{2}}\frac{1}{\left\langle(\cdots)_{\rm
LQCD}\right\rangle}_{q_{i}^{2}}\right]^{1/2},$ (66)

where the quantity in the denominator is understood to be the binwise average
(i.e., integrated over the bin) of the lattice-QCD form factors together with
the appropriate kinematic factors appearing in Eq. 2,

$\displaystyle(\cdots)_{\rm
LQCD}\equiv(1-\epsilon)^{2}\left(\left|\bm{p}\right|^{3}(1+\epsilon/2)f_{+}(q^{2})+\left|\bm{p}\right|M_{H}^{2}\left(1-\frac{M_{L}^{2}}{M_{H}^{2}}\right)^{2}\frac{3}{8}\epsilon
f_{0}(q^{2})\right).$ (67)

This expression depends on the lepton mass via $\epsilon\equiv
m_{\ell}^{2}/q^{2}$ as well as the experimentally measured hadron masses
$M_{H}$ and $M_{L}$ for each mode (e.g., $D^{0}$ and $\pi^{-}$ or $D^{+}$ and
$\pi^{0}$ for $D\to\pi$). A weighted, correlated average (i.e., a fit to a
constant) then gives $|V_{cx}|_{\rm Binwise}$. The binned method is entirely
general and makes no assumptions about the relative size of the vector and
scalar contributions. Results for $|V_{cd}|$ from $D\to\pi$ and $|V_{cs}|$
from $D\to K$ for each $q^{2}$ bin and experiment, as well as the correlated
average over bins including only semielectronic (blue lines) or only
semimuonic data (red lines), are shown in Fig. 28. The semimuonic results lie
roughly $1\sigma$ below the semielectronic results for both $|V_{cd}|$ and
$|V_{cs}|$, so below we report the values in each channel as well as the
combined results. Those combined extractions, including all leptonic channels,
lie between the two bands in Fig. 28, and are statistically consistent with
the individual determinations, as shown in Fig. 29. For $|V_{cd}|$ from
$D_{s}\to K$, results are shown in Fig. 26. As argued above, with present
statistical precision, the presence of the scalar form factor is
quantitatively important for the differential rate $d\Gamma/dq^{2}$,
especially for semimuonic channels. Figure 30 shows the effect of dropping the
contribution from $f_{0}$ for $D\to K\mu\nu$. Values for $|V_{cs}|^{\rm
binned}$ are observed to shift by a few percent and, when considered as a
function of $q^{2}$, become statistically inconsistent with a constant.
Similar few-percent shifts occur for $D\to\pi\mu\nu$.

Figure 28:  The binwise estimate of the CKM matrix element
$\left[|V_{cx}(q_{i}^{2})|\right]_{\rm Binned}$ from the decays $D\to\pi$ (top
rows) and $D\to K$ (bottom row) in the semielectronic (left) and semimuonic
(right) channels. The horizontal bands show the resulting values for
$|V_{cx}|^{\rm Binned}$ from correlated fits to a constant in each channel.
The result for $|V_{cd}|_{\rm Binned}^{D\to\pi\mu^{+}\nu}$ (red) lies slightly
below $|V_{cd}|_{\rm Binned}^{D\to\pi e^{+}\nu}$ (blue). The combined
extraction using both channels lies between the two bands and is statistically
consistent with each. A comparison of the different extractions of $|V_{cx}|$
is given in Fig. 29. For $D\to\pi$, experimental data are taken from BaBar
[122], CLEO [123], and BES III [124, 126, 127]. For $D\to K$, experimental
data are taken from BaBar [121], CLEO [123], and BES III [124, 126, 128].
Although all the correlated fits have good quality ($\chi^{2}/{\rm DOF}\approx
1$, $p\gtrsim 0.05$), the residuals for $D\to K$ are visually larger near
$q^{2}_{\rm max}$.

Because the joint-fit and binned methods explicitly account for (potentially)
percent-level contributions from the scalar form factor, they constitute our
main extractions for $|V_{cx}|$. For continuity with previous studies, we also
consider the _endpoint method_ , in which $|V_{cx}|$ is defined according to

$\displaystyle[|V_{cx}|]_{\rm Endpoint}\equiv\frac{[|V_{cx}|\eta_{\rm
EW}f_{+}(0)]_{\rm Expt}}{\eta_{\rm EW}[f_{+}(0)]_{\rm LQCD}}.$ (68)

The experimental values are taken from the HFLAV world averages:
$|V_{cd}|\eta_{\rm EW}f_{+}^{D\to\pi}(0)=0.1426(18)$, $|V_{cs}|\eta_{\rm
EW}f_{+}^{D\to K}(0)=0.7180(33)$ [44]. The resulting values for
$[|V_{cx}|]_{\rm Endpoint}$ are shown in Fig. 29 and given in Table 19.
Although these endpoint results give a statistical precision comparable to our
preferred extractions, it’s worth emphasizing that our precise values for
$f_{+}(0)$ were made possible by leveraging information about the form factor
across the full kinematic range of the decays. The final errors can
potentially be much larger in a simulation that works directly at the endpoint
($q^{2}=0$). For example, preliminary work by our collaboration has focused on
$q^{2}\approx 0$ on many of the same ensembles and with comparable statistics
[60]. Using the preliminary values of $f_{+}(0)$ from these proceedings gives
values for $|V_{cd}|$ and $|V_{cs}|$ with errors that are roughly 2.5 to 3.5
larger than the final errors in the present work.

Figure 29:  Determinations of $|V_{cd}|$ and $|V_{cs}|$ using experimental
measurements of the decays $D\to\pi$ and $D\to K$. The outer and inner error
bars and bands show the results with and without QED uncertainties,
respectively.  Figure 30:  The effect of neglecting the scalar form factor
(setting $f_{0}(q^{2})\equiv 0$ in Eq. 2) when computing $|V_{cs}|^{\rm
binned}$ for $D\to K\mu\nu$. The red and blue horizontal lines and the black
data are reproduced from the bottom-right panel of Fig. 28. Similar few-
percent shifts occur for $D\to\pi\mu\nu$.

The results for $|V_{cd}|$ and $|V_{cs}|$ from the different methods described
above and for different leptons in the final states are summarized in Fig. 29
and Table 19. Since, as shown in the plot, the binned and joint-fit
extractions give statistically consistent values well within 1$\sigma$, we
take the joint-fit extractions to define our preferred results:

$\displaystyle|V_{cd}|^{D\to\pi\ell^{+}\nu}$ $\displaystyle=0.2238(11)^{\rm
Expt}(15)^{\rm QCD}(04)^{\rm EW}(02)^{\rm SIB}[22]^{\rm QED},$ (69)
$\displaystyle|V_{cd}|^{D_{s}\to Ke^{+}\nu}$ $\displaystyle=0.258(15)^{\rm
Expt}(01)^{\rm QCD}[03]^{\rm QED},$ (70) $\displaystyle|V_{cs}|^{D\to
K\ell^{+}\nu}$ $\displaystyle=0.9589(23)^{\rm Expt}(40)^{\rm QCD}(15)^{\rm
EW}(05)^{\rm SIB}[95]^{\rm QED},$ (71)

where the first error comes from the experimental differential decay rate
uncertainty, the second error comes from our form factor calculation (see
Table 15), the third error shows the uncertainty in $\eta_{EW}$, and the
fourth and fifth from our estimate of SIB and long-distance QED corrections
described in Sec. VII.2. The errors in these expressions combine in quadrature
to give the total errors in Table 19. Since our preferred extraction of
$|V_{cs}|$ includes both $e^{+}$ and $\mu^{+}$ final states, the experimental
contribution to the error is smaller by roughly a factor of two than in Ref.
[62].

We also repeated our analysis separating charged-hadron and neutral-final
states (e.g., $\pi^{-}e^{+}$ versus $\pi^{0}e^{+}$). No statistically
significant difference was observed within the uncertainties, consistent with
what was observed in Ref. [62].

Table 19: Summary of results for $|V_{cd}|$ and $|V_{cs}|$ from different decays and different extraction methods. The final column gives the result when errors from QED are neglected.  | Process | Method | $|V_{cx}|$ |  $|V_{cx}|$ (no QED)
---|---|---|---|---
$|V_{cd}|$ | $D\to\pi e^{+}\nu$ |  $z$-expansion | 0.2251(30) | 0.2251(20)
$|V_{cd}|$ | $D\to\pi e^{+}\nu$ | binned | 0.2250(31) | 0.2250(21)
$|V_{cd}|$ | $D\to\pi\mu^{+}\nu$ |  $z$-expansion | 0.2199(35) | 0.2199(27)
$|V_{cd}|$ | $D\to\pi\mu^{+}\nu$ | binned | 0.2209(36) | 0.2209(28)
$|V_{cd}|$ | $D\to\pi\ell^{+}\nu$ |  $z$-expansion | 0.2238(29) | 0.2238(19)
$|V_{cd}|$ | $D\to\pi\ell^{+}\nu$ | binned | 0.2238(30) | 0.2238(19)
$|V_{cd}|$ | $D\to\pi\ell^{+}\nu$ | endpoint | 0.2243(41) | 0.2243(34)
$|V_{cs}|$ | $D\to Ke^{+}\nu$ |  $z$-expansion | 0.9653(110) | 0.9653(53)
$|V_{cs}|$ | $D\to Ke^{+}\nu$ | binned | 0.9631(113) | 0.9631(58)
$|V_{cs}|$ | $D\to K\mu^{+}\nu$ |  $z$-expansion | 0.9537(111) | 0.9537(56)
$|V_{cs}|$ | $D\to K\mu^{+}\nu$ | binned | 0.9543(116) | 0.9543(65)
$|V_{cs}|$ | $D\to K\ell^{+}\nu$ |  $z$-expansion | 0.9589(108) | 0.9589(49)
$|V_{cs}|$ | $D\to K\ell^{+}\nu$ | binned | 0.9582(110) | 0.9582(54)
$|V_{cs}|$ | $D\to K\ell^{+}\nu$ | endpoint | 0.9549(110) | 0.9549(61)
$|V_{cd}|$ | $D_{s}\to Ke^{+}\nu$ |  $z$-expansion | 0.2582(155) | 0.2582(153)
$|V_{cd}|$ | $D_{s}\to Ke^{+}\nu$ | binned | 0.2583(157) | 0.2583(155)

For the first time, our calculation provides a value of $|V_{cd}|$ from
$D\to\pi$ for which lattice QCD errors are at the same level as the
experimental errors, $\sim 0.5\%$ each. This represents an improvement by
roughly a factor of six from the existing state of the art [57, 139]. For
$|V_{cd}|^{D_{s}\to K}$, experimental errors dominate and are substantially
larger than for $D\to\pi$. Since the theoretical uncertainty is actually the
smallest for $D_{s}\to K$, additional experimental measurements of this
channel would be particularly welcome. On the other hand, theoretical error
exceeds the experimental error by roughly a factor of two in the extraction of
$|V_{cs}|$ from $D\to K$, leaving room for improvements in the theory side.
Experimental errors also dominate the CKM extractions from the semimuonic
channels, where we have only included recent results from BES-III. Another key
ingredient for improved semileptonic extractions of $|V_{cd}|$ and $|V_{cs}|$
would be the calculation of long-distance structure-dependent EM corrections
or a more robust estimate of their effect on these decays, since our lack of
knowledge of these corrections currently dominates the uncertainty of the most
precise determinations.

A comparison of our final results for $|V_{cd}|$ and $|V_{cs}|$ with existing
results in the literature appears in Fig. 31, including leptonic decays,
global fits assuming CKM unitarity fits, and scattering. Our determinations of
$|V_{cd}|$ and $|V_{cs}|$ agree well, at the level of 1–2 standard deviations,
with previous leptonic [140, 141, 142, 31, 143, 32] and semileptonic [48, 49,
57, 139, 62] determinations reported in FLAG [36].

Our correlated results for $|V_{cd}|$ and $|V_{cs}|$ also yield the ratio,

$|V_{cd}|/|V_{cs}|=0.2334(13)^{\rm Expt}(16)^{\rm QCD}(02)^{\rm SIB}[11]^{\rm
QED}$ (72)

where the correlation coefficient between $|V_{cd}|$ and $|V_{cs}|$,
neglecting QED, is 0.18. As described in Sec. VII.2, we have taken a
conservative $0.5\%$ systematic uncertainty for QED effects in the ratio.

Using the latest measurements $f_{D^{+}}|V_{cd}|$ and $f_{D_{s}}|V_{cs}|$
reported by HFLAV [44] and the ratio of decay constants $f_{D_{s}}/f_{D^{+}}$
computed by our collaboration in a similar set of ensembles and with the same
action in Ref. [32], one finds $[|V_{cd}|/|V_{cs}|]^{\rm
leptonic}=0.2212(58)$, where the error is dominated by the experimental
uncertainty. Both values are plotted in Fig. 32 together with previous
leptonic [140, 141, 142, 31, 143, 32] semileptonic [48, 49, 57, 139, 62]
determinations combined in averages by FLAG [36] and the result from the PDG
global unitarity fit [25] (the global-fit methodologies of CKMfitter [144] and
UTfit [145] give very similar results). The leptonic extraction above agrees
with our semileptonic result within roughly $2\sigma$, although, as plotted in
Fig. 32, leptonic determinations tend to give smaller values of the ratio. The
error in our result is more than a factor of two smaller than the leptonic
one, with similar uncertainties from lattice QCD and experiment. Results for
$|V_{cd}|/|V_{cs}|$ from the PDG global fit assuming unitarity and from the
ratio $|V_{us}|/|V_{ud}|$ (see Sec. VII.4 below for more details) are also
shown in Fig. 32. Our result agrees well with both of them.

Figure 31:  Comparison of our preferred determinations of $|V_{cd}|^{D\to\pi}$
and $|V_{cs}|^{D\to K}$ (blue bands) with existing results in the literature.
The outer and inner error bands show our preferred result with and without QED
uncertainties, respectively. The world’s first determination
$|V_{cd}|^{D_{s}\to K}$ is also given. Results from FLAG are taken from Ref
[36]. Results from the PDG appear in Ref. [25]. We emphasize that FLAG uses
slightly different conventions for the semileptonic extraction of
$|V_{cd(cs)}|$ as we used here; for instance they do not include short-
distance electroweak corrections to $G_{F}$ or an error from QED. For the
leptonic results, we combine the latest experimental averages reported in
HFLAV [44] with the FLAG averages for $f_{D}$ and $f_{D_{s}}$ [36]. “CKM
unitarity” denotes the global fit result reported by the PDG, which includes
all available measurements (for all nine matrix elements) imposing three-
generation unitarity.  Figure 32: Comparison of different extractions of the
ratio $|V_{cd}|/|V_{cs}|$. The blue point and band show the value from the
preferred extractions of the present work. Error bands with and without QED
error are indistinguishable. The red and green points denote semileptonic and
leptonic extractions given by FLAG [36]. The points below the dashed line are
constraints from unitarity. The orange point is computed using values from
CKMFitter’s global fit (as reported in the PDG), assuming CKM unitarity. The
brown points comes from two different extractions of $|V_{us}|/|V_{ud}|$
which, as explained in the text, are related to $|V_{cd}|/|V_{cs}|$ by CKM
unitarity.

###  VII.4 Tests of CKM unitarity

Our results for $|V_{cd}|$ and $|V_{cs}|$ enable a test of unitary in the
second row of the CKM matrix, including theoretical correlations between
$|V_{cd}|$ and $|V_{cs}|$. Using our preferred extractions in Eq. 69 and Eq.
71, and $|V_{cb}|^{\rm incl+excl}=(40.8\pm 1.4)\times 10^{-3}$ from a combined
average of inclusive and exclusive semileptonic $B$-decays [25]111111 In
particular, see the review “Semileptonic $b$-Hadron Decays, Determination of
$V_{cb}$, $V_{ub}$” yields the following result for the deviation from
unitarity in the second row:

$\displaystyle|V_{cd}|^{2}+$
$\displaystyle|V_{cs}|^{2}+|V_{cb}|^{2}-1-0.0286(44)^{\rm Expt}(78)^{\rm
QCD}[194]^{\rm QED}(28)^{\rm EW}=-0.029(22).$ (73)

Because $|V_{cb}|$ is so small compared to $|V_{cd}|$ and $|V_{cs}|$,
numerically indistinguishable results are obtained (within current precision)
if inclusive or exclusive values are taken for $|V_{cb}|$. This result is
compatible with three-generation CKM unitary within approximately one standard
deviation. The precision of this test is roughly $2\%$ and is limited by the
systematic uncertainty from QED in our extractions of $|V_{cd}|$ and
$|V_{cs}|$. We show the constraints on $|V_{cd}|$ and $|V_{cs}|$ from our
calculation in Fig. 33, together with constraints coming from leptonic decays
[32, 44] and second-row unitarity. The leptonic inputs used for the green
ellipse are summarized in Table 20. As the figure shows, semileptonic tests of
second-row CKM unitarity are now slighty more precise than leptonic tests. The
leptonic and semileptonic results are consistent at the level of roughly 1-2
standard deviations.

Figure 33: Constraints on $|V_{cd}|$ and $|V_{cs}|$ from our results,
$D-$meson leptonic decays, and unitarity. The blue ellipse shows the preferred
values of the present work from semileptonic decays in Eq. 69 and Eq. 71. The
green ellipse is the result of combining the latest results for the products
$\eta_{\rm EW}|V_{cd}|f_{D^{+}}$ and $\eta_{\rm EW}|V_{cs}|f_{D_{s}}$ with
leptonic decay constants from lattice-QCD calculations; the inputs values are
summarized in Table 20. The dotted line comes from assuming unitarity of the
second row, taking $|V_{cb}|^{\rm incl+excl}=(40.8\pm 1.4)\times 10^{-3}$
[25]. In all cases, the ellipses shows the correlated $1\sigma$ (68%)
confidence intervals. The inner blue ellipse shows our result without the QED
uncertainty.

One can perform further tests of the unitarity of the CKM matrix using the
fact that in the Standard Model,
$|V_{cd}|=|V_{us}|+\mathcal{O}(A^{2}\lambda^{5})$ and
$|V_{cs}|=|V_{ud}|+\mathcal{O}(A^{2}\lambda^{4})$. Including the dominant
corrections [146] with the Wolfenstein parameters taken from global unitarity
fits by CKMFitter [144] (using values from the January 2022 update) gives

$\displaystyle|V_{cs}|$ $\displaystyle=0.97282(32)\text{ from
}|V_{ud}|^{0^{+}\to 0^{+}},$ (74) $\displaystyle|V_{cd}|$
$\displaystyle=0.22317(53)\text{ from }|V_{us}|^{K\ell 3},$ (75)
$\displaystyle|V_{cd}|/|V_{cs}|$ $\displaystyle=0.22941(55)\text{ from
}|V_{us}|^{K\ell 3}/|V_{ud}|^{0^{+}\to 0^{+}},$ (76)

using $|V_{ud}|=0.97367(32)$ from superallowed $0^{+}\to 0^{+}$ nuclear
$\beta$ decays [17, 37] and $|V_{us}|=0.22330(53)$ from $K_{\ell 3}$ decays
[36, 37]. Alternatively, the ratio of $K_{\ell 2}$ to $\pi_{\ell 2}$ decays
yields [36, 37]

$|V_{cd}|/|V_{cs}|=0.23135(51)\text{ from }|V_{us}/V_{ud}|^{K_{\ell
2}/\pi_{\ell 2}}.$ (77)

As shown in Fig. 32, our preferred value in Eq. 72 lies roughly $1\sigma$
above the result coming from $|V_{us}/V_{ud}|^{K_{\ell 2}/\pi_{\ell 2}}$ and
roughly $2\sigma$ above that from $|V_{us}|^{K_{\ell 3}}/|V_{ud}|^{0^{+}\to
0^{+}}$. Our preferred value for $|V_{cd}|$ in Eq. 69 shows excellent
agreement with $|V_{cd}|$ from $|V_{us}|^{K\ell 3}$. Our preferred value for
$|V_{cs}|$ in Eq. 71 lies somewhat below $|V_{cs}|$ from $|V_{ud}|^{0^{+}\to
0^{+}}$ but is consistent at 1-2 standard deviations.

Table 20: Leptonic inputs used for comparison in Fig. 33. HFLAV reports the product $\eta_{\rm EW}|V_{cx}|f_{D_{(s)}}$ [44]. Following the prescription of the PDG [25], we include an EW+QED error of $0.7\%$ for the product $|V_{cx}|f_{D_{(s)}}$.  Value | Source
---|---
$\eta_{\rm EW}|V_{cd}|f_{D^{+}}=46.2(1.0)(0.3)^{\rm EW+QED}\leavevmode\nobreak\ \text{MeV}$ | HFLAV [44]
$\eta_{\rm EW}|V_{cs}|f_{D_{s}}=245.4(2.4)(1.7)^{\rm EW+QED}\leavevmode\nobreak\ \text{MeV}$ | HFLAV [44]
$f_{D^{+}}=212.7(0.6)\leavevmode\nobreak\ \text{MeV}$ | FNAL-MILC 2018 [32]
$f_{D_{s}}=249.9(0.4)\leavevmode\nobreak\ \text{MeV}$ | FNAL-MILC 2018 [32]
$f_{D_{s}}/f_{D^{+}}=1.1749(16)$ | FNAL-MILC 2018 [32]

###  VII.5 Lepton flavor universality

For a given semileptonic decay $H\to L\ell\nu$, the lepton flavor universality
(LFU) ratio $R_{\mu/e}$ is defined as the ratio of branching fractions into
muon versus electron final states

$\displaystyle R_{\mu/e}^{H\to L}\equiv\frac{\mathcal{B}(H\to
L\mu\nu)}{\mathcal{B}(H\to Le\nu)}=\frac{\Gamma_{\mu}}{\Gamma_{e}},$ (78)

where the total rates to each final state are defined in the usual way,

$\displaystyle\Gamma_{\ell}\equiv\int_{m_{\ell}^{2}}^{q^{2}_{\rm
max}}dq^{2}\,\left(\frac{d\Gamma}{dq^{2}}\right).$ (79)

In the SM, the LFU ratios are close but not identically equal to unity. This
difference from unity arises from at least three effects. First, the lower
boundary of the integration region in Eq. 79 depends on the lepton mass.
Second, the differential decay rate in Eq. 2 itself depends on the lepton
mass, with the scalar form factor contributing more for larger masses.
Finally, QED corrections depend in principle on both the charges of the final
state and the lepton mass. The coefficients
$\frac{G_{F}^{2}}{24\pi^{3}}\left(\eta_{\rm EW}|V_{cx}|\right)^{2}$ are
independent of $q^{2}$ and cancel in the ratio, meaning that predictions for
$R_{\mu/e}$ are entirely calculable using our lattice-QCD form factors, up to
corrections from QED and SIB. The rates $d\Gamma/dq^{2}$ for the decay
$D\to\pi$, using as inputs our form factors $f_{0}(q^{2})$ and $f_{+}(q^{2})$
together with the estimates of systematic uncertainties from QED and SIB (see
Sec. VII.2), are shown in Fig. 34 for both semielectronic and semimuonic final
states. When computing the rates, the meson masses were taken to be the
average of the experimentally measured masses for the charged and neutral
states (e.g., $D^{0}$ and $D^{+}$ or $\pi^{0}$ and $\pi^{+}$). The final
results for the SM predictions of the ratios $R_{\mu/e}$ are

$\displaystyle R_{\mu/e}^{D\to\pi}$ $\displaystyle=0.98671(17)^{\rm
QCD}[500]^{\rm QED},$ (80) $\displaystyle R_{\mu/e}^{D\to K}$
$\displaystyle=0.97606(16)^{\rm QCD}[500]^{\rm QED},$ (81) $\displaystyle
R_{\mu/e}^{D_{s}\to K}$ $\displaystyle=0.98099(10)^{\rm QCD}[500]^{\rm QED}.$
(82)

The dominant error is the systematic uncertainty from QED corrections, which
we conservatively take to be $0.5\%$, as described in Sec. VII.2. Our
prediction for $R_{\mu/e}^{D\to K}$ is in good agreement with a recent
calculation by HPQCD, which found $R_{\mu/e}^{D\to K}=0.97594(19)^{\rm
QCD}[500]^{\rm QED}$ and used the same estimate of the QED uncertainty
[62].121212The central value we quote here differs slightly from the published
value in Ref. [62]. We thank HPQCD for providing the correct central value
(William Parrott, private communication, 16 Dec 2022). We also find good
agreement with previous lattice QCD results by ETMC and experimental
measurements of $R_{\mu/e}^{D\to\pi}$ and $R_{\mu/e}^{D\to K}$, as shown in
Fig. 35. The measurement of $R_{\mu/e}^{D\to\pi}$ by BES III for the channel
$D^{0}\to\pi^{-}$ lies below our result but is consistent at the $2\sigma$
level. Because the QED error is dominant for the lattice-QCD predictions of
the LFU ratios, the insets in Fig. 35 compare the lattice-QCD results with the
QED uncertainty removed.

Figure 34:  Our prediction for the decay rates $d\Gamma/dq^{2}$ for the decays
$D\to\pi$. The majority of the total rate comes from small $q^{2}$, where
$(d\Gamma/dq^{2})_{\mu}<(d\Gamma/dq^{2})_{e}$. The Standard Model therefore
predicts $R_{\mu/e}<1$.

Figure 35: Comparison of $R_{\mu/e}^{D\to\pi}$ and $R_{\mu/e}^{D\to K}$ with
experimental HFLAV averages [44], which are dominated by measurements from BES
III [125, 128, 127], and other SM predictions from lattice QCD [139, 62]. In
the main body of both figures, all lattice QCD results are presented with a
QED uncertainty of $0.5\%$. The results from ETMC 18 were reported in the
isospin-symmetric limit of QCD, without including QED or SIB uncertainties
[139], so we have added the QED uncertainty for a like-to-like comparison. The
insets compare lattice QCD results when QED uncertainty is removed.

##  VIII Conclusions

We have calculated the hadronic form factors $f_{+}(q^{2})$ and $f_{0}(q^{2})$
relevant for the semileptonic decays $D\to\pi\ell\nu$, $D\to K\ell\nu$, and
$D_{s}\to K\ell\nu$ using lattice QCD. These decays occur at tree level in the
SM and are important channels for determining the CKM matrix elements
$|V_{cd}|$ and $|V_{cs}|$. Our calculation uses $N_{f}=2+1+1$ flavors of
dynamical staggered quarks and includes several ensembles with all quarks near
their physical masses. The use of the HISQ action permits all the quarks to be
treated with the same relativistic light-quark action and allows for
nonperturbative renormalization using PCVC, Eq. 10. Our results improve
significantly on the previous precision for the form factors for $D\to\pi$ and
$D_{s}\to K$, and have precision comparable to that of recent $N_{f}=2+1+1$
calculations by HPQCD for $D\to K$ [62, 63]. We agree well with HPQCD’s $D\to
K$ form factors over the entire kinematic range, especially with their latest
results in Ref. [63], while for both $D\to\pi$ and $D\to K$, our form factors
are significantly larger near $q^{2}_{\rm max}$ than the $N_{f}=2+1+1$ results
of ETMC [57]. Table 13 shows the $z$-expansion parameters from which our final
results for the form factors, computed in isospin-symmetric QCD where
$m_{u}=m_{d}$, can be reconstructed, while a complete error budget, including
all statistical and systematic uncertainties, is given in Table 15 for the
edges of the kinematic range.

Our results suggest a very mild spectator dependence for $D\to\pi$ and
$D_{s}\to K$, with close agreement at $\lesssim 2\%$ level throughout the
kinematic range between the respective form factors (cf. Fig. 18). This
picture was also recently confirmed, within experimental uncertainty, by the
first measurement of the decay $D_{s}\to K$ by BES III [109].

When combined with the available experimental data for the corresponding decay
rates, summarized in Table 17, our form factors enable the extraction of the
CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ with percent-level
uncertainties. These extractions include correlations between all the lattice
form factors and between the different experimental channels.131313In the
supplementary material, we provide correlated values for all the $z$-expansion
coefficients needed to reproduce our final results for all three decays. The
values obtained from our preferred extractions are

$\displaystyle|V_{cd}|^{D\to\pi\ell^{+}\nu}$ $\displaystyle=0.2238(11)^{\rm
Expt}(15)^{\rm QCD}(04)^{\rm EW}(02)^{\rm SIB}[22]^{\rm QED},$
$\displaystyle|V_{cd}|^{D_{s}\to Ke^{+}\nu}$ $\displaystyle=0.258(15)^{\rm
Expt}(01)^{\rm QCD}[03]^{\rm QED},$ $\displaystyle|V_{cs}|^{D\to
K\ell^{+}\nu}$ $\displaystyle=0.9589(23)^{\rm Expt}(40)^{\rm QCD}(15)^{\rm
EW}(05)^{\rm SIB}[95]^{\rm QED}.$ $\displaystyle|V_{cd}|/|V_{cs}|$
$\displaystyle=0.2329(13)^{\rm Expt}(16)^{\rm QCD}(02)^{\rm SIB}[11]^{\rm
QED}$

For $|V_{cd}|$ we obtain the most precise determination to date, with lattice-
QCD form factors errors that, for the first time in a semileptonic extraction,
are commensurate with experimental uncertainties. The improved determination
of $D\to\pi$ form factors, together with the fact that we account for
theoretical correlations among channels, also allows us to provide the most
precise determination of the ratio $|V_{cd}|/|V_{cs}|$, around a factor of two
more precise than the leptonic determination. The rate for $D_{s}\to K$ was
only recently measured for the first time by BES III [109], and our
calculation delivers the first extraction of $|V_{cd}|^{D_{s}\to K}$. Although
this determination is not yet competitive with the one from $D\to\pi$, the
error is dominated by the statistics-limited experimental uncertainty. Our
result for $|V_{cd}|^{D_{s}\to K}$ lies roughly $2\sigma$ above
$|V_{cd}|^{D\to\pi}$, albeit with large uncertainty. Experimental improvements
for this Cabibbo-suppressed decay would immediately give improved precision
for $|V_{cd}|^{D_{s}\to K}$ and help clarify the situation.

Our determinations of $|V_{cd}|$ and $|V_{cs}|$, combined with the value of
$|V_{cb}|$ from Ref. [25], give a precise test of second-row CKM unitarity. We
find consistency with unitarity at the level of roughly $2\%$ and one standard
deviation, with an uncertainty dominated by the systematic effect of QED. As
shown in Fig. 33, the precision of the semileptonic constraint is now slightly
better than the corresponding leptonic one.

After demonstrating consistency between the form factor shapes from our
calculations and those measured in experiments, we computed the SM prediction
for the lepton flavor universality ratios $R_{\mu/e}$ with sub-percent
precision for all three decays:

$\displaystyle R_{\mu/e}^{D\to\pi}$ $\displaystyle=0.98671(17)^{\rm
QCD}[500]^{\rm QED},$ $\displaystyle R_{\mu/e}^{D\to K}$
$\displaystyle=0.97606(16)^{\rm QCD}[500]^{\rm QED},$ $\displaystyle
R_{\mu/e}^{D_{s}\to K}$ $\displaystyle=0.98099(10)^{\rm QCD}[500]^{\rm QED}.$

These results agree with previous $N_{f}=2+1+1$ lattice calculations,
considerably improving the precision for $D\to\pi$, and with experimental
measurements within $2\sigma$, for $D\to\pi$ and $D\to K$.

With the total precision for $|V_{cd}|$ and $|V_{cs}|$ approaching the
subpercent level, the effects of the scalar form factor in the differential
rate, Eq. 2, become quantitatively important. For semielectronic decays,
contributions from $f_{0}$ enter at roughly the $1\%$ level in the lowest
$q^{2}$ bin. For semimuonic decays, the effect is much larger, a roughly
$10\%$ effect in the lowest $q^{2}$ bin and a few-percent effect throughout
the rest of the kinematic range. Figure 30 showed that naively neglecting
contributions from $f_{0}$ can shift values for $|V_{cs}|$ by a few percent in
the case of $D\to K\mu\nu$ (similar results hold for $D\to\pi\mu\nu$).

Future progress in the precision of $|V_{cd}|$, $|V_{cs}|$, and the LFU ratios
will depend crucially on improved understanding of QED corrections to these
decays, which are already the dominant source of uncertainty. The one
exception is the decay $|V_{cs}|^{D_{s}\to K\ell^{+}\nu}$, for which the
experimental error is still large. One avenue for improvement is through EFT
calculations in the spirit of those for $K\to\pi\ell\nu$ [133, 134, 41, 42,
43], which were used in Sec. VII.2 to estimate our systematic uncertainties
(cf. Sec. VII.2). As usual, the intermediate mass of the charm quark (which is
simultaneously too heavy for $\chi$PT to apply and too light for reliable
application of HQET) may present a challenge for robust treatment with EFT.
Another possibility is carrying out lattice simulations to compute the
structure-dependent QED corrections to the semileptonic decay amplitudes. Such
calculations have not yet reached a mature state, but the field is progressing
rapidly, particularly for the QED corrections to leptonic decays [147, 39, 40,
148, 149, 150, 151].

Regarding the pure QCD calculation, it should be straightforward to improve
the precision of our form factor results. A leading contribution to the error
budget is statistics (cf. Table 15 and Figs. 22, 49 and 50), for which the
physical mass ensembles at $a\approx 0.06\leavevmode\nobreak\ \text{fm}$ and
$0.09\leavevmode\nobreak\ \text{fm}$ play the largest role. As part of our
ongoing work toward $B$-meson semileptonic decays, we are simulating on a
finer physical-mass ensemble with $a\approx 0.04\leavevmode\nobreak\
\text{fm}$. We expect that new data from this ensemble will reduce the
uncertainties both from statistics and from the continuum extrapolation.
Future calculations will also benefit from ongoing work in the community to
improve scale-setting measurements (e.g., $w_{0}$ or the $\Omega$-baryon mass)
on the HISQ ensembles used in this work.

## Acknowledgments

We thank Claude Bernard, Urs Heller, Javad Komijani, and Jack Laiho for
collaboration and essential contributions to previous projects, which paved
the way for this work. We also thank Claude Bernard for helpful advice about
scale setting and about the chiral expansion and Javad Komijani for useful
correspondence regarding $\alpha_{s}$ on the HISQ ensembles. We thank Jake
Bennett and Alan Schwartz for answering questions about the Belle data. We
thank the BES III collaboration, and especially Lei Li and Hailong Ma, for
providing us with their data for $D_{s}\to Ke\nu$ as well as correlation data
for $D\to\pi\mu\nu$. We thank Ryan Mitchell for useful comments about the CLEO
and BES III detectors. We thank William Parrott for answering questions about
HPQCD’s evaluation of LFU ratios.

This material is based upon work supported in part by the U.S. Department of
Energy, Office of Science under grant Contract Numbers DE-SC0010120 (S.G.),
DE-SC0011090 (W.J.), DE-SC0021006 (W.J.), DE-SC0015655 (A.X.K., Z.G., A.T.L.),
and DE-SC0010005 (E.T.N.); by the U.S. National Science Foundation under
Grants No. PHY17-19626 and PHY20-13064 (C.D., A.V.); by the Simons Foundation
under their Simons Fellows in Theoretical Physics program (A.X.K.); by SRA
(Spain) under Grant No. PID2019-106087GB-C21 / 10.13039/501100011033 (E.G.);
by the Junta de Andalucía (Spain) under Grants No. FQM-101, A-FQM-467-UGR18
(FEDER), and P18-FR-4314 (E.G.); by AEI (Spain) under Grant No.
RYC2020-030244-I / AEI / 10.13039/501100011033 (A.V.). This document was
prepared by the Fermilab Lattice and MILC Collaborations using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.

Computations for this work were carried out in part on facilities of the USQCD
Collaboration, which are funded by the Office of Science of the U.S.
Department of Energy. An award of computer time was provided by the Innovative
and Novel Computational Impact on Theory and Experiment (INCITE) program. This
research used resources of the Argonne Leadership Computing Facility, which is
a DOE Office of Science User Facility supported under contract DE-
AC02-06CH11357. This research also used resources of the Oak Ridge Leadership
Computing Facility, which is a DOE Office of Science User Facility supported
under Contract No. DE-AC05-00OR22725. This research used resources of the
National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility located at Lawrence
Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231.
The authors acknowledge support from the ASCR Leadership Computing Challenge
(ALCC) in the form of time on the computers Summit and Theta. The authors
acknowledge the Texas Advanced Computing Center (TACC) at The University of
Texas at Austin for providing HPC resources that have contributed to the
research results reported within this paper. This research is part of the
Frontera computing project at the Texas Advanced Computing Center. Frontera is
made possible by National Science Foundation award OAC-1818253 [152]. This
work used the Extreme Science and Engineering Discovery Environment (XSEDE),
which is supported by National Science Foundation Grant No. ACI-1548562. This
work used XSEDE Ranch through the allocation TG-MCA93S002 [153].

##  Appendix A Analysis of staggered correlation functions

As demonstrated in Ref. [154], averaging over adjacent time slices can
dramatically suppress contributions from oscillating states. Consider a two-
point correlation function $C_{2}(t)$ Let $E$ denote the ground state energy.
Then the averaged two-point function is $\overline{C}_{2}(t)$:

$\displaystyle\overline{C}_{2}(t)$
$\displaystyle=\frac{e^{-Et}}{4}\left[\frac{C_{2}(t)}{e^{-Et}}+\frac{2C_{2}(t+1)}{e^{-E(t+1)}}+\frac{C_{2}(t+2)}{e^{-E(t+2)}}\right],$
(83)
$\displaystyle=\frac{\left|\left<\emptyset\vphantom{\mathcal{O}E}\right|\mathcal{O}\left|E\vphantom{\emptyset\mathcal{O}}\right>\right|^{2}}{2E}e^{-Et}+\mathcal{O}({\Delta
E^{2}}),$ (84)

where $\mathcal{O}$ is an interpolating operator as given in Table 3 and
$|\emptyset\rangle$ is the QCD vacuum. Similarly, consider a three-point
correlation function $C_{3}(t,T)$ with ground states $E_{L}$ and $E_{H}$ at
the source and sink, respectively, and connected by the current $J$. The
averaged three-point function is $\overline{C}_{3}(t,T)$:

$\displaystyle\overline{C}_{3}(t,T)=$
$\displaystyle\frac{e^{-E_{L}t}e^{-E_{H}(T-t)}}{8}\times$
$\displaystyle\left[\frac{C_{3}(t,T)}{e^{-E_{L}t}e^{-E_{H}(T-t)}}+\frac{2C_{3}(t+1,T)}{e^{-E_{L}(t+1)}e^{-E_{H}(T-t-1)}}+\frac{C_{3}(t+2,T)}{e^{-E_{L}(t+2)}e^{-E_{H}(T-t-2)}}\right.$
(85)
$\displaystyle+\left.\frac{C_{3}(t,T+1)}{e^{-E_{L}t}e^{-E_{H}(T+1-t)}}+\frac{2C_{3}(t+1,T+1)}{e^{-E_{L}(t+1)}e^{-E_{H}(T-t)}}+\frac{C_{3}(t+2,T+1)}{e^{-E_{L}(t+2)}e^{-E_{H}(T-t-1)}}\right]$
$\displaystyle=$
$\displaystyle\frac{\left<\emptyset\vphantom{\mathcal{O}_{L}E_{L}}\right|\mathcal{O}_{L}\left|E_{L}\vphantom{\emptyset\mathcal{O}_{L}}\right>\left<E_{L}\vphantom{JE_{H}}\right|J\left|E_{H}\vphantom{E_{L}J}\right>\left<E_{H}\vphantom{\mathcal{O}_{H}\emptyset}\right|\mathcal{O}_{H}\left|\emptyset\vphantom{E_{H}\mathcal{O}_{H}}\right>}{4E_{L}E_{H}}e^{-E_{L}t}e^{-E_{H}(T-t)}+\mathcal{O}(\Delta
E_{H}^{2},\Delta E_{L}^{2}).$ (86)

These averaged two- and three-point functions are used in Eqs. 22, 23 and 24.

##  Appendix B Discretization errors for HISQ

The discussion in this appendix follows closely that of Ref. [32]. Let
$am_{0}$ and $am_{1}$ denote a quark’s bare and rest masses, respectively. The
two quantities are related by the transcendental equation

$am_{0}=a\widetilde{\mathcal{S}h}(am_{1})=\sinh(am_{1})\left(1-\frac{1}{6}\mathbb{N}(am_{1})\sinh^{2}(am_{1})\right).$
(87)

In this expression, $\mathbb{N}(am_{1})$ denotes the coefficient of the Naik
improvement term appearing in the HISQ action

$\displaystyle\mathbb{N}(am_{1})$
$\displaystyle=\frac{4-2\sqrt{1+3X(am_{1})}}{\sinh^{2}(am_{1})},$ (88)
$\displaystyle X(am_{1})$ $\displaystyle=\frac{2am_{1}}{\sinh(2am_{1})}.$ (89)

When bare masses are not small, $am_{0}\centernot\ll 1$, quark bilinears can
lose their conventional normalization. This phenomenon has been discussed in
the literature for both Wilson [155, 156] and staggered fermions. Consider a
quark bilinear containing a heavy quark $h$ and a generic (heavy or light)
quark $x$. Arguments from leading-order HQET [32] show that the conventional
normalization can be restored, at leading order, by multiplying matrix
elements containing the bilinear by the factor $Z^{\rm HQET,\,LO}_{hx}$

$\displaystyle\widetilde{\mathcal{C}h}(am_{1})$
$\displaystyle=\cosh(am_{1})\left(1-\frac{1}{2}\mathbb{N}(am_{1})\sinh^{2}(am_{1})\right),$
(90) $\displaystyle Z^{\rm HQET,\,LO}_{hx}$
$\displaystyle=\begin{cases}\sqrt{\widetilde{\mathcal{C}h}(am_{1,h})\,\widetilde{\mathcal{C}h}(am_{1,x})},\quad
x\text{ nonrelativistic}\\\
\sqrt{\widetilde{\mathcal{C}h}(am_{1,h})},\phantom{\,\widetilde{\mathcal{C}h}(am_{1,x})}\quad
x\text{ ultrarelativistic}\end{cases}.$ (91)

Residual discretization effects from next-to-leading HQET appear at order
$x_{h}^{4}$ and $\alpha_{s}x_{h}^{2}$, where $x_{h}$ is the parameter linear
in the heavy quark mass defined in Eq. 45.

##  Appendix C Shrinkage of covariance and correlation matrices

Analysis of highly correlated Monte Carlo data encountered in lattice gauge
theory presents a formidable statistical challenge. Many problems are phrased
in terms of least-squares minimization of a suitable $\chi^{2}$ function.
Examples in the present work include the correlator analysis in Sec. IV to
extract energies and matrix elements and the chiral-continuum fits of Sec. V.
The essential difficulty is that covariance matrix appearing in the $\chi^{2}$
functions can be nearly singular, and the small eigenvalues in the sample
covariance matrix are poorly determined. Shrinkage estimators, which we review
here, are a class of tools for improving the sample covariance matrix by
“regulating” the small eigenvalues. For motivation, we follow closely the
discussion and notation of Ref. [157], beginning with a technical result.

######  Lemma C.1 (Ledoit and Wolf).

Let $M$ be a real, symmetric matrix. The eigenvalues are the most dispersed
diagonal elements obtainable by rotation.

###### Proof.

Consider a real, symmetric $p\times p$ matrix $M$. Let $R\in\text{SO}(p)$ be a
rotation, under which $M$ transforms into $R^{T}MR$. The average of the
eigenvalue spectrum
$\left\langle\lambda\right\rangle\equiv(1/p)\operatorname{Tr}[M]$ is clearly
invariant under rotations. Let $\bm{v}_{i}$ denote the $i^{\text{th}}$ column
of the rotation $R$. The $i^{\text{th}}$ diagonal element of $R^{T}MR$ is
$\bm{v}_{i}^{T}M\bm{v}_{i}$, and the dispersion of the diagonal elements
around the average of the spectrum is defined via

$\frac{1}{p}\sum_{i}^{p}\left(\bm{v}_{i}^{T}M\bm{v}_{i}-\left\langle\lambda\right\rangle\right)^{2}.$
(92)

This expression is not invariant under rotations, but a closely related
quantity is:

$\operatorname{Tr}[(R^{T}MR-\left\langle\lambda\right\rangle
I)^{2}]=\frac{1}{p}\sum_{i}^{p}\left(\bm{v}_{i}^{T}M\bm{v}_{i}-\left\langle\lambda\right\rangle\right)^{2}+\sum_{i=1}^{p}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{p}\left(\bm{v}_{i}^{T}M\bm{v}_{j}\right)^{2}.$ (93)

The second term on the right-hand side is non-negative and vanishes precisely
when the rotation $R$ diagonalizes $M$. In other words, since the left-hand
side is constant, the dispersion is maximized when the eigenvalues of $M$
appear on the diagonals of $R^{T}MR$, which was to be shown. ∎

This result has important consequence for the near-singular covariance
matrices encountered in practical problems. Let
$\operatorname{diag}({\bm{\lambda}})=U^{T}\Sigma U$ denote the spectral
decomposition of the “true” population covariance matrix of a statistical
distribution, where $U$ contains the eigenvectors and $\bm{\lambda}$ are the
eigenvalues. The corresponding sample covariance matrix has decomposition
$\operatorname{diag}({\bm{\lambda}_{n}})=U_{n}^{T}S_{n}U_{n}$. As usual,
$S_{n}$ is an unbiased estimator of $\Sigma$. Therefore, $U^{T}S_{n}U$ is also
an unbiased estimator of $\operatorname{diag}{\bm{\lambda}}$. Unfortunately,
one does not typically have access to the population eigenvectors of $U$ and
is instead obliged to work with the sample estimates of $U_{n}$. As the
preceding lemma makes clear, the sample eigenvalues $\bm{\lambda}_{n}$ will be
more widely dispersed than those of the population $\bm{\lambda}$. Indeed,
$\bm{\lambda}_{n}$ is _not_ an unbiased estimator of $U^{T}\Sigma U$ due to
correlations between the eigenvectors in $U_{n}$ and the eigenvalues in
$\bm{\lambda}_{n}$. The general idea behind shrinkage estimators is to apply
some function which decreases the dispersion of the sample eigenvalues
$\bm{\lambda}_{n}$ to better approximate the population $\bm{\lambda}$.

The remainder of this appendix is organized as follows. Section C.1 describes
linear shrinkage, which is used in the chiral-continuum analysis (cf. Sec.
V.2). Section C.2 describes nonlinear shrinkage, which is used in the
correlator fits (cf. Sec. IV).

###  C.1 Linear shrinkage

Linear shrinkage was introduced by Ledoit and Wolf in Ref. [157]. There seems
to be some knowledge of this technique in the recent lattice-gauge-theory
literature [158]. Because lattice data often vary over many orders of
magnitude, it is common to invert the correlation matrix instead of the
covariance matrix, with shrinkage techniques being applied to them instead.

The linear shrinkage estimator $\hat{C}_{n}$ is defined as the convex sum of
two matrices:

$\hat{C}_{n}=(1-\lambda)C_{n}+\lambda C_{\rm target},$ (94)

with $\lambda\in[0,1]$. As the parameter $\lambda$ is varied, the shrinkage
estimator smoothly interpolates between the sample correlation matrix $C_{n}$
and the target matrix $C_{\rm target}$. Many options are possible for $C_{\rm
target}$. Examples in the literature [157, 158] advocate using the identity
matrix as the shrinkage target. The idea is that suppressing the correlations
by a small amount (say, $\lambda=0.05$ or $0.1$) will correct the small
eigenvalues while preserving the rest of the correlated structure to the
data.141414 As discussed in the main text, the preferred value of
$\lambda=0.1$ was chosen to regulate the small eigenvalues (thus giving good
fits) with unnecessarily discarding correlations, which can also cause fit
quality to degrade. Ultimately, our results are insensitive to the precise
choice of $\lambda$, as shown in Figs. 21, 47 and 48.  Besides using the
identity matrix, our analysis also experimented with block-diagonal matrices
(e.g., to retain the full correlations between different momenta at fixed
valence mass). The more complicated choices did not improve fit results
compared with the simpler choice of the identity matrix. The preferred chiral-
continuum analysis of Sec. V therefore uses only the identity matrix. Once a
shrinkage estimator for the correlation matrix has been chosen, the
corresponding covariance matrix follows in the usual way,

$\hat{S}_{n}=\operatorname{diag}(\bm{\sigma})\hat{C}_{n}\operatorname{diag}(\bm{\sigma}),$
(95)

where $\bm{\sigma}$ is a vector containing the standard deviations. The
shrinkage estimator, which enjoys a smaller condition number and approximates
the population covariance matrix better than the sample estimate, is then
inverted to give $\hat{S}_{n}^{-1}$, which is used in our fits.

###  C.2 Nonlinear shrinkage

Nonlinear shrinkage has been described by Ledoit and Wolf [79], whose notation
and presentation we follow closely. A complete theoretical justification
exceeds the scope of the work; the interested reader is invited to consult the
original paper for proofs, additional references, and numerical evidence
supporting the applicability in realistic finite data. To keep the present
work self-contained, we restrict ourselves to reproducing the required
formulae with some discussion.

Suppose the sample covariance matrix $S_{n}$ is computed from $n$ observations
of $p$ total random variables. Consider the diagonalization of this matrix,
$S_{n}=U_{n}^{T}\operatorname{diag}(\bm{\lambda}_{n})U_{n}$. Individual
eigenvalues are denoted $\lambda_{n,i}$, $i\in\\{1,\dots,p\\}$ and, without
loss of generality, may be supposed to be sorted in ascending order. For large
$p$ and $n$, suppose the eigenvalues follow some asymptotic cumulative
distribution function $F(x)$ with associated spectral density
$f(x)=F^{\prime}(x)$. Nonlinear shrinkage is a method for adjusting the
empirical spectral density locally to improve the spread in eigenvalues for
finite $n$.

Nonlinear shrinkage is based on the Hilbert transform, which maps continuous
real functions $g(x)$ to $\mathcal{H}_{g}(x)$ via

$\mathcal{H}_{g}(x)\equiv\frac{1}{\pi}\operatorname{PV}\int_{-\infty}^{\infty}dx^{\prime}\,\frac{g(x^{\prime})}{x^{\prime}-x}\,,$
(96)

where $\operatorname{PV}$ denotes the Cauchy principal value. Conceptually,
and as described at length in Ref. [79], the Hilbert transform acts like a
local attractor, pulling eigenvalues towards regions of greater density.
Define the oracle function

$\displaystyle d(x)$ $\displaystyle\equiv\frac{x}{[\pi cxf(x)]^{2}+[1-c-\pi
cx\mathcal{H}_{f}(x)]}$
$\displaystyle=\frac{x}{1+c^{2}[\varphi(x)^{2}+\mathcal{H}_{\varphi}(x)^{2}]-2c\mathcal{H}_{\varphi}(x)}\,,$
(97)

where $c\equiv p/n$ is the concentration ratio, $\varphi(x)=\pi xf(x)$ and
$\mathcal{H}_{\varphi}(x)=1+\pi x\mathcal{H}_{f}(x)$ is its Hilbert transform.
Given a set of sample eigenvalues $\bm{\lambda}_{n}$, $d(\bm{\lambda}_{n})$
provides a shrinkage estimator. To see this, first observe that as the number
of samples becomes large ($c\to 0$), no shrinkage occurs ($d(x)\to x$), in
agreement with intuition. For small but finite concentration, the linear term
in the denominator will dominate:

$d(x)\approx x\left[1+2c\mathcal{H}_{\varphi}(x)+\mathcal{O}(c^{2})\right].$
(98)

Since the Hilbert transform attracts eigenvalues, anomalously large or small
eigenvalues will be pulled locally toward regions of higher density, shrinking
the spectrum. The same qualitative behavior is also present for generic $c$,
as described in Ref. [79]. For some given finite data set, the underlying
distributions $F(x)$ and $f(x)$ are typically unknown. Moreover, since neither
the empirical density nor the empirical CDF are continuous (the former is a
sum of $\delta$ functions, one at each eigenvalue), the necessary Hilbert
transform does not exist. Instead, one works with a kernel estimator for
$f(x)$, for which the necessary derivatives do exist:

$\displaystyle\tilde{f}_{n}(x)$
$\displaystyle=\frac{1}{p}\sum_{i=1}^{p}\frac{1}{h_{n,i}}k\left(\frac{x-\lambda_{n,i}}{h_{n,i}}\right),$
(99) $\displaystyle\mathcal{H}_{\tilde{f}_{n}}(x)$
$\displaystyle=\frac{1}{p}\sum_{i=1}^{p}\frac{1}{h_{n,i}}\mathcal{H}_{k}\left(\frac{x-\lambda_{n,i}}{h_{n,i}}\right),$
(100)

where $h_{n,i}\equiv\lambda_{n,i}h_{n}$ for some suitable choice of bandwidth
$h_{n}$. In principle, many possibilities exist for the choice of the kernel
function $k$. In practice, it is advantageous to take a kernel with finite
support and an analytically calculable Hilbert transform. Reference [79]
advocates choosing the Wigner semicircle distribution,

$\displaystyle k(x)$ $\displaystyle=\frac{\sqrt{[4-x^{2}]^{+}}}{2\pi},$ (101)
$\displaystyle\mathcal{H}_{k}(x)$
$\displaystyle=\frac{\operatorname{sgn}(x)\sqrt{[4-x^{2}]^{+}}-x}{2\pi},$
(102)

where $[x]^{+}\equiv\max\\{0,x\\}$ for any $x\in\mathbb{R}$. With this choice,
the kernel estimators $\tilde{f}_{n}$ and $\mathcal{H}_{\tilde{f}_{n}}$ take
the following form when evaluated at the eigenvalues:

$\displaystyle\tilde{f}_{n}(\lambda_{n,i})$
$\displaystyle=\frac{1}{p}\sum_{i=1}^{p}\frac{\sqrt{4\lambda_{n,j}^{2}h_{n}^{2}-(\lambda_{n,i}-\lambda_{n,j})^{2}}}{2\pi\lambda_{n,j}^{2}h_{n}^{2}},$
(103) $\displaystyle\mathcal{H}_{\tilde{f}_{n}}(\lambda_{n,i})$
$\displaystyle=\frac{1}{p}\sum_{i=1}^{p}\frac{\operatorname{sgn}(\lambda_{n,i}-\lambda_{n,j})\sqrt{[(\lambda_{n,i}-\lambda_{n,j})^{2}-4\lambda_{n,j}^{2}h_{n}^{2}]^{+}}-\lambda_{n,i}+\lambda_{n,j}}{2\pi\lambda_{n,j}^{2}h_{n}^{2}}.$
(104)

Likewise, the sample estimator for the oracle function becomes

$\tilde{d}_{n,i}=\frac{\lambda_{n,i}}{\left[\pi
c\lambda_{n,i}\tilde{f}_{n}(\lambda_{n,i})\right]^{2}+\left[1-c+\pi
c\lambda_{n,i}\mathcal{H}_{\tilde{f}_{n}}(\lambda_{n,i})\right]^{2}}.$ (105)

Some freedom exists in the choice of bandwidth. For reasons of statistical
convergence, i.e., so that $\tilde{f}_{n}(x)\to f(x)$ and
$\mathcal{H}_{\tilde{f}_{n}}(x)\to\mathcal{H}_{f}(x)$ uniformly in
probability, Ref. [79] argues that the bandwidth should vanish for large $n$
($\lim_{n\to\infty}h_{n}=0$) but not decrease too quickly
($\lim_{n\to\infty}nh_{n}^{5/2}=0$). We follow their recommendation of
choosing $h_{n}\equiv n^{-0.35}$.

After shrinkage is applied, the new “eigenvalues” $\tilde{d}_{n,i}$ computed
from $\lambda_{n,i}$ are not guaranteed to maintain their ascending order. For
this reason, the penultimate step is to restore ascending order by applying
the pool adjacent violators (PAV) algorithm [159, 79]. Finally, the shrinkage
estimator for the sample covariance matrix is given by

$\displaystyle\hat{\bm{d}}_{n}$
$\displaystyle\equiv\operatorname{PAV}(\tilde{\bm{d}}_{n}),$ (106)
$\displaystyle\hat{S}_{n}$ $\displaystyle\equiv
U_{n}\operatorname{diag}(\hat{\bm{d}}_{n})U_{n}^{T}.$ (107)

As above, the shrinkage estimator is then inverted, and $\hat{S}_{n}^{-1}$ is
used in our fits.

The PAV algorithm is as follows. Given an input set of data $\bm{d}$, the
algorithm iteratively updates the values, locally pooling adjacent values
which violate $d_{i}\geq d_{i+1}$, and replacing them with their average. The
process is repeated until the monotonicity condition is satisfied everywhere,
yielding $\operatorname{PAV}(\bm{d})$.

###  C.3 Numerical examples of shrinkage

Figure 36: Comparison of eigenvalue spectra resulting before and after
shrinkage or an SVD cut, for the correlation matrices for $C_{D_{s}}^{P}(t)$
(left) and $C_{K}^{P}(t,\bm{0})$ (right) on the physical-mass $a\approx
0.06\leavevmode\nobreak\ \text{fm}$ ensemble. Linear shrinkage was applied
with $\lambda=0.1$. An SVD cut of $10^{-3}$ was chosen to have an effect on
the spectrum similar to shrinkage.

In this section, we present representative examples of correlation matrices
appearing in our analysis. For concreteness, we consider the correlation
matrices for the two-point functions $C_{D_{s}}^{P}(t)$ and
$C_{K}^{P}(t,\bm{0})$ (cf. Eqs. 13 and 15) associated with the $D_{s}$ and $K$
mesons on the physical-mass $a\approx 0.06\leavevmode\nobreak\ \text{fm}$
ensemble. The eigenvalue spectra associated with the correlation matrices are
shown in Fig. 36, for raw data, nonlinear shrinkage, linear shrinkage with
$\lambda=0.1$, and an SVD cut of $10^{-3}$ The raw spectra, shown in blue,
span a range of roughly eight orders of magnitude. (In fact, not displayed are
the last few eigenvalues, which are consistent with zero at double precision).
For the given parameter choices, linear shrinkage and the SVD cut give similar
results. With nonlinear shrinkage, the shape of the small-eigenvalue region of
the spectrum retains some of its original curvature. In the case of the kaon
(left in Fig. 36), the small eigenvalues from nonlinear shrinkage vary by
approximately an order of magnitude over the region where they are roughly
constant for linear shrinkage and SVD cut.

These methods all alter the covariance between pairs of data. Figures 37 and
38 show heat maps for the corresponding correlation matrices. As with the
eigenvalue spectra in Fig. 36, the results for linear shrinkage and SVD cut
are qualitatively similar. Compared with the other methods, nonlinear
shrinkage tends to smooth the far off-diagonal correlation coefficients. All
three correction methods suppress the near-diagonal correlations which are
nearly unity in the raw data.

Figure 37: Comparison of correlation matrices resulting from different
correction techniques applied to the zero-momentum $K$ two-point function
$C_{K}^{P}(t,\bm{0})$ on the physical-mass $a\approx 0.06\leavevmode\nobreak\
\text{fm}$ ensemble. The associated eigenvalue spectra are shown in Fig. 36.
Figure 38: Comparison of correlation matrices resulting from different
correction techniques applied to the $D_{s}$ two-point function
$C_{D_{s}}^{P}(t)$ on the physical-mass $a\approx 0.06\leavevmode\nobreak\
\text{fm}$ ensemble. The associated eigenvalue spectra are shown in Fig. 36.

##  Appendix D Fits: additional details and figures

This appendix gives additional details concerning the correlator, chiral-
continuum, and $z$ expansion fits described in Secs. IV, V.2 and V.4, and
compiles figures that illustrate the robustness of our chiral-continuum
analysis for $D\to K$ and $D_{s}\to K$ decays. Analogous figures are included
in the main text for $D\to\pi$.

###  D.1 Correlator fits

As introduced in Sec. IV, the correlator fits must satisfy checks related to
the ratios Eqs. 24, 22 and 23. Figure 10 shows tests based on the ratio
$R_{0}^{D\to\pi}$ for the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble with the charm-quark mass approximately tuned to its physical value.
Similar figures are shown for the other decays and form factors in Figs. 39,
40, 41, 42, 43 and 44. The first test concerns the approach of the ratios
$R_{0,\parallel,\perp}(t,T,\bm{p})$ to the asymptotic plateau region. This
behavior is examined in the top row of Fig. 10 (and in Figs. 39, 40 and 41) by
considering the ratios at fixed momentum as the source-sink separation is
increased. As $T$ increases, the data tend to flatten out as the ratio
approaches the asymptotic limit. In the right-hand panes, the data show the
highest point,

$\displaystyle\max_{t}R_{0,\parallel,\perp}(t,T,\bm{p}=2\pi(1,0,0)/N_{s}a),$
(108)

as a convenient proxy for the value of “plateau.” As $T$ is increased, theses
points gradually approach the form factor’s fit posterior value, indicated by
the horizontal band in both the left and right panes. It bears emphasizing
that the value of the form factor itself emerges from a fit to the spectral
decomposition, Eqs. 19, 20 and 21, and therefore explicitly includes excited-
state effects.

The third visual test checks the momentum dependence and is shown in the
bottom row of Fig. 10 (and in Figs. 42, 43 and 44). The left panel shows the
ratio $R_{0}^{D\to\pi}$, with each color corresponding to a different
momentum. The horizontal lines with matching colors show the central values of
the posteriors for $f_{0}^{D\to\pi}(\bm{p}^{2})$. For visual clarity, data are
only shown for fixed $T_{\rm max}$, but all available source-sink separations
$T$ were included in the fits. Moving from top to bottom, the form factors
fall monotonically with momentum, and the effects of excited states tend to
decrease. The bottom right panel shows the corresponding posterior values for
$f_{0}^{D\to\pi}(\bm{p}^{2})$, which exhibit smooth dependence on the
momentum.

Figure 39:  Comparing the ratios $R_{0,\parallel,\perp}^{D\to\pi}$, Eqs. 24,
22 and 23, with the form factor’s fit posterior result at fixed momentum
$\bm{p}=(1,0,0)$ on the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble. Left: The data are the ratios
$R_{0,\parallel,\perp}^{D\to\pi}(t,T,\hat{p}^{2}=1)$, with each color
corresponding to a different source-sink separation $T$. Right: The approach
to the asymptotic plateau. Each point corresponds to the maximum point in the
curves on the left,
$\max_{t}R_{0,\parallel,\perp}^{D\to\pi}(t,T,\hat{p}^{2}=1)$. As the source-
sink separation is increased, the data gradually approaches the form factor’s
posterior value given by the band.

Figure 40:  Comparing the ratios $R_{0,\parallel,\perp}^{D\to K}$, Eqs. 24, 22
and 23, with the form factor’s fit posterior result at fixed momentum
$\bm{p}=(1,0,0)$ on the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble. See the caption of Fig. 39 for a detailed explanation.

Figure 41:  Comparing the ratios $R_{0,\parallel,\perp}^{D_{s}\to K}$, Eqs.
24, 22 and 23, with the form factor’s fit posterior result at fixed momentum
$\bm{p}=(1,0,0)$ on the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble. See the caption of Fig. 39 for a detailed explanation.

Figure 42: Comparing the ratios $R_{0,\parallel,\perp}^{D\to\pi}$, Eqs. 24, 22
and 23, with fit results for the form factors coming from the spectral
decomposition on the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble. Left: The data are the ratios
$R_{0,\parallel,\perp}^{D\to\pi}(t,T_{\rm max},\bm{p})$, with each color
corresponding to a different momentum. In each case, only the largest source-
sink separation $T_{\rm max}$ is displayed. Horizontal lines denote the
central values form factor’s fit posterior values, coming from fits including
all source-sink separations $T$. Right: The momentum dependence of the form
factor’s fit posterior values.

Figure 43: Comparing the ratios $R_{0,\parallel,\perp}^{D\to K}$, Eqs. 24, 22
and 23, with fit results for the form factors coming from the spectral
decomposition on the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble. See the caption of Fig. 42 for a detailed explanation.

Figure 44: Comparing the ratios $R_{0,\parallel,\perp}^{D_{s}\to K}$, Eqs. 24,
22 and 23, with fit results for the form factors coming from the spectral
decomposition on the physical-mass $0.12\leavevmode\nobreak\ \text{fm}$
ensemble. See the caption of Fig. 42 for a detailed explanation.

###  D.2 Chiral-continuum fits: Results for $D\to K$ and $D_{s}\to K$

Figure 45:  The result of the chiral-continuum fit for the $D\to K$ form
factors constructed using Eqs. 7, 6 and 8 in units of the gradient-flow scale
$w_{0}$. For visual clarity, only the physical-mass ensembles with heavy
valence masses $m_{h}/m_{c}\in\\{0.9,1.0,1.1\\}$ are shown, although all
ensembles in Table 1 were included in the fit.  Figure 46:  The result of the
chiral-continuum fit for the $D_{s}\to K$ form factors constructed using Eqs.
7, 6 and 8 in units of the gradient-flow scale $w_{0}$. For visual clarity,
only the physical-mass ensembles with heavy valence masses
$m_{h}/m_{c}\in\\{0.9,1.0,1.1\\}$ are shown, although all ensembles in Table 1
were included in the fit.

###  D.3 Chiral-continuum fits: Stability plots for $D\to K$ and $D_{s}\to K$

Figure 47:  Stability of the $D\to K$ form factors $f_{\perp/\parallel/0}$ at
$q^{2}=0$ under variations to the EFT model, the model for discretization
effects, to the choice of data included in the fit, and other analysis choices
as described in the main body. The central values have been normalized by the
central value of preferred fit in green. All variations are statistically
consistent with the preferred fit, highlighted by the green band in each
panel. The statistical significance of the fits is indicated by the marker
size, with larger points denoting better fits.  Figure 48:  Stability of the
$D_{s}\to K$ form factors $f_{\perp/\parallel/0}$ at $q^{2}=0$ under
variations to the EFT model, the model for discretization effects, to the
choice of data included in the fit, and other analysis choices as described in
the main body. The central values have been normalized by the central value of
preferred fit in green. All variations are statistically consistent with the
preferred fit, highlighted by the green band in each panel. The statistical
significance of the fits is indicated by the marker size, with larger points
denoting better fits.

###  D.4 Chiral continuum fits: Error breakdowns for $D\to K$ and $D_{s}\to K$

Figure 49:  Final error budget for the form factors $f^{D\to K}_{+}$ and
$f^{D\to K}_{0}$ after the fit to the $z$ expansion. Contributions less than
$0.01\%$ are not shown.  Figure 50:  Final error budget for the form factors
$f^{D_{s}\to K}_{+}$ and $f^{D_{s}\to K}_{0}$ after the fit to the $z$
expansion. Contributions less than $0.01\%$ are not shown.

###  D.5 $z$-expansion fits: Joint fits to lattice-QCD form factors and
experimental data

Tables 21, 22 and 23 compare the results of the $z$-expansion fits for the
decays $D\to\pi$, $D\to K$, and $D_{s}\to K$. The fits enforce the kinematic
identity $f_{+}(0)=f_{0}(0)$ by imposing $a_{0}=b_{0}$ [cf. Eqs. 54 and 55].
For the scalar form factor, the higher parameters $b_{1}$, $b_{2}$, and
$b_{3}$ are unconstrained by the fits including experimental data. In the
joint fit, the lattice QCD form factors include a systematic from SIB, as
described in Sec. VII.2. No uncertainty from QED is included in the fit, since
this is applied directly to $|V_{cx}|$ as a final $1\%$ uncertainty.

Table 21:  Comparison of $z$-expansion fit results for the decay $D\to\pi$.  $D\to\pi$ | LQCD only | Joint LQCD and Expt | Expt only
---|---|---|---
$a_{0}\equiv b_{0}$ | $0.6300(51)$ | $0.6306(47)$ | $0.1426(17)$
$a_{1}$ | $-0.610(99)$ | $-0.574(83)$ | $-0.157(45)$
$a_{2}$ | $-0.20(30)$ | $-0.009(393)$ | $-0.15(32)$
$a_{3}$ | $0.30(19)$ | $0.32(94)$ | $0.12(94)$
$b_{1}$ | $0.330(51)$ | $0.379(52)$ | $-$
$b_{2}$ | $-0.31(25)$ | $0.22(36)$ | $-$
$b_{3}$ | $-1.90(39)$ | $-0.54(84)$ | $-$
Table 22:  Comparison of $z$-expansion fit results for the decay $D\to K$.  $D\to K$ | LQCD only | Joint LQCD and Expt | Expt only
---|---|---|---
$a_{0}\equiv b_{0}$ | $0.7452(31)$ | $0.7450(31)$ | $0.7246(26)$
$a_{1}$ | $-0.948(97)$ | $-1.036(73)$ | $-1.049(89)$
$a_{2}$ | $0.14(40)$ | $0.18(73)$ | $0.10(92)$
$a_{3}$ | $0.07(12)$ | $-0.03(1.00)$ | $-0.03(1.00)$
$b_{1}$ | $0.776(62)$ | $0.772(66)$ | $-$
$b_{2}$ | $0.14(34)$ | $0.08(56)$ | $-$
$b_{3}$ | $0.03(13)$ | $-0.02(99)$ | $-$
Table 23:  Comparison of $z$-expansion fit results for the decay $D_{s}\to K$.  $D_{s}\to K$ | LQCD only | Joint LQCD and Expt | Expt only
---|---|---|---
$a_{0}\equiv b_{0}$ | $0.6307(20)$ | $0.6306(20)$ | $0.164(18)$
$a_{1}$ | $-0.562(65)$ | $-0.557(72)$ | $-0.14(29)$
$a_{2}$ | $-0.19(20)$ | $-0.20(42)$ | $-0.03(98)$
$a_{3}$ | $0.33(29)$ | $0.04(98)$ | $0.008(1.000)$
$b_{1}$ | $0.347(27)$ | $0.346(35)$ | $-$
$b_{2}$ | $0.44(18)$ | $0.45(30)$ | $-$
$b_{3}$ | $-0.21(43)$ | $-0.11(96)$ | $-$

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