Stefan Meinel

# Quark flavor physics with lattice QCD

###### Abstract

This is an overview of quark flavor physics as presented in a plenary talk at
Lattice 2023. In the first part, I discuss the main processes and lattice-QCD
inputs used to determine the Wolfenstein parameters of the Cabibbo-Kobayashi-
Maskawa matrix. In the second part, I review selected further processes that
are being used to search for physics beyond the Standard Model. New results
presented at Lattice 2023 are referenced throughout, but detailed discussions
are limited to selected work published prior to the conference. QED
corrections, inclusive decays on the lattice, and processes involving hadronic
resonances are not discussed in detail here, as they were covered in other
plenary talks at Lattice 2023 and Lattice 2022.

##  1 Introduction

The Standard Model of particle physics has the fascinating property that each
of the five different fermion representations of the gauge group $SU(3)\times
SU(2)\times U(1)$ comes in three copies, the generations [1, 2, 3, 4, 5]:

$\displaystyle\hskip 17.22217pt1\hskip 30.1388pt2\hskip 30.1388pt3$
$\displaystyle Q^{\prime}_{L}$ $\displaystyle=$
$\displaystyle\left(\left(\begin{array}[]{c}u^{\prime}_{L}\\\
d^{\prime}_{L}\end{array}\right),\left(\begin{array}[]{c}c^{\prime}_{L}\\\
s^{\prime}_{L}\end{array}\right),\left(\begin{array}[]{c}t^{\prime}_{L}\\\
b^{\prime}_{L}\end{array}\right)\right),$ (7) $\displaystyle U^{\prime}_{R}$
$\displaystyle=$ $\displaystyle\left(\hskip 10.76385ptu^{\prime}_{R},\hskip
19.37494ptc^{\prime}_{R},\hskip 19.37494ptt^{\prime}_{R}\hskip
8.61108pt\right),$ $\displaystyle D^{\prime}_{R}$ $\displaystyle=$
$\displaystyle\left(\hskip 10.76385ptd^{\prime}_{R},\hskip
19.37494pts^{\prime}_{R},\hskip 19.37494ptb^{\prime}_{R}\hskip
6.45831pt\right),$ $\displaystyle L^{\prime}_{L}$ $\displaystyle=$
$\displaystyle\left(\\!\left(\begin{array}[]{c}{\nu_{e}}^{\prime}_{L}\\\
e^{\prime}_{L}\end{array}\\!\right),\left(\\!\begin{array}[]{c}{\nu_{\mu}}^{\prime}_{L}\\\
\mu^{\prime}_{L}\end{array}\\!\right),\left(\\!\begin{array}[]{c}{\nu_{\tau}}^{\prime}_{L}\\\
\tau^{\prime}_{L}\end{array}\\!\right)\right),$ (14) $\displaystyle
E^{\prime}_{R}$ $\displaystyle=$ $\displaystyle\left(\hskip
10.76385pte^{\prime}_{R},\hskip 19.37494pt\mu^{\prime}_{R},\hskip
19.37494pt\tau^{\prime}_{R}\hskip 8.61108pt\right).$ (15)

The species labels $u$, $c$, $t$, $d$, $s$, $b$, $e$, $\mu$, $\tau$ are called
flavors. The ′ indicates the use of the gauge basis. In the absence of flavor-
violating interactions, we would have a $U(3)^{5}$ global flavor symmetry. In
the Standard Model, the only origin of flavor symmetry violation (and CP
violation) is the Yukawa interaction of the fermions with the Higgs field
$\phi$ [6, 4]:

$\mathcal{L}_{\rm
Yukawa}\>=\>-\overline{Q^{\prime}_{Li}}Y^{U}_{ij}U^{\prime}_{Rj}\tilde{\phi}-\overline{Q^{\prime}_{Li}}Y^{D}_{ij}D^{\prime}_{Rj}\phi-\overline{L^{\prime}_{Li}}Y^{E}_{ij}E^{\prime}_{Rj}\phi\>+\>{\rm
h.c.}\,.$ (16)

When $\phi$ acquires its vacuum expectation value
$\langle\phi\rangle=(0,v/\sqrt{2})$, these couplings produce the fermion mass
terms. In the quark sector, the unitary field transformations that diagonalize
the mass matrices,

$\begin{array}[]{ll}U^{\prime}_{L}=V^{U}_{L}U_{L},&U^{\prime}_{R}=V^{U}_{R}U_{R},\\\
D^{\prime}_{L}=V^{D}_{L}D_{L},&D^{\prime}_{R}=V^{D}_{R}D_{R},\\\ \end{array}$
(17)

do not cancel in the charged current coupling to the $W$ field,

$\displaystyle\mathcal{L}_{\rm
c.c.}\>\>=\>\>-\frac{g}{\sqrt{2}}\overline{U^{\prime}_{Li}}\gamma^{\mu}D^{\prime}_{Li}\>W^{+}_{\mu}\>+\>{\rm
h.c.}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{U_{Li}}\underbrace{(V^{U{\dagger}}_{L}V^{D}_{L})_{ij}}_{=\displaystyle
V_{ij}}\gamma^{\mu}D_{Lj}\>W^{+}_{\mu}\>+\>{\rm h.c.},$ (18)

giving rise to the unitary Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix

$V=\left(\\!\begin{array}[]{lll}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\end{array}\\!\right).$ (19)

After eliminating unobservable phase factors, $V$ can be written in terms of
four parameters.

Some of the fundamental questions in flavor physics are:

  * •

What is the origin of the three generations?

  * •

What is the origin of the hierarchies in the fermion masses and mixing
matrices?

  * •

Are there other sources of flavor-violating interactions and CP violation
beyond the Standard Model?

In most of the more fundamental theories that have been proposed to address
the deficiencies of the Standard Model, the answer to the third question is
“yes”. The precision study of flavor-changing processes is therefore a
powerful tool for discovering new physics. Lattice-QCD calculations play an
essential role in this effort, as discussed in the following.

##  2 Main processes used to determine the CKM parameters

An insightful parametrization of the CKM matrix was proposed by Lincoln
Wolfenstein 40 years before this Lattice conference [7]. This parametrization
is based on the observation that $V$ is close to the identity matrix. Among
the off-diagonal elements, $V_{us}$ was already well determined to be close to
$0.22$, and Wolfenstein estimated $V_{cb}\approx 0.06$ from the $B$ meson
lifetime that had just been measured at the PEP storage ring [8]. This
hierarchy suggested an expansion in powers of a parameter $\lambda$, where
$V_{us}$ is of first order and $V_{cb}$ is of second order. Writing
$V_{us}=\lambda$ and $V_{cb}=A\lambda^{2}$ and imposing unitarity gives the
two-parameter approximation

$V\>=\>\left(\\!\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}&\lambda&0\\\
-\lambda&1-\frac{1}{2}\lambda^{2}&A\lambda^{2}\\\
0&-A\lambda^{2}&1\end{array}\\!\right)\>+\>\mathcal{O}(\lambda^{3}).$ (20)

At higher orders, two additional parameters, denoted as $\rho$ and $\eta$, are
needed. Imposing unitarity yields the form

$V\>=\>\left(\\!\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\frac{1}{2}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{array}\\!\right)\>+\>\mathcal{O}(\lambda^{4}).$
(21)

In 1983, only upper limits on $\rho$ and $\eta$ were available [7], and there
has been a lot of progress since then. In the next few subsections, I will
discuss the current status of determinations of the CKM parameters and the
relevant inputs from lattice QCD, following the hierarchy encoded in the
Wolfenstein parametrization.

###  2.1 Determination of ${\boldsymbol{\lambda}}$

The exact definition of this Wolfenstein parameter is
$\lambda=\frac{|V_{us}|}{\sqrt{|V_{ud}|^{2}+|V_{us}|^{2}}}$ [9]. Assuming the
CKM unitarity relation $|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1$ and
neglecting the very small $|V_{ub}|^{2}$, a determination of $|V_{ud}|$ alone
already gives us $|V_{us}|$ as well, and hence $\lambda$. The most precise
direct result for $|V_{ud}|$ comes from the study of superallowed $0^{+}\to
0^{+}$ nuclear $\beta$ decays, which are pure vector transitions and therefore
fairly insensitive to nuclear/nucleon structure, yielding [10]

$|V_{ud}|=0.97373(11)_{\rm exp.}(9)_{\rm RC}(27)_{\rm NS}.$ (22)

This result alone would give $\lambda=0.2277(13)$. But is the unitarity
relation actually satisfied?

Figure 1: A schematic diagram of kaon semileptonic decay in the Standard
Model.

We also have the following experimental results for kaon decays [10]:

$\displaystyle\displaystyle\frac{\Gamma(K^{\pm}\to\mu^{\pm}\nu[\gamma])}{\Gamma(\pi^{\pm}\to\mu^{\pm}\nu[\gamma])}\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!1.3367(28),$ (23)
$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}(K\to\pi\ell\nu[\gamma])\hskip
8.61108pt\underset{\text{non-lattice theory}}{\Rightarrow}\hskip
8.61108ptf_{+}(K\to\pi,q^{2}=0)|V_{us}|\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!0.21635(38)(3).$ (24)

Here, $q$ is the 4-momentum transfer, defined as shown in Fig. 1. To get
$|V_{us}/V_{ud}|$ and $|V_{us}|$ from these results, we need lattice-QCD
calculations of the ratio of decay constants $f_{K^{\pm}}/f_{\pi^{\pm}}$ and
of the form factor $f_{+}(K\to\pi,q^{2}=0)$; these quantities parametrize the
QCD matrix elements

$\displaystyle\langle
0|\>\bar{u}\gamma^{\mu}\gamma_{5}d\>{|\pi^{-}(p)\rangle}$ $\displaystyle=$
$\displaystyle ip^{\mu}{f_{\pi^{-}}},$ (25) $\displaystyle\langle
0|\>\bar{u}\gamma^{\mu}\gamma_{5}s\>{|K^{-}(p)\rangle}$ $\displaystyle=$
$\displaystyle ip^{\mu}{f_{K^{-}}},$ (26)
$\displaystyle{\langle\pi^{+}(p^{\prime})}|\>\bar{u}\gamma^{\mu}s\>{|\bar{K}^{0}(p)\rangle}$
$\displaystyle=$
$\displaystyle\left[(p+p^{\prime})^{\mu}-\frac{m_{K}^{2}-m_{\pi}^{2}}{q^{2}}q^{\mu}\right]{f_{+}(K\to\pi,q^{2})}$
(27)
$\displaystyle+\frac{m_{K}^{2}-m_{\pi}^{2}}{q^{2}}q^{\mu}{f_{0}(K\to\pi,q^{2})}.$

As shown in Fig. 2, lattice calculations of $f_{K^{\pm}}/f_{\pi^{\pm}}$ and
$f_{+}(K\to\pi,q^{2}=0)$ have reached impressive precision, resulting in
Flavour Lattice Averaging Group (FLAG) averages with uncertainties of
approximately 0.2% [11, 12]. Note that the decay rates measured by the
experiments are fully photon-inclusive. These QED corrections can be
calculated in chiral perturbation theory or on the lattice, as done for the
leptonic decays in Ref. [13]. Using the latter, the FLAG determination of
$(|V_{us}|,|V_{ud}|)$ from lattice calculations with $N_{f}=2+1+1$ dynamical
quark flavors, shown as the red ellipse in Fig. 2, is in significant tension
with CKM unitarity and with the result for $|V_{ud}|$ from nuclear beta
decays. It is interesting to note that there are viable new-physics models
that can resolve these tensions; for example, TeV-scale vector-like quarks can
introduce small right-handed couplings that will do the job and can also
explain the $W$-boson-mass anomaly [14].

At this conference, Takeshi Yamazaki presented a new preliminary result for
$f_{+}(K\to\pi,q^{2}=0)$ from calculations on impressively large lattices of
size greater than $(10\text{ fm})^{4}$ for three different lattice spacings
and with physical pion masses [15]. The corresponding value for $|V_{us}|$ is
shown in yellow in Fig. 2 and would bring the $(|V_{us}|,|V_{ud}|)$
combination into better agreement with unitarity and with $|V_{ud}|$ from
nuclear beta decays. Another possible way to determine $|V_{us}|$ and
$|V_{ud}|$ is though inclusive hadronic $\tau$ decays. Antonio Evangelista
presented the first fully nonperturbative lattice calculation of the inclusive
$\tau$ decay rate in the $\bar{u}d$ flavor channel using spectral
reconstruction [16], resulting in a $|V_{ud}|$ value with 0.4% uncertainty
that is also shown in Fig. 2. While this result is not competitive with the
extremely precise determination from nuclear beta decays, the method can also
be applied in the $\bar{u}s$ flavor channel where it will likely be
competitive with the meson decays. Further work related to $|V_{ud}|$ and
$|V_{us}|$ shown at this conference includes a calculation of $f_{\pi}$ and
$f_{K}$ presented by Zack Hall [17] and investigations of QED corrections
presented by Norman Christ [18], Antonin Portelli [19], Nils Hermansson
Truedsson [20], and Jun-sik Yoo [21]. A separate plenary talk, given by Matteo
Di Carlo, was devoted to the topic of QED corrections [22].

Figure 2: Summary of lattice results for the ratio of kaon and pion decay
constants (left), and of both lattice and non-lattice results for the kaon
semileptonic form factor at $q^{2}=0$ (right) by FLAG [11, 12]. Figure 3:
Constraints in the $|V_{us}|$-$|V_{ud}|$ plane from leptonic and semileptonic
meson decays (red, green, and yellow), nuclear $\beta$ decays (blue), and
inclusive hadronic $\tau$ decays [11, 12, 15, 16].

###  2.2 Determination of ${\boldsymbol{A}}$

The exact definition of the Wolfenstein parameter $A$ is given by
$A\lambda^{2}=\frac{\lambda}{|V_{us}|}|V_{cb}|$ [9]. The next task is
therefore to determine $|V_{cb}|$. The most important processes currently used
to determine $|V_{cb}|$ are semileptonic $B_{(s)}$ meson decays:

  * •

Inclusive $B\to X_{c}\ell\nu$ ($\ell=e,\mu$; BaBar, Belle, Belle II, and older
experiments),

  * •

Exclusive $B\to D\ell\nu$ ($\ell=e,\mu$; BaBar, Belle, Belle II, and older
experiments),

  * •

Exclusive $B\to D^{*}\ell\nu$ ($\ell=e,\mu$; BaBar, Belle, Belle II, and older
experiments),

  * •

Exclusive $B_{s}\to D_{s}\mu\nu$ (LCHb),

  * •

Exclusive $B_{s}\to D^{*}_{s}\mu\nu$ (LCHb).

The exclusive determinations use form factors from lattice QCD. The most
precise inclusive determinations use the heavy-quark/operator-product
expansion in powers of $1/m_{b}$ and $\alpha_{s}$, where hadronic matrix
elements of $\Delta B=0$ matrix elements are fitted to experimental data [23];
these calculations use lattice input for $m_{b}$, $m_{c}$, $\alpha_{s}$. There
is also substantial progress with direct lattice calculations of inclusive
rates. This was covered thoroughly in the Lattice 2022 plenary talks by
Takashi Kaneko [24] and John Bulava [25].

| Fermilab/MILC [26] | HPQCD [27] | JLQCD [28]
---|---|---|---
$\int_{a}^{b}$ $u,d,s,(c)$-quark action | AsqTad (2+1) | HISQ (2+1+1) | domain wall (2+1)
$\int_{a}^{b}$ $b$-quark action | Fermilab clover | HISQ | domain wall
$\int_{a}^{b}$ $B$-meson mass | $m_{\rm kin}\approx m_{\rm phys}$ | $m\lesssim 0.93\>m_{\rm phys}$ | $m\lesssim 0.74\>m_{\rm phys}$
$\int_{a}^{b}$ $m_{\pi}$ (MeV) | 180 - 560∗ | 135 - 329∗ | 230 - 500
$\int_{a}^{b}$ $a$ (fm) | 0.045 - 0.15 | 0.044 - 0.090 | 0.044 - 0.080
$\int_{a}^{b}$ $\\#$(source-sink separations) | 2 ($T$, $T\\!+\\!1$) | 3 | 4

∗These are the masses of the lightest pion (taste $\gamma_{5}$)

Table 1: Lattice calculations of the $B\to D^{*}$ form factors at nonzero
recoil. Figure 4: Comparison of lattice predictions [26, 27, 28] and
experimental results [29, 30] for the combination of form factors that appears
in the $B\to D^{*}\ell\nu$ differential decay rate. The black and green curves
are from BGL fits to the experimental data [plot by Alejandro Vaquero]. Figure
5: Comparison of lattice predictions [26, 27, 28] and experimental results
[29, 30] for the $B\to D^{*}\ell\nu$ decay rate differential with respect to
$w=v\cdot v^{\prime}$ (top left plot) and with respect to the decay angles
(other three plots). The black and green curves are from BGL fits to the
experimental data [plots by Alejandro Vaquero]. Figure 6: Comparison of
selected results for $|V_{cb}|$ as of July 2023: Inclusive, M. Bordone et al.
[31]; Inclusive, F. Bernlochner et al. [32] (the first extraction using
$q^{2}$ moments); $B\to D\ell\nu$, FLAG [11] using form factors from
Fermilab/MILC [33] and HPQCD [34] and experimental data from Refs. [35, 36];
$B\to D^{*}\ell\nu$ using form factors from Fermilab/MILC [26] and
experimental data from Refs. [29, 30], $B_{(s)}\to D_{(s)}^{*}\ell\nu$ using
form factors from HPQCD [27] and experimental data from Refs. [30, 37]; $B\to
D^{*}\ell\nu$ using form factors from JLQCD [28] and experimental data from
Ref. [30]; $B_{s}\to D_{s}^{(*)}\mu\nu$, LHCb [37, 38] using form factors from
HPQCD [39, 40].

In 2023, two new lattice calculations of the $B\to D^{*}$ form factors were
published, by the HPQCD [27] and JLQCD [28] collaborations. Like the 2021
Fermilab/MILC calculation [26], these works provide the full kinematic
dependence of the vector and axial-vector form factors (past calculations for
$B\to D^{*}$ were performed only at the zero-recoil point, where
$q^{2}=q^{2}_{\rm max}$ and $w=v\cdot v^{\prime}=1$, meaning that experimental
data had to be extrapolated to that point in order to extract $|V_{cb}|$).
Table 1 shows a comparison of the parameters of the three calculations.
Notably, the HPQCD and JLQCD calculations use the same “fully relativistic”
lattice action for the $b$ quark as used for the light, strange, and charm
quarks, which simplifies the renormalization but requires extrapolations from
unphysically light $b$-quark masses up to the physical value.

Comparisons of the $B\to D^{*}\ell\nu$ decay observables predicted using the
lattice calculations in the Standard Model to experimental measurements [29,
30] are shown in Figs. 4 and 5. There is good agreement between the
predictions using the JLQCD form factors and the measurements. The
Fermilab/MILC and HPQCD calculations predict a somewhat steeper slope with
respect to $w$ for the combination of form factors that appears in the
differential decay rate. In addition, there are deviations between the SM
predictions and measurements of the angular distributions, most significantly
for HPQCD. Improved lattice calculations and new measurements by Belle II
should help clarify the situation in the future.

A summary of results for $|V_{cb}|$ is shown in Fig. 6. The values from $B\to
D^{*}\ell\nu$ using combined fits to the lattice and experimental data
performed by the three different lattice collaborations are consistent with
each other and still significantly below the inclusive values [31, 32].
Resolving this long-standing exclusive-inclusive discrepancy is of critical
importance for flavor physics, since the uncertainty of the Wolfenstein
parameter $A$ is the dominant source of uncertainty in Standard-Model
predictions for many flavor-changing neutral-current processes. As is also
shown in Fig. 6, extracting $|V_{cb}|$ from the total rate measured in
experiment and predicted by HPQCD’s form factors instead of the combined fit
to the differential distribution gives a much higher value of $|V_{cb}|$
(albeit with larger uncertainties); this is due to the discrepancy in the
predicted slope.

At this conference, Andrew Lytle presented preliminary results for the
$B_{s}\to D_{s}$ form factors using the fully relativistic all-HISQ approach
from the Fermilab/MILC collaboration [41], and Benjamin Choi and Seungyeob Jwa
from the LANL-SWME collaboration reported on their progress toward a
calculation of the $B\to D^{*}$ form factors using the Oktay-Kronfeld action
for the heavy quarks on the MILC HISQ ensembles [42, 43]. Aurora Melis
presented a lattice determination of the susceptibilities that enter in
dispersive bounds on $b\to c$ form factors [44]. Alessandro Barone showed a
comparison of the Chebyshev and Backus-Gilbert reconstruction methods applied
to inclusive $B_{s}\to X_{c}\ell\nu$ decays, demonstrating that these methods
yield consistent results [45]. Ryan Kellerman reported on a study of finite-
volume effects in the Chebyshev approach [46].

###  2.3 Determination of ${\boldsymbol{\bar{\rho}}}$ and
${\boldsymbol{\bar{\eta}}}$

The remaining two Wolfenstein parameters are $\rho$ and $\eta$, or, to ensure
exact unitarity, $\bar{\rho}$ and $\bar{\eta}$ [9]:

$V^{*}_{ub}=A\lambda^{3}(\rho+i\eta)=\frac{\sqrt{1-A^{2}\lambda^{4}}}{\sqrt{1-\lambda^{2}}[1-A^{2}\lambda^{4}(\bar{\rho}+i\bar{\eta})]}A\lambda^{3}(\bar{\rho}+i\bar{\eta}).$
(28)

Also note that
$\bar{\rho}+i\bar{\eta}=-\displaystyle\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}$,
and the orthogonality of the first and third columns of the CKM matrix,
$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0$, can be represented as
a triangle in the complex plane with apex $\bar{\rho}+i\bar{\eta}$ as shown in
Fig. 7.

Figure 7: The triangle, in the complex plane, representing the orthogonality
of the first and third columns of the CKM matrix.

####  2.3.1 Determination of ${\boldsymbol{|V_{ub}|}}$

The magnitude $|V_{ub}|=A\lambda^{3}\sqrt{\rho^{2}+\eta^{2}}$ can be
determined from $b$-hadron semileptonic decays. The most important processes
currently used to determine $|V_{ub}|$ are

  * •

Inclusive $B\to X_{u}\ell\nu$ ($\ell=e,\mu$; BaBar, Belle, Belle II, and older
experiments)

  * •

Exclusive $B\to\pi\ell\nu$ ($\ell=e,\mu$; BaBar, Belle, Belle II, and older
experiments)

  * •

Exclusive $B\to\rho\ell\nu$ and $B\to\omega\ell\nu$ ($\ell=e,\mu$; BaBar,
Belle, Belle II, and older experiments, still using light-cone sum rules)

  * •

Exclusive $B_{s}\to K\mu\nu$ (LCHb)

  * •

Exclusive $\Lambda_{b}\to p\mu\bar{\nu}$ (LCHb)

  * •

Exclusive $B\to\tau\nu$ (BaBar, Belle, Belle II, and older experiments)

The inclusive determination of $|V_{ub}|$ is more difficult compared to
$|V_{cb}|$ due to the large $b\to c\ell\bar{\nu}$ background. Cutting away
this contribution with a requirement on the lepton energy leaves only the
endpoint region with $2E_{\ell}/m_{b}\sim 1$, where the local HQE breaks down.
In this region, one needs to use a light-cone OPE, such that the HQE
parameters are replaced by nonlocal matrix elements, the so-called shape
functions [23]. The exclusive determinations using $B\to\pi\ell\nu$, $B_{s}\to
K\mu\nu$, $\Lambda_{b}\to p\mu\bar{\nu}$, $B\to\tau\nu$ use form factors and
the $B$ decay constant from lattice QCD. Calculating the $B\to\rho$ and
$B\to\omega$ form factors in lattice QCD in a rigorous way is more complicated
because the $\rho$ and $\omega$ are resonances that are unstable under the
strong interactions. As a two-body resonance, the $\rho$ is more tractable
than the $\omega$ and work is already underway to calculate the
$B\to\rho(\to\pi\pi)\ell\nu$ form factors in lattice QCD using the Lellouch-
Lüscher method [47, 48], as shown in Luka Leskovec’s plenary talk [49].

In 2023, there have been updates on the $B\to\pi$ and $B_{s}\to K$ form
factors from lattice QCD. For $B\to\pi$, FLAG updated the average of lattice
results by including the 2022 JLQCD results [50] in their fit. The JLQCD
calculation is the first to use the fully relativistic approach (with Möbius
domain-wall fermions in this case) and was already reviewed at Lattice 2022
[24]. For $B_{s}\to K$, the RBC/UKQCD collaboration published an improved
calculation [51, 52] that supersedes their 2015 results [53], and I will
discuss a preliminary update of the FLAG average below. Some of the plots in
the following will show the form factors as a function of the variable $z$,
which is defined as

$z(q^{2})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}\,,$
(29)

where $t_{+}$ and $t_{0}$ are constants. The properties of this change of
variable are illustrated in Fig. 8. Furthermore, some of the plots show
$B(q^{2})f(q^{2})$ instead of $f(q^{2})$, where $B(q^{2})=(1-m_{\rm
pole}^{2}/q^{2})$.

The effect of including the 2022 JLQCD results [50] in the FLAG average of
$B\to\pi$ form factors is shown in Fig. 9 [11, 12]. The updated BCL fit has a
large $\chi^{2}/{\rm d.o.f.}$ caused by slight tensions between the results
from the different collaborations; in particular in the slopes of $f_{0}$,
which are very constrained due to strong correlations between the data points.
The uncertainties of the fit parameters have therefore been rescaled by
$\sqrt{\chi^{2}/{\rm d.o.f.}}$. Overall, the uncertainty of the $z$-expansion
parameters $a_{0}^{0,+}$ has increased, while the uncertainty of $a_{1}^{0}$,
$a_{1,2}^{+}$ has decreased. The result for $|V_{ub}|$ extracted from a
combined fit of the lattice results and the BaBar [54, 55] and Belle [56, 57]
data has changed from $3.74(17)\times 10^{-3}$ to $3.64(16)\times 10^{-3}$
[11, 12].

Figure 8: The mapping of the complex $q^{2}$ plane to the unit disk in the
complex $z$ plane through Eq. (29); see Ref. [58] for explanations. Note that
Refs. [59, 51] denote $t_{+}$ by $t^{*}$ and $t_{\rm th}$ by $t_{+}$.

HPQCD 06 |  2+1 Asqtad, NRQCD $b$ (not included in fit)
---|---
FNAL/MILC 15 |  2+1 Asqtad, Fermilab $b$
RBC/UKQCD 15 |  2+1 DWF, RHQ $b$
JLQCD 22 |  2+1 DWF, DWF $b$

Old

$\chi^{2}/{\rm dof}=0.82$

All uncertainties rescaled by $\sqrt{\chi^{2}/{\rm dof}}$

New, including JLQCD 22

$\chi^{2}/{\rm dof}=3.63$

All uncertainties rescaled by $\sqrt{\chi^{2}/{\rm dof}}$

Figure 9: The effect of including the 2022 JLQCD results [50] in the FLAG
average of $B\to\pi$ form factors [11, 12].

Coming to the $B_{s}\to K$ form factors, the new 2023 calculation by RBC/UKQCD
[51, 52] uses $N_{f}=2+1$ domain-wall fermions, RHQ (a.k.a. Columbia)
anisotropic-clover $b$ quarks, and “mostly nonperturbative” renormalization
like the 2015 calculation [53]. The new calculation includes one additional
ensemble with a third, finer lattice spacing that also has a lower pion mass
than the other ensembles, uses updated scale setting and updated tuning of
$m_{s}$ and of the RHQ parameters, and uses a different form-factor basis for
the chiral-continuum extrapolations. These extrapolations use the fit
functions [51]

$\displaystyle f_{X}^{B_{s}\to K}(M_{\pi},E_{K},a^{2})$
$\displaystyle=\frac{\Lambda}{E_{K}+\Delta_{X}}\left[c_{X,0}\bigg{(}1+\frac{\delta
f(M^{s}_{\pi})-\delta f(M^{p}_{\pi})}{(4\pi
f_{\pi})^{2}}\bigg{)}+c_{X,1}\frac{\Delta
M_{\pi}^{2}}{\Lambda^{2}}+c_{X,2}\frac{E_{K}}{\Lambda}+c_{X,3}\frac{E_{K}^{2}}{\Lambda^{2}}+c_{X,4}(a\Lambda)^{2}\right],$
(30)

where the first factor describes the pole from a $\bar{b}u$ bound state
coupling to the weak current. The quantum numbers of this bound state are
$J^{P}=1^{-}$ for the form factor $f_{+}$ and $J^{P}=0^{+}$ for the form
factor $f_{0}$, so that $\Delta_{+}=-42.1\>{\rm MeV}$ and
$\Delta_{0}=263\>{\rm MeV}$ [51]. In the new RBC/UKQCD calculation, as well as
in the 2014 HPQCD calculation [60], the chiral/continuum extrapolations were
performed in this basis. In contrast, in the 2015 RBC/UKQCD calculation, as
well as in the 2019 Fermilab/MILC calculation [61], the extrapolations were
performed for $f_{\perp}$ and $f_{\parallel}$, each of which are linear
combinations of $f_{+}$ and $f_{0}$, while still using the pole factor as in
Eq. (30) with $\Delta_{\perp}=\Delta_{+}$ and $\Delta_{\parallel}=\Delta_{0}$;
the justification for this choice was that $f_{\perp}$ is dominated by $f_{+}$
and $f_{\parallel}$ by $f_{0}$. The authors of the new RBC/UKQCD paper [51]
compared both prescriptions and found a significant dependence on this basis
choice in the final results for $f_{0}$, as shown here in Fig. 10.

Figure 11 shows the effect of replacing the 2015 RBC/UKQCD results by the 2023
RBC/UKQCD results in the FLAG average of $B_{s}\to K$ form factors. The upward
shift in the RBC/UKQCD results for $f_{0}$ leads to an increased tension
between the lattice results from the different collaborations for this form
factor. The increase in the scale factor $\sqrt{\chi^{2}/{\rm d.o.f.}}$ leads
to an overall increase in the uncertainty of $\frac{\Gamma(B_{s}\to
K\mu\nu)}{|V_{ub}|^{2}}$ calculated from the FLAG fit. The discrepancies in
$f_{0}$ indicate that the uncertainties have been underestimated by at least
some of the collaborations. It is possible that other sources of error besides
the problem with extrapolating $f_{\perp}$ and $f_{\parallel}$ discussed above
(which applies to the Fermilab/MILC results only), such as insufficiently
controlled excited-state contamination, also contribute to the tension.
Excited-state contamination can affect multiple steps of the analysis; it
appears to have caused order-10% biases in the $B_{s}$ kinetic-mass
determinations in the tuning of the RHQ parameters by RBC/UKQCD, as discussed
in Ref. [62].

A summary of $|V_{ub}|$ results is shown in Fig. 12. Note that the result from
$B_{s}\to K\mu\nu$, RBC/UKQCD 2023 is based on secondary extrapolations of the
form factors to the full $q^{2}$ range using a novel approach with unitarity
bounds [59, 63], taking into account that the dispersive integral ranges only
over an arc of the unit circle (in the $z$ plane) instead of the full circle
for this process [58]. However, these secondary extrapolations were not used
in generating the inputs [51] to the FLAG fits in Fig. 11, and the FLAG fits
also do not include unitarity bounds.

Figure 10: Results for the $B_{s}\to K$ form factors from chiral/continuum
extrapolations performed directly for $f_{+}$ and $f_{0}$, compared to results
obtained by performing the extrapolations for $f_{\perp}$ and $f_{\parallel}$
and then converting to $f_{+}$ and $f_{0}$ [51].

Old

$\chi^{2}/{\rm dof}=1.54$

All uncertainties rescaled by $\sqrt{\chi^{2}/{\rm dof}}$

$\frac{\Gamma(B_{s}\to K\mu\nu)}{|V_{ub}|^{2}}=6.28(0.67)\>\>{\rm ps}^{-1}$

New

$\chi^{2}/{\rm dof}=3.82$

All uncertainties rescaled by $\sqrt{\chi^{2}/{\rm dof}}$

$\frac{\Gamma(B_{s}\to K\mu\nu)}{|V_{ub}|^{2}}=6.5(1.1)\>\>{\rm ps}^{-1}$

Figure 11: The effect of replacing the 2015 RBC/UKQCD results [53] by the 2023
RBC/UKQCD results [51] in the FLAG average [11] of $B_{s}\to K$ form factors.
Figure 12: Comparison of selected results for $|V_{ub}|$ as of July 2023: from
inclusive semileptonic $B$ decays [64], from $B\to\pi\ell\nu$ [12] using form
factors of Refs. [53, 65, 50] and experimental data from Refs. [54, 55, 56,
57], from $B_{s}\to K\mu\nu$ using the form factors of Ref. [51], from
$B_{s}\to K\mu\nu$ using my update of the FLAG average of the $B_{s}\to K$
form factors, from $\Lambda_{b}\to p\mu\bar{\nu}$ [66] using the form factors
of Ref. [67], and from $B\to\rho\ell\nu$, $B\to\omega\ell\nu$ [68] using form
factors from light-cone sum rules [69]. Here, the determinations from
$B_{s}\to K\mu\nu$ use the ratios $\mathcal{B}(B_{s}\to
K\mu\nu)/\mathcal{B}(B_{s}\to D_{s}\mu\nu)$ and $\mathcal{B}(B_{s}\to
D_{s}\mu\nu)/\mathcal{B}(B\to D\mu\nu)$ from LHCb [70, 37] combined with
$\mathcal{B}(B\to D\mu\nu)$ from PDG [71], and the determination from
$\Lambda_{b}\to p\mu\bar{\nu}$ uses the ratio $\mathcal{B}(\Lambda_{b}\to
p\mu\bar{\nu})/\mathcal{B}(\Lambda_{b}\to\Lambda_{c}\mu\bar{\nu})$ from LHCb
[66] and $|V_{cb}|=40.8(1.4)\times 10^{-3}$ from PDG [64].

Several groups reported on their progress with new calculations of $b\to u$
semileptonic form factors at this conference. Protick Mohanta discussed
JLQCD’s study of the $B_{s}\to K$ form factors using the Möbius domain-wall
action for all quarks [72]. Andrew Lytle from the Fermilab/MILC collaboration
presented preliminary results for the same form factors using an all-HISQ
approach [41], while Hwancheol Jeong reported on Fermilab/MILC’s calculation
of the $B\to\pi$ and $B_{s}\to K$ form factors using HISQ light quarks and
Fermilab $b$ quarks [73]. I gave an update on a next-generation determination
of the $\Lambda_{b}\to p$ form factors with domain-wall light quarks and RHQ
$b$ quarks [62]. Luka Leskovec presented preliminary results for the
$B\to\pi\pi$ ($I=1,L=1$) vector form factor at a heavier-than physical pion
mass [49]. Rainer Sommer discussed an approach for $b$-physics calculations on
the lattice based on interpolations between relativistic and static
computations and the step-scaling method [74], generalizing earlier work [75]
to the case of semileptonic form factors and other observables. Alessandro
Conigli presented preliminary numerical results for the $b$-quark mass and
$B^{(*)}$-meson decay constants using this approach [76].

####  2.3.2 Other constraints on the Wolfenstein parameters
${\boldsymbol{\bar{\rho}}}$ and ${\boldsymbol{\bar{\eta}}}$

  * •

$\alpha$ from $CP$ violation in e.g.
$B^{0}(\overline{B^{0}})\to\pi\pi,\pi\rho,\rho\rho$

  * •

$\beta$ from $CP$ violation in e.g. $B^{0}(\overline{B^{0}})\to J/\psi K_{S}$

  * •

$\gamma$ from $CP$ violation in e.g. $B^{-}\to D^{0}(\overline{D^{0}})(\to
f)K^{-}$

  * •

$\Delta m_{d}$, $\displaystyle\frac{\Delta m_{d}}{\Delta m_{s}}$:
$B^{0}/\overline{B^{0}}$, $B_{s}^{0}/\overline{B_{s}^{0}}$ mixing mass
differences – uses hadronic matrix elements from lattice QCD

  * •

$\epsilon_{K}$: indirect $CP$ violation in the neutral kaon system – uses
hadronic matrix elements from lattice QCD

  * •

$\epsilon^{\prime}_{K}$ (not shown): direct $CP$ violation in the neutral kaon
system – uses hadronic matrix elements from lattice QCD

Figure 13: Constraints on the Wolfenstein parameters $\bar{\rho}$ and
$\bar{\eta}$. Figure from UTfit [77] (see also Ref. [78] for a similar
analysis by the CKMfitter Group).

The determinations of $|V_{ub}|$ discussed in the previous section set the
length of the left side of the unitarity triangle and constrain the apex to
lie within the yellow ring in Fig. 13. We will now discuss the other
constraints shown in the figure.

The determinations of the angles $\alpha$, $\beta$, and $\gamma$ are based
from CP-violation measurements in nonleptonic decays, where all necessary
combinations of hadronic matrix elements can be determined by fitting
experimental data.

The length of the right side of the triangle is constrained by the
$B^{0}_{(s)}$-$\overline{B^{0}_{(s)}}$ oscillation frequencies, which have
been measured very precisely to be [79, 80]

$\displaystyle\Delta m_{d}$ $\displaystyle=0.5065(19)\>{\rm ps}^{-1},$
$\displaystyle\Delta m_{s}$ $\displaystyle=17.765(6)\>{\rm ps}^{-1}.$ (31)

Examples of diagrams contributing to $B_{s}^{0}$-$\overline{B_{s}^{0}}$ mixing
in the Standard Model are shown in Fig. 14. The hadronic matrix elements
currently taken from lattice QCD to relate the oscillation frequencies to the
CKM parameters are

$\langle\overline{B^{0}_{q}}|O^{\Delta
B=2}_{q}|B^{0}_{q}\rangle=\frac{8}{3}f_{B_{q}}^{2}m_{B_{q}}^{2}B_{B_{q}},\hskip
8.61108pt\text{where}\hskip 8.61108ptO^{\Delta
B=2}_{q}=[\bar{b}\gamma_{\mu}(1-\gamma_{5})q][\bar{b}\gamma^{\mu}(1-\gamma_{5})q].$
(32)

Figure 14: Examples of diagrams contributing to
$B_{s}^{0}$-$\overline{B_{s}^{0}}$ mixing in the Standard Model.

The kaon CP violation parameters $\epsilon_{K}$ (indirect) and
$\epsilon^{\prime}_{K}$ (direct) are related to the ratios of $K\to\pi\pi$
amplitudes through [81]

$\displaystyle\frac{A(K^{0}_{L}\to\pi^{+}\pi^{-})}{A(K^{0}_{S}\to\pi^{+}\pi^{-})}\approx\epsilon_{K}+\epsilon^{\prime}_{K},\hskip
17.22217pt\frac{A(K^{0}_{L}\to\pi^{0}\pi^{0})}{A(K^{0}_{S}\to\pi^{0}\pi^{0})}\approx\epsilon_{K}-2\epsilon^{\prime}_{K}.$
(33)

The measured values of these parameters are [81]

$\displaystyle\epsilon_{K}$ $\displaystyle=2.228(11)\times 10^{-3},$
$\displaystyle\mathrm{Re}(\epsilon^{\prime}_{K}/\epsilon_{K})$
$\displaystyle=1.66(23)\times 10^{-3}.$ (34)

The hadronic matrix elements currently taken from lattice QCD for the
Standard-Model calculation of $\epsilon_{K}$ and $\epsilon^{\prime}_{K}$ (and
hence to constrain the CKM parameters using these observables) are

$\langle\pi\pi|O_{i}^{\Delta S=1}|K^{0}\rangle$ (35)

for seven different four-quark operators $O_{i}^{\Delta S=1}$ and

$\langle\overline{K^{0}}|O^{\Delta
S=2}|K^{0}\rangle=\frac{8}{3}f_{K}^{2}m_{K}^{2}B_{K},\hskip
8.61108pt\text{where}\hskip 8.61108ptO^{\Delta
S=2}=[\bar{s}\gamma_{\mu}(1-\gamma_{5})d][\bar{s}\gamma^{\mu}(1-\gamma_{5})d].$
(36)

The kaon CP-violation parameters also receive contributions from nonlocal two-
current matrix elements; the UTfit collaboration [77] currently takes these
matrix elements from a chiral-perturbation-theory calculation [82], but they
can, in principle, also be calculated in lattice QCD [83, 84, 85, 86].

It is worth noting that a major source of uncertainty in the Standard-Model
predictions of $\epsilon_{K}$ and $\Delta m_{d}$, and hence in the
uncertainties of the corresponding constraints on $\bar{\rho}$ and
$\bar{\eta}$ (and constraints on new physics), is the uncertainty of the
Wolfenstein parameter $A$, i.e., the uncertainty of $|V_{cb}|$ [87, 88].

A summary of lattice results for the renormalization-group-independent
versions [11] of the bag parameters defined in Eqs. (32) and (36) is shown in
Fig. 15. No new complete lattice results for the kaon bag parameter have been
published since 2016. A major challenge and source of systematic uncertainty
in lattice calculations of four-quark-operator matrix elements is the
renormalization and matching, and it is good to see progress in this area at
this conference, as discussed later. The $K\to(\pi\pi)_{I=2}$ and
$K\to(\pi\pi)_{I=0}$ matrix elements were computed in Ref.s [89] and [90],
respectively. The combination of the isospin-2 and isospin-0 amplitudes give
the Standard-Model prediction
$\mathrm{Re}(\epsilon^{\prime}_{K}/\epsilon_{K})_{\rm
SM}=2.17(26)(62)(50)\times 10^{-3}$ [90]. In these calculations, another big
challenge is the precise determination of the $\pi\pi$ finite-volume spectrum
and the corresponding Lellouch-Lüscher factors. Reference [90] used so-called
$G$-parity boundary conditions to obtain physical kinematics.

Figure 15: FLAG summary of lattice results for the $B$, $B_{s}$, and $K$ meson
renormalization-group-independent bag parameters [11].

At this conference, Justus Tobias Tsang presented an ongoing new calculation
of the $B_{s}^{0}$-$\overline{B_{s}^{0}}$ mixing matrix elements using a
domain-wall action for all quarks and RI/sMOM renormalization [91]. The
calculation uses both RBC/UKQCD and JLQCD ensembles. Matthew Black discussed
how the gradient flow and short-flow-time expansion can be used to renormalize
four-quark operators, including both the $\Delta B=2$ operators relevant for
neutral-meson mixing and the $\Delta B=0$ operators relevant for the theory of
heavy-hadron lifetimes [92]. Joshua Lin presented a position-space
renormalization scheme for $\Delta B=2$ and $\Delta B=0$ four-quark operators
in HQET and reported on progress with the matching from this scheme to
$\overline{\text{MS}}$ at order $\alpha_{s}$ [93]. Riccardo Marinelli
presented a nonperturbative calculation of the renormalization-group running
of the complete basis of $\Delta S=2$ four-quark operators in the chirally
rotated Schrödinger functional in the region of energies between the $W$ mass
and the switching scale of 4 GeV; this running is traditionally done
perturbatively, and the new method can eliminate the associated uncertainty
[94]. Weonjong Lee gave an update on the Standard-Model prediction for
$\epsilon_{K}$ using lattice inputs, where a significant tension with the
experimental value is observed when using $|V_{cb}|$ from exclusive decays,
and this tension disappears when using the inclusively determined $|V_{cb}|$
instead [87]. Masaaki Tomii presented a new RBC/UKQCD calculation of the
$K\to\pi\pi$ matrix elements with periodic instead of $G$-parity boundary
conditions, using the variational method to extract the matrix elements for
excited $\pi\pi$ finite-volume states to cover the energy region around the
physical kinematic point [95, 96].

###  2.4 Global fit of the Wolfenstein parameters

To conclude this part, I quote the final results for all four Wolfenstein
parameters from a global fit to the observables discussed in the previous
sections. The 2022 Standard-Model fit by UTfit gives [77]

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle 0.22519(83),$
$\displaystyle A$ $\displaystyle=$ $\displaystyle 0.828(11),$
$\displaystyle\bar{\rho}$ $\displaystyle=$ $\displaystyle 0.161(10),$
$\displaystyle\bar{\eta}$ $\displaystyle=$ $\displaystyle 0.347(10),$ (37)

which corresponds to the CKM matrix elements

$\displaystyle\left(\\!\begin{array}[]{lll}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\end{array}\\!\right)$ (41)
$\displaystyle=\left(\\!\\!\begin{array}[]{lll}\phantom{-}0.97431(19)&\phantom{-}0.22517(81)&0.003715(93)\,e^{-i(65.1(1.3))^{\circ}}\\\
-0.22503(83)\,e^{+i(0.0351(1))^{\circ}}&\phantom{-}0.97345(20)\,e^{-i(0.00187(5))^{\circ}}&0.0420(5)\\\
\phantom{-}0.00859(11)\,e^{-i(22.4(7))^{\circ}}&-0.04128(46)\,e^{+i(1.05(3))^{\circ}}&0.999111(20)\end{array}\right).$
(45)

See also Refs. [78, 97, 98] for a similar analysis by the CKMfitter Group.

##  3 Selected further processes in quark flavor physics

###  3.1 Tests of lepton-flavor universality in ${\boldsymbol{b\to
c\ell\bar{\nu}}}$ decays

In the Standard Model, the different generations of leptons couple to $W$
bosons with the same strength, and it is interesting to search for deviations
from this lepton-flavor universality. Important observables for such tests are
the ratios

$R(H_{c})=\frac{\Gamma(H_{b}\to H_{c}\tau\nu)}{\Gamma(H_{b}\to
H_{c}\ell\nu)},$ (46)

where $H_{b}$ denotes a bottom hadron, $H_{c}$ a charm hadron, and $\ell$ an
electron or muon. The SM predictions of $R(H_{c})$ depend on the $H_{b}\to
H_{c}$ form factors, some of which were already discussed in Sec. 2.2. A
comparison of experimental measurements and SM predictions using form factors
from lattice QCD is shown in Fig. 16. For $R(D^{*})$, the pure SM predictions
that do not rely on experimental information on the $B\to D^{*}\ell\nu$ decay
distribution became available only relatively recently [26, 27, 28]. The
measurements of $R(H_{c})$ with mesons show hints for excesses with respect to
the SM predictions; these hints were first seen in 2012 [99] and many possible
new-physics explanations have been proposed. In contrast, the recent LHCb
result for the baryon-decay ratio $R(\Lambda_{c})$ [100] has a central value
below the SM prediction [67]. No common explanation of the measured values of
all ratios within $2\sigma$ using heavy new physics is possible [101].

Figure 16: Comparison of lattice-only SM predictions (blue points with error
bars) and experimental measurements (red lines with error bands) of the ratios
$R(H_{c})$. For $R(D)$, the SM prediction from FLAG [11] uses the average of
form factors from Fermilab/MILC [33] and HPQCD [34]; the experimental value is
the HFLAV average [102] of measurement by BaBar [99], Belle [103, 104], and
LHCb [105]. For $R(D^{*})$, the SM predictions are from Fermilab/MILC [26],
HPQCD [27], and JLQCD [28], and the experimental value is the HFLAV average
[102] of measurements by BaBar [99], Belle [103, 106, 104], Belle II [107],
and LHCb [105, 108]. For $R(J/\psi)$, the SM prediction is from HPQCD [109]
and the measurement from LHCb [110]. For $R(\Lambda_{c})$, the SM prediction
is from Detmold, Lehner, and Meinel [67] and the measurement is from LHCb
[100].

###  3.2 Direct determinations of ${\boldsymbol{|V_{cd}|}}$ and
${\boldsymbol{|V_{cs}|}}$

With $|V_{cd}|$ and $|V_{cs}|$ predicted precisely by the global fit (without
inputs from charm decays) as shown in Eq. (45), it is interesting to check
whether direct determinations are compatible with these values. Experimental
data are more precise for semileptonic $D_{(s)}$ decays compared to leptonic
$D_{(s)}$ decays. In the past, leptonic decays nevertheless gave the most
precise $|V_{cs}|$ and $|V_{cd}|$ because lattice results for decay constants
are more precise than for form factors. Recently, lattice results for
$D_{(s)}$ semileptonic decays have reached high precision [111, 112], and
semileptonic decays currently yield the most precise direct $|V_{cs}|$ and
$|V_{cd}|$ determinations (but there are some tensions with older lattice
results as discussed below).

In the year leading up to the Lattice conference, the Fermilab/MILC
collaboration published a new calculation of the $D\to\pi$ and $D_{(s)}\to K$
form factors using the $N_{f}=2+1+1$ all-HISQ approach [112]. A comparison of
these results with the ones from HPQCD [111] and ETMC [113] is shown in Fig.
17. Good agreement is seen for the form factor $f_{+}(D\to K)$ that is most
relevant for the $D\to K\ell\nu$ decay rate between Fermilab/MILC and HPQCD,
while for $f_{0}(D\to K)$, there appears to be a small but statistically
significant tension between the Fermilab/MILC and HPQCD results. Compared to
the older ETMC results that are based on computations with the twisted-mass
and Osterwalder-Seiler actions [113], the Fermilab/MILC results for all form
factors are higher in the high-$q^{2}$ region, most dramatically for
$f_{+}(D\to\pi)$. Even though this kinematic region contributes less to the
decay rate than the low-$q^{2}$ region, it would be good to understand the
origin of this large discrepancy.

The Fermilab/MILC collaboration also investigated the dependence of the form-
factor results on the choice of basis used for the chiral and continuum
extrapolation (cf. Sec. 2.3.1), as shown in Fig. 18. Similarly to the findings
for $B_{s}\to K$ by RBC/UKQCD [51], doing the extrapolation directly for
$f_{0}$ gives slightly higher values for this form factor compared to
constructing $f_{0}$ from the extrapolated $f_{\perp}$ and $f_{\parallel}$ (I
thank Andreas Jüttner for pointing this out to me).

A comparison of the most recent direct $|V_{cd}|$ and $|V_{cs}|$
determinations with the values predicted by the global CKM fit using unitarity
is shown in Fig. 19. The results from the semileptonic decays are consistent
with unitarity within 1-2$\sigma$.

Figure 17: Comparison of Fermilab/MILC’s new results for the $D\to K$ and
$D\to\pi$ form factors [112] to previous results from HPQCD [111] and ETMC
[113]. Figure from Ref. [112].  Figure 18: Dependence of the Fermilab/MILC
$D\to\pi$ form-factor results on the choice of basis for the chiral-continuum
extrapolations [112]. Figure modified from Ref. [112]; I added the
magnification box.

Figure 19: Comparison of direct $|V_{cd}|$ and $|V_{cs}|$ determinations from
meson decays and neutrino scattering [112, 11, 71] with the values predicted
by the global CKM fit using unitarity [77]. The experimental data for the
semileptonic decays are from BaBar [114, 115], CLEO [116], and BES III [117,
118, 119, 120, 121]. Figure modified from Ref. [112]; I removed the blue bands
and added the orange unitarity bands. In addition to what is shown here, there
are also $|V_{cs}|$ results from $\Lambda_{c}\to\Lambda\ell\nu$, most recently
$|V_{cs}|=0.937(29)$ [122, 123].

The $c\to s\ell^{+}\nu$ transition can also be observed in baryon decays. In
2023, the BESIII collaboration published a combined analysis of the
$\Lambda_{c}\to\Lambda e^{+}\nu$ and $\Lambda_{c}\to\Lambda\mu^{+}\nu$
differential branching fractions and angular distributions [123], updating
their 2022 analysis of $\Lambda_{c}\to\Lambda e^{+}\nu$ only [124]. The
results are compared to the SM predictions using lattice QCD [122] in Fig. 20.
From the total rates, the result $|V_{cs}|=0.937\pm 0.014_{\mathcal{B}}\pm
0.024_{\rm LQCD}\pm 0.007_{\tau_{\Lambda_{c}}}$ is obtained [123], where the
uncertainty from the lattice calculation of the form factors now dominates. In
addition, BESIII fitted a simplified form-factor parametrization to the
$\Lambda_{c}\to\Lambda(\to p\pi)\ell^{+}\nu$ angular distributions, as shown
in the lower panels in Fig. 20. Compared to my lattice-QCD predictions [122],
the BESIII fit yields slightly steeper slopes for the vector form factors and
less steep slopes for the axial-vector form factors (there is also a recent
independent lattice calculation of the $\Lambda_{c}\to\Lambda$ form factors
[125], but it used only a single $N_{f}=2$ ensemble on a $16^{3}\times 32$
lattice with $a\approx 0.16$ fm, $m_{\pi}\approx 550$ MeV).

Figure 20: Upper five panels: BESIII binned measurements and fits of the
$\Lambda_{c}\to\Lambda e^{+}\nu$ and $\Lambda_{c}\to\Lambda\mu^{+}\nu$
differential decay rates, lepton-side forward-backward asymmetries, and
hadron-side forward-backward asymmetries [123], compared to Standard-Model
predictions (only central values shown) using form factors from lattice QCD
[122]. Lower four panels: the form-factor parametrization fitted to the BESIII
$\Lambda_{c}\to\Lambda(\to p\pi)\ell^{+}\nu$ angular-distribution data [123]
compared to the predictions from lattice QCD [122]. All plots from Ref. [123].

At this conference, Callum Farrell presented preliminary lattice-QCD results
for another charm-baryon semileptonic decay, $\Xi_{c}\to\Xi\ell^{+}\nu$ [126].
The calculation points toward a branching fraction in the Standard Model that
is substantially higher than the current experimental results, and also higher
than predicted by a previous lattice calculation [127]. Davide Giusti
presented a lattice calculation of the structure-dependent form factors in the
radiative leptonic decays $D_{s}\to\ell\nu\gamma$ [128, 129]. Ryan Kellermann
reported on a study of finite-volume effects in a lattice determination of the
inclusive $D_{s}\to X_{s}\ell\nu$ rate [46]. Antonin Portelli discussed the
occurrence of finite-volume collinear divergences in the $\text{QED}_{L}$
treatment of electromagnetic corrections to $D_{(s)}$ leptonic decays [19].
Justus Kuhlmann [130] presented an exploration of stabilized Wilson fermions
[131] for charm physics.

###  3.3 Rare ${\boldsymbol{b\to s\ell^{+}\ell^{-}}}$ decays

Decays of $b$-hadrons involving the transition $b\to s\ell^{+}\ell^{-}$ are
among the most important modes used to search for physics beyond the Standard
Model [132], and there have been significant new developments since Lattice
2022 that I will discuss further below. At hadronic energy scales, $b\to
s\ell^{+}\ell^{-}$ decays are described by the weak effective Hamiltonian
[133]

$\mathcal{H}_{\rm
eff}\>\>=\>\>-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\sum_{i}C_{i}{O_{i}},$
(47)

where the most important operators are

$\displaystyle{O_{1}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\bar{c}^{b}\gamma^{\mu}b_{L}^{a}\>\>\>\bar{s}^{a}\gamma_{\mu}c_{L}^{b},\phantom{\big{]}}$
$\displaystyle{O_{2}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\bar{c}^{a}\gamma^{\mu}b_{L}^{a}\>\>\>\bar{s}^{b}\gamma_{\mu}c_{L}^{b},\phantom{\big{]}}$
$\displaystyle{O_{7}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!(e\>m_{b})/(16\pi^{2})\>\>\bar{s}\sigma^{\mu\nu}b_{R}\>\>\>F_{\mu\nu}^{(\rm
e.m.)},\phantom{\big{]}}$ $\displaystyle{O_{9\ell}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!e^{2}/(16\pi^{2})\>\>\bar{s}\gamma^{\mu}b_{L}\>\>\>\bar{\ell}\gamma_{\mu}\ell,\phantom{\big{]}}$
$\displaystyle{O_{10\ell}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!e^{2}/(16\pi^{2})\>\>\bar{s}\gamma^{\mu}b_{L}\>\>\>\bar{\ell}\gamma_{\mu}\gamma_{5}\ell.$
(48)

The SM values of the Wilson coefficients, evaluated using EOS [134] in the
$\overline{\text{MS}}$ scheme and at $\mu=4.2$ GeV, are $C_{1}\approx-0.288$,
$C_{2}\approx 1.010$, $C_{7}\approx-0.336$, $C_{9\ell}\approx 4.275$,
$C_{10\ell}\approx-4.160$ (independent of the lepton flavor $\ell$).

Figure 21: Diagrams illustrating the contributions from different weak
effective operators to the $\bar{B}^{0}\to\bar{K}^{*0}\ell^{+}\ell^{-}$
amplitude.

Figure 22: Selected experimental results for $b\to s\ell^{+}\ell^{-}$ decays
compared to SM predictions: the $\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$
angular observable $P_{5}^{\prime}$ [135] (SM prediction from Refs. [136,
137]), the $\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$ differential branching
fraction [138] (SM prediction from Refs. [69, 139, 140]), the
$\bar{B}_{s}^{0}\to\phi\mu^{+}\mu^{-}$ differential branching fraction [141]
(SM predictions from Refs. [142, 69, 143, 144, 140]), the
$\bar{B}\to\bar{K}\ell^{+}\ell^{-}$ differential branching fractions for both
$\ell=\mu$ and $\ell=e$ (plot by Patrick Koppenburg with experimental results
from Refs. [145, 146, 147, 148, 149, 150] and SM prediction from Ref. [151]),
the $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ differential branching fraction (my
plot with LHCb data from [152], with an updated normalization as discussed in
Ref. [153] and SM prediction from [154]), and the
$\Lambda_{b}\to\Lambda^{*}(1520)\mu^{+}\mu^{-}$ differential branching
fraction [155] (SM predictions from Refs. [156, 157]; I removed the quark-
model predictions from the figure).

The way in which the above operators contribute to, for example, the
$\bar{B}^{0}\to\bar{K}^{*0}\ell^{+}\ell^{-}$ decay amplitude is illustrated in
Fig. 21. For a generic decay $H_{b}\to H_{s}\ell^{+}\ell^{-}$, evaluating the
contributions from $O_{7}$, $O_{9}$, $O_{10}$ requires the hadronic matrix
elements $\langle H_{s}(p^{\prime})|\>\bar{s}\Gamma b\>|H_{b}(p)\rangle$ that
correspond to the local form factors and have been calculated using lattice
QCD for several different combinations of $H_{b}$ and $H_{s}$ hadrons [158,
139, 140, 159, 154, 156, 160]. Note that some of the $H_{s}$ considered are
unstable under the strong interactions [$K^{*}$, $\phi$, $\Lambda^{*}(1520)$],
and the calculations published to date neglected this property. Going beyond
this approximation with the proper Lellouch-Lüscher approach [48] is needed
and feasible for the $B\to K^{*}(\to K\pi)$ form factors [49]. For the
$B_{s}\to\phi$ form factors with the much narrower $\phi$, I think it is worth
doing new calculations even with the simple single-hadron treatment.

The operators $O_{1,...,6}$, $O_{8}$ contribute through matrix elements with
an additional nonlocal insertion of the quark electromagnetic current: in
Minkowski spacetime,

$\int\mathrm{d}^{4}x\>\>e^{iq\cdot x}\,\langle
H_{s}(p^{\prime})|\>\mathsf{T}\>O_{i}(0)\>J^{\mu}_{\rm
e.m.}(x)\>|H_{b}(p)\rangle.$ (49)

Computing these matrix elements is very challenging for Euclidean lattice QCD
(see Ref. [161] for first steps), and they are currently instead being
approximated in the continuum using a local operator-product expansion (OPE)
at high $q^{2}$ [162, 163] and QCD factorization or a light-cone OPE at low
$q^{2}$ [164, 165]. This usually means staying away from the region with
$q^{2}\sim m_{J/\psi,\psi^{\prime}}^{2}$, where the matrix elements of
$O_{1,2}$ are enhanced dramatically though the charmonium resonances, but
recent work using dispersion relations has also taken advantage of the
experimental measurements of $\mathcal{B}(H_{b}\to H_{s}\,J/\psi^{(\prime)})$
[166, 167].

There has been substantial excitement about $b\to s\ell^{+}\ell^{-}$ decays
since 2013, when deviations between measurements and SM predictions were first
seen in $\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$ at both low $q^{2}$ [168]
and high $q^{2}$ [144] (also for $\bar{B}_{s}^{0}\to\phi\mu^{+}\mu^{-}$ [144]
and $\bar{B}\to\bar{K}\mu^{+}\mu^{-}$ [169]) that can be explained by a
negative shift in the Wilson coefficient $C_{9\mu}$. These deviations, with a
general trend of branching fractions below the SM predictions, persist to
date, as shown in Fig. 22. Moreover, LHCb measurements [170, 171, 172, 173,
174, 175] of muon-versus-electron ratios such as $R_{K}\>\equiv\>\int_{1\>{\rm
GeV}^{2}}^{6\>{\rm GeV}^{2}}\frac{\mathrm{d}\mathcal{B}(B^{+}\to
K^{+}{\mu^{+}\mu^{-}})}{\mathrm{d}q^{2}}\mathrm{d}q^{2}/\int_{1\>{\rm
GeV}^{2}}^{6\>{\rm GeV}^{2}}\frac{\mathrm{d}\mathcal{B}(B^{+}\to
K^{+}{e^{+}e^{-}})}{\mathrm{d}q^{2}}\mathrm{d}q^{2}$ gave values below the
theoretically clean SM predictions equal to $1+\mathcal{O}(10^{-3})$,
consistent with the scenario $C_{9\mu}<C_{9}^{\rm SM}$ and $C_{9e}=C_{9}^{\rm
SM}$. The big news from December 2022 [149] is that the LHCb results for these
ratios had an uncontrolled systematic error (hadrons misidentified as
electrons), and the corrected LHCb analysis gives $R_{K^{(*)}}$ consistent
with unity. It is important to note, however, that
$\frac{\mathrm{d}\mathcal{B}(B\to K^{(*)}{\mu^{+}\mu^{-}})}{\mathrm{d}q^{2}}$
is unchanged and $\frac{\mathrm{d}\mathcal{B}(B\to
K^{(*)}{e^{+}e^{-}})}{\mathrm{d}q^{2}}$ has moved lower, _farther away_ from
SM predictions. As a result, a new-physics scenario with both
$C_{9\mu}<C_{9}^{\rm SM}$ and $C_{9e}<C_{9}^{\rm SM}$ is now favored by global
fits (see, e.g., Ref. [176]); viable models with this effect are discussed,
for example, in Ref. [177]. These findings critically depend on QCD
calculations, and improving these calculations is therefore now more important
than ever.

On that note, at this conference, progress with new lattice-QCD calculations
of $b\to s$ form factors was reported by Hwancheol Jeong for $B\to K$, using
using HISQ light quarks and Fermilab $b$ quarks [73], and by me for
$\Lambda_{b}\to\Lambda$, using domain-wall light quarks and RHQ $b$ quarks
[62].

###  3.4 Rare kaon and hyperon decays

Rare decays of _strange_ hadrons also provide powerful constraints on physics
beyond the Standard Model [178]. For some decay modes, theory predictions are
far more precise than current experimental results, while for others, the
situation is the opposite. Theoretically cleanest are the dineutrino modes
$K_{L}\to\pi^{0}\bar{\nu}\nu$ and $K^{+}\to\pi^{+}\bar{\nu}\nu$ [179, 180],
for which the QCD uncertainties of the SM predictions of the decay rates are
1.5% and 4% respectively [178], while the current experimental results are
[181]

$\mathcal{B}(K_{L}\to\pi^{0}\bar{\nu}\nu)<3.0\times 10^{-9}\>\>(90\%\>\>{\rm
CL})$ (50)

and [182]

$\mathcal{B}(K^{+}\to\pi^{+}\bar{\nu}\nu)=(10.6_{-3.4}^{+4.0}|_{\rm stat}\pm
0.9|_{\rm syst})\times 10^{-11}.$ (51)

The 4% QCD uncertainty in the SM prediction of
$\mathcal{B}(K^{+}\to\pi^{+}\bar{\nu}\nu)$ is dominated by the contribution
from nonlocal hadronic matrix elements, which can be calculated with lattice
QCD [183] and will become more relevant as the experimental precision improves
in the future.

Some of the experimental results for charged-lepton modes are [71]

$\displaystyle\mathcal{B}(K^{+}\to\pi^{+}e^{+}e^{-})$
$\displaystyle=3.00(9)\times 10^{-7},$
$\displaystyle\mathcal{B}(K^{+}\to\pi^{+}\mu^{+}\mu^{-})$
$\displaystyle=9.17(14)\times 10^{-8},$
$\displaystyle\mathcal{B}(\Sigma^{+}\to p^{+}\mu^{+}\mu^{-})$
$\displaystyle=2.4^{+1.7}_{-1.3}\times 10^{-8}.$ (52)

The SM predictions for these processes are dominated by nonlocal hadronic
matrix elements, which can, in principle, be calculated on the lattice [184,
185]. The current lattice results still have uncertainties larger than the
central values of the experimental measurements (see Ref. [186] for a
discussion of future prospects).

Another interesting rare kaon decay is $K_{L}\to\mu^{+}\mu^{-}$. Here,
diagrams with exchanges of two weak bosons as well as diagrams with one weak
boson and two photons (see Fig. 23) are important. The contribution of the
former has a 5% QCD uncertainty, mainly from the diagrams with charm quarks
[187]. The absorptive part of the two-photon contribution, which nearly
saturates the SM prediction of $\mathcal{B}(K_{L}\to\mu^{+}\mu^{-})$, can be
obtained with high precision from the measured
$\mathcal{B}(K_{L}\to\gamma\gamma)$, but the remaining dispersive part of the
two-photon contribution is of comparable magnitude to the two-weak-boson
contribution [187] and, while very challenging, is a possible target for
lattice-QCD calculations [188]. A first step in this direction, a lattice-QCD
calculation of $\pi^{0}\to e^{+}e^{-}$, was recently completed [189].

At this conference, En-Hung Chao discussed a framework to compute the two-
photon contribution to $K_{L}\to\mu^{+}\mu^{-}$ (which requires identifying
and subtracting unwanted contributions that appear with Euclidean time) and
presented preliminary numerical results for the quark-connected diagrams
[190]. Bai-Long Hoid reviewed the application of the continuum, dispersive
approach to $\pi^{0}\to e^{+}e^{-}$ and $K_{L}\to\mu^{+}\mu^{-}$ and its
relation to lattice QCD [191]. Raoul Hodgson gave an update [192] on the
exploratory lattice calculation [185] of the rare hyperon decay $\Sigma^{+}\to
p\ell^{+}\ell^{-}$. Amarjit Soni presented an overview of rare kaon decays and
emphasized the role of $K^{0}\to\pi^{0}\ell^{+}\ell^{-}$ [193].

Figure 23: Examples of diagrams contributing to $K_{L}\to\mu^{+}\mu^{-}$ in
the Standard Model.

###  3.5 CP violation in charm decays

CP violation in charm decays was discovered in 2019 by LHCb, with the time-
averaged result [194]

$\Delta A_{CP}=A_{CP}(K^{+}K^{-})-A_{CP}(\pi^{+}\pi^{-})=(-15.4\pm 2.9)\times
10^{-4},$ (53)

where

$A_{CP}(f;t)=\frac{\Gamma(D^{0}(t)\to f)-\Gamma(\overline{D}^{0}(t)\to
f)}{\Gamma(D^{0}(t)\to f)+\Gamma(\overline{D}^{0}(t)\to f)}.$ (54)

The time-dependent analysis shows that $\Delta A_{CP}$ is dominated by direct
CP violation. More recently, LHCb also determined the individual asymmetries
[195]. It is an open question whether the result (53) is compatible with the
Standard Model or a signal of new physics, because theoretical predictions for
$\Delta A_{CP}$ vary substantially depending on the methods used to estimate
the nonperturbative QCD contributions. For example,

$\displaystyle\Delta A_{CP}^{\rm SM}\approx 2\times 10^{-4}\hskip
8.61108pt\text{\cite[cite]{[\@@bibref{Number}{Khodjamirian:2017zdu}{}{}]}},$
$\displaystyle\Delta A_{CP}^{\rm SM}\approx-4\times 10^{-4}\hskip
8.61108pt\text{\cite[cite]{[\@@bibref{Number}{Pich:2023kim}{}{}]}},$
$\displaystyle\Delta A_{CP}^{\rm SM}\approx-16\times 10^{-4}\hskip
8.61108pt\text{\cite[cite]{[\@@bibref{Number}{Schacht:2021jaz}{}{}]}}.$ (55)

A lattice-QCD calculation of the relevant $D\to\pi\pi$ and $D\to K\bar{K}$
amplitudes is therefore an important long-term goal. This is even more
challenging than $K\to\pi\pi$ due to the several coupled channels (including
$\pi\pi\pi\pi$) that need to be considered to understand the finite-volume
final states in the $D$-meson energy region for physical quark masses. A
necessary first step was the generalization of the Lellouch-Lüscher approach
to coupled two-body channels [199].

At this conference, Maxwell Hansen presented a pilot lattice study of hadronic
$D$ decays using stabilized Wilson fermions [200]. This work considers $D\to
K\pi$ at the $SU(3)$ flavor-symmetric point, which avoids power-divergent
operator mixing otherwise present with Wilson fermions and allows the use of
the single-channel Lellouch-Lüscher factor. He showed preliminary results for
the renormalization of the four-quark operators and for the finite-volume
spectra and the corresponding scattering phase shifts.

##  4 Conclusions

Quark flavor physics is exciting and may lead to the discovery of physics
beyond the Standard Model. We already see interesting deviations between
measurements and SM predictions that have inspired substantial model-building
work and demonstrate possible routes to discovery. Lattice-QCD calculations
are essential for quark flavor physics. There has been excellent progress, and
we need to continue and expand this work to make the best use of existing
precise measurements and to keep up with the expected experimental progress
(see, e.g., Refs. [201, 178]) in the coming years. It is very valuable to have
multiple calculations from different groups with different methods. Tensions
between some of the lattice results for semileptonic form factors have
emerged, indicating that uncertainties were underestimated in some cases.
Thankfully, many new calculations with improved methods are already underway.

## References

  * [1] N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Phys. Rev. Lett. 10 (1963) 531–533.
  * [2] J. D. Bjorken and S. L. Glashow, “Elementary Particles and SU(4),” Phys. Lett. 11 (1964) 255–257.
  * [3] S. L. Glashow, J. Iliopoulos, and L. Maiani, “Weak Interactions with Lepton-Hadron Symmetry,” Phys. Rev. D 2 (1970) 1285–1292.
  * [4] M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory of Weak Interaction,” Prog. Theor. Phys. 49 (1973) 652–657.
  * [5] H. Harari, “Three Generations of Quarks and Leptons,” in 5th International Conference on Meson Spectroscopy, p. 0170.  7, 1977.
  * [6] S. Weinberg, “A Model of Leptons,” Phys. Rev. Lett. 19 (1967) 1264–1266.
  * [7] L. Wolfenstein, “Parametrization of the Kobayashi-Maskawa Matrix,” Phys. Rev. Lett. 51 (1983) 1945.
  * [8] N. Lockyer et al., “Measurement of the Lifetime of Bottom Hadrons,” Phys. Rev. Lett. 51 (1983) 1316.
  * [9] Particle Data Group Collaboration, A. Ceccucci, Z. Ligeti, Y. Sakai, et al., “CKM Quark-Mixing Matrix (Review of Particle Physics, Sec. 12),” PTEP 2022 (2022) 083C01.
  * [10] Particle Data Group Collaboration, E. C. Blucher, W. J. Marciano, et al., “$|V_{ud}|$, $|V_{us}|$, the Cabibbo Angle, and CKM Unitarity (Review of Particle Physics, Sec. 67),” PTEP 2022 (2022) 083C01.
  * [11] Flavour Lattice Averaging Group (FLAG) Collaboration, Y. Aoki et al., “FLAG Review 2021,” Eur. Phys. J. C 82 no. 10, (2022) 869, arXiv:2111.09849 [hep-lat].
  * [12] Flavour Lattice Averaging Group (FLAG) Collaboration, Y. Aoki et al., “2023 web update of the FLAG Review 2021.”  http://flag.unibe.ch/2021/.
  * [13] M. Di Carlo, D. Giusti, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula, and N. Tantalo, “Light-meson leptonic decay rates in lattice QCD+QED,” Phys. Rev. D 100 no. 3, (2019) 034514, arXiv:1904.08731 [hep-lat].
  * [14] B. Belfatto and S. Trifinopoulos, “Cabibbo angle anomalies and oblique corrections: The remarkable role of the vectorlike quark doublet,” Phys. Rev. D 108 no. 3, (2023) 035022, arXiv:2302.14097 [hep-ph].
  * [15] PACS Collaboration, T. Yamazaki, K.-i. Ishikawa, N. Ishizuka, Y. Kuramashi, Y. Namekawa, Y. Taniguchi, N. Ukita, and T. Yoshié, “$|V_{us}|$ from kaon semileptonic form factor in $N_{f}=2+1$ QCD at the physical point on $(10\>{\rm fm})^{4}$,” in 40th International Symposium on Lattice Field Theory.  arXiv:2311.16755 [hep-lat].
  * [16] Extended Twisted Mass Collaboration, A. Evangelista, R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfilippo, S. Simula, and V. Lubicz, “Inclusive hadronic decay rate of the $\tau$ lepton from lattice QCD,” Phys. Rev. D 108 no. 7, (2023) 074513, arXiv:2308.03125 [hep-lat].
  * [17] Z. Hall, “Light meson decay constants from Möbius domain-wall fermions on gradient flowed HISQ ensembles,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271070/.
  * [18] N. Christ, “Lattice Calculation of Electromagnetic Corrections to $K_{\ell 3}$ decay,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/270656/.
  * [19] A. Portelli, “Finite-volume collinear divergences in radiative corrections to meson leptonic decays,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/270783/.
  * [20] M. Di Carlo, M. T. Hansen, N. Hermansson-Truedsson, and A. Portelli, “Structure-dependent electromagnetic finite-volume effects through order $1/L^{3}$,” in 40th International Symposium on Lattice Field Theory.  10, 2023.  arXiv:2310.13358 [hep-lat].
  * [21] J.-S. Yoo, T. Bhattacharya, R. Gupta, S. Mondal, and B. Yoon, “Electroweak box diagram contribution for pion and kaon decay from lattice QCD,” Phys. Rev. D 108 no. 3, (2023) 034508, arXiv:2305.03198 [hep-lat].
  * [22] M. Di Carlo, “Isospin-breaking and electromagnetic corrections to weak decays,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/270656/.
  * [23] P. Gambino et al., “Challenges in semileptonic $B$ decays,” Eur. Phys. J. C 80 no. 10, (2020) 966, arXiv:2006.07287 [hep-ph].
  * [24] T. Kaneko, “Heavy flavor physics from lattice QCD,” PoS LATTICE2022 (2023) 238, arXiv:2304.01618 [hep-lat].
  * [25] J. Bulava, “The spectral reconstruction of inclusive rates,” PoS LATTICE2022 (2023) 231, arXiv:2301.04072 [hep-lat].
  * [26] Fermilab Lattice, MILC, Fermilab Lattice, MILC Collaboration, A. Bazavov et al., “Semileptonic form factors for $B\rightarrow D^{*}\ell\nu$ at nonzero recoil from $2+1$-flavor lattice QCD: Fermilab Lattice and MILC Collaborations,” Eur. Phys. J. C 82 no. 12, (2022) 1141, arXiv:2105.14019 [hep-lat]. [Erratum: Eur.Phys.J.C 83, 21 (2023)].
  * [27] J. Harrison and C. T. H. Davies, “$B\rightarrow D^{*}$ vector, axial-vector and tensor form factors for the full $q^{2}$ range from lattice QCD,” arXiv:2304.03137 [hep-lat].
  * [28] JLQCD Collaboration, Y. Aoki, B. Colquhoun, H. Fukaya, S. Hashimoto, T. Kaneko, R. Kellermann, J. Koponen, and E. Kou, “$B\to D^{*}\ell\nu_{\ell}$ semileptonic form factors from lattice QCD with Möbius domain-wall quarks,” arXiv:2306.05657 [hep-lat].
  * [29] BaBar Collaboration, J. P. Lees et al., “Extraction of form Factors from a Four-Dimensional Angular Analysis of $\overline{B}\rightarrow D^{\ast}\ell^{-}\overline{\nu}_{\ell}$,” Phys. Rev. Lett. 123 no. 9, (2019) 091801, arXiv:1903.10002 [hep-ex].
  * [30] Belle Collaboration, E. Waheed et al., “Measurement of the CKM matrix element $|V_{cb}|$ from $B^{0}\to D^{*-}\ell^{+}\nu_{\ell}$ at Belle,” Phys. Rev. D 100 no. 5, (2019) 052007, arXiv:1809.03290 [hep-ex]. [Erratum: Phys.Rev.D 103, 079901 (2021)].
  * [31] M. Bordone, B. Capdevila, and P. Gambino, “Three loop calculations and inclusive Vcb,” Phys. Lett. B 822 (2021) 136679, arXiv:2107.00604 [hep-ph].
  * [32] F. Bernlochner, M. Fael, K. Olschewsky, E. Persson, R. van Tonder, K. K. Vos, and M. Welsch, “First extraction of inclusive $V_{cb}$ from $q^{2}$ moments,” JHEP 10 (2022) 068, arXiv:2205.10274 [hep-ph].
  * [33] MILC Collaboration, J. A. Bailey et al., “$B\to D\ell\nu$ form factors at nonzero recoil and $|V_{cb}|$ from 2+1-flavor lattice QCD,” Phys. Rev. D 92 no. 3, (2015) 034506, arXiv:1503.07237 [hep-lat].
  * [34] HPQCD Collaboration, H. Na, C. M. Bouchard, G. P. Lepage, C. Monahan, and J. Shigemitsu, “$B\rightarrow Dl\nu$ form factors at nonzero recoil and extraction of $|V_{cb}|$,” Phys. Rev. D 92 no. 5, (2015) 054510, arXiv:1505.03925 [hep-lat]. [Erratum: Phys.Rev.D 93, 119906 (2016)].
  * [35] BaBar Collaboration, B. Aubert et al., “Measurement of $|V_{cb}|$ and the Form-Factor Slope in $\bar{B}\to D\ell^{-}\bar{\nu}$ Decays in Events Tagged by a Fully Reconstructed B Meson,” Phys. Rev. Lett. 104 (2010) 011802, arXiv:0904.4063 [hep-ex].
  * [36] Belle Collaboration, R. Glattauer et al., “Measurement of the decay $B\to D\ell\nu_{\ell}$ in fully reconstructed events and determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$,” Phys. Rev. D 93 no. 3, (2016) 032006, arXiv:1510.03657 [hep-ex].
  * [37] LHCb Collaboration, R. Aaij et al., “Measurement of $|V_{cb}|$ with $B_{s}^{0}\to D_{s}^{(*)-}\mu^{+}\nu_{\mu}$ decays,” Phys. Rev. D 101 no. 7, (2020) 072004, arXiv:2001.03225 [hep-ex].
  * [38] LHCb Collaboration, R. Aaij et al., “Precise measurement of the $f_{s}/f_{d}$ ratio of fragmentation fractions and of $B^{0}_{s}$ decay branching fractions,” Phys. Rev. D 104 no. 3, (2021) 032005, arXiv:2103.06810 [hep-ex].
  * [39] E. McLean, C. T. H. Davies, A. T. Lytle, and J. Koponen, “Lattice QCD form factor for $B_{s}\to D_{s}^{*}l\nu$ at zero recoil with non-perturbative current renormalisation,” Phys. Rev. D 99 no. 11, (2019) 114512, arXiv:1904.02046 [hep-lat].
  * [40] E. McLean, C. T. H. Davies, J. Koponen, and A. T. Lytle, “$B_{s}\to D_{s}\ell\nu$ Form Factors for the full $q^{2}$ range from Lattice QCD with non-perturbatively normalized currents,” Phys. Rev. D 101 no. 7, (2020) 074513, arXiv:1906.00701 [hep-lat].
  * [41] A. Lytle, “$B$-meson semileptonic decays from highly improved staggered quarks,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271146/.
  * [42] T. Bhattacharya et al., “Current progress on the semileptonic form factors for $\bar{B}\to D^{\ast}\ell\bar{\nu}$ decay using the Oktay-Kronfeld action,” PoS LATTICE2023 (2023) 245, arXiv:2401.01561 [hep-lat].
  * [43] T. Bhattacharya et al., “Progress report on testing robustness of the Newton method in data analysis on 2-point correlation function using a MILC HISQ ensemble,” PoS LATTICE2023 (2023) 155, arXiv:2401.01584 [hep-lat].
  * [44] A. Melis, F. Sanfilippo, and S. Simula, “Hadronic susceptibilities for $b$ to $c$ transitions from two point correlation functions,” in 40th International Symposium on Lattice Field Theory.  1, 2024.  arXiv:2401.03920 [hep-lat].
  * [45] A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko, and R. Kellermann, “Chebyshev and Backus-Gilbert reconstruction for inclusive semileptonic $B_{(s)}$-meson decays from Lattice QCD,”  12, 2023.  arXiv:2312.17401 [hep-lat].
  * [46] R. Kellermann, A. Barone, S. Hashimoto, A. Jüttner, and T. Kaneko, “Studies on finite-volume effects in the inclusive semileptonic decays of charmed mesons,” arXiv:2312.16442 [hep-lat].
  * [47] L. Lellouch and M. Luscher, “Weak transition matrix elements from finite volume correlation functions,” Commun. Math. Phys. 219 (2001) 31–44, arXiv:hep-lat/0003023.
  * [48] R. A. Briceño, M. T. Hansen, and A. Walker-Loud, “Multichannel 1 $\rightarrow$ 2 transition amplitudes in a finite volume,” Phys. Rev. D 91 no. 3, (2015) 034501, arXiv:1406.5965 [hep-lat].
  * [49] L. Leskovec, “Electroweak Transitions Involving Resonances,” in 40th International Symposium on Lattice Field Theory.  1, 2024.  arXiv:2401.02495 [hep-lat].
  * [50] JLQCD Collaboration, B. Colquhoun, S. Hashimoto, T. Kaneko, and J. Koponen, “Form factors of $B\to\pi\ell\nu$ and a determination of $|V_{ub}|$ with Möbius domain-wall fermions,” Phys. Rev. D 106 no. 5, (2022) 054502, arXiv:2203.04938 [hep-lat].
  * [51] RBC/UKQCD Collaboration, J. M. Flynn, R. C. Hill, A. Jüttner, A. Soni, J. T. Tsang, and O. Witzel, “Exclusive semileptonic $B_{s}\to K\ell\nu$ decays on the lattice,” Phys. Rev. D 107 no. 11, (2023) 114512, arXiv:2303.11280 [hep-lat].
  * [52] R. Hill, “Semileptonic form factors for exclusive $B_{s}\to K\ell\nu$ decays,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271079/.
  * [53] J. M. Flynn, T. Izubuchi, T. Kawanai, C. Lehner, A. Soni, R. S. Van de Water, and O. Witzel, “$B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors and $|V_{ub}|$ from 2+1-flavor lattice QCD with domain-wall light quarks and relativistic heavy quarks,” Phys. Rev. D 91 no. 7, (2015) 074510, arXiv:1501.05373 [hep-lat].
  * [54] BaBar Collaboration, P. del Amo Sanchez et al., “Study of $B\to\pi\ell\nu$ and $B\to\rho\ell\nu$ Decays and Determination of $|V_{ub}|$,” Phys. Rev. D 83 (2011) 032007, arXiv:1005.3288 [hep-ex].
  * [55] BaBar Collaboration, J. P. Lees et al., “Branching fraction and form-factor shape measurements of exclusive charmless semileptonic B decays, and determination of $|V_{ub}|$,” Phys. Rev. D 86 (2012) 092004, arXiv:1208.1253 [hep-ex].
  * [56] Belle Collaboration, H. Ha et al., “Measurement of the decay $B^{0}\to\pi^{-}\ell^{+}\nu$ and determination of $|V_{ub}|$,” Phys. Rev. D 83 (2011) 071101, arXiv:1012.0090 [hep-ex].
  * [57] Belle Collaboration, A. Sibidanov et al., “Study of Exclusive $B\to X_{u}\ell\nu$ Decays and Extraction of $|V_{ub}|$ using Full Reconstruction Tagging at the Belle Experiment,” Phys. Rev. D 88 no. 3, (2013) 032005, arXiv:1306.2781 [hep-ex].
  * [58] T. Blake, S. Meinel, M. Rahimi, and D. van Dyk, “Dispersive bounds for local form factors in $\Lambda_{b}\to\Lambda$ transitions,” Phys. Rev. D 108 no. 9, (2023) 094509, arXiv:2205.06041 [hep-ph].
  * [59] J. M. Flynn, A. Jüttner, and J. T. Tsang, “Bayesian inference for form-factor fits regulated by unitarity and analyticity,” JHEP 12 (2023) 175, arXiv:2303.11285 [hep-ph].
  * [60] C. M. Bouchard, G. P. Lepage, C. Monahan, H. Na, and J. Shigemitsu, “$B_{s}\to K\ell\nu$ form factors from lattice QCD,” Phys. Rev. D 90 (2014) 054506, arXiv:1406.2279 [hep-lat].
  * [61] Fermilab Lattice, MILC Collaboration, A. Bazavov et al., “$B_{s}\to K\ell\nu$ decay from lattice QCD,” Phys. Rev. D 100 no. 3, (2019) 034501, arXiv:1901.02561 [hep-lat].
  * [62] S. Meinel, “Status of next-generation $\Lambda_{b}\to p,\Lambda,\Lambda_{c}$ form-factor calculations,” in 40th International Symposium on Lattice Field Theory.  9, 2023.  arXiv:2309.01821 [hep-lat].
  * [63] J. M. Flynn, A. Jüttner, and J. T. Tsang, “Extrapolating semileptonic form factors using Bayesian-inference fits regulated by unitarity and analyticity,”  12, 2023.  arXiv:2312.14631 [hep-lat].
  * [64] Particle Data Group Collaboration, A. X. El-Khadra, P. Urquijo, et al., “Semileptonic $b$-Hadron Decays, Determination of $V_{cb}$, $V_{ub}$ (Review of Particle Physics, Sec. 76),” PTEP 2022 (2022) 083C01.
  * [65] Fermilab Lattice, MILC Collaboration, J. A. Bailey et al., “$|V_{ub}|$ from $B\to\pi\ell\nu$ decays and (2+1)-flavor lattice QCD,” Phys. Rev. D 92 no. 1, (2015) 014024, arXiv:1503.07839 [hep-lat].
  * [66] LHCb Collaboration, R. Aaij et al., “Determination of the quark coupling strength $|V_{ub}|$ using baryonic decays,” Nature Phys. 11 (2015) 743–747, arXiv:1504.01568 [hep-ex].
  * [67] W. Detmold, C. Lehner, and S. Meinel, “$\Lambda_{b}\to p\ell^{-}\bar{\nu}_{\ell}$ and $\Lambda_{b}\to\Lambda_{c}\ell^{-}\bar{\nu}_{\ell}$ form factors from lattice QCD with relativistic heavy quarks,” Phys. Rev. D 92 no. 3, (2015) 034503, arXiv:1503.01421 [hep-lat].
  * [68] F. U. Bernlochner, M. T. Prim, and D. J. Robinson, “$B\to\rho\ell\nu$ and $B\to\omega\ell\nu$ in and beyond the Standard Model: Improved predictions and $|V_{ub}|$,” Phys. Rev. D 104 no. 3, (2021) 034032, arXiv:2104.05739 [hep-ph].
  * [69] A. Bharucha, D. M. Straub, and R. Zwicky, “$B\to V\ell^{+}\ell^{-}$ in the Standard Model from light-cone sum rules,” JHEP 08 (2016) 098, arXiv:1503.05534 [hep-ph].
  * [70] LHCb Collaboration, R. Aaij et al., “First observation of the decay $B_{s}^{0}\to K^{-}\mu^{+}\nu_{\mu}$ and Measurement of $|V_{ub}|/|V_{cb}|$,” Phys. Rev. Lett. 126 no. 8, (2021) 081804, arXiv:2012.05143 [hep-ex].
  * [71] Particle Data Group Collaboration, R. L. Workman et al., “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * [72] JLQCD Collaboration, P. Mohanta, T. Kaneko, and S. Hashimoto, “$B_{s}\to K\ell\nu$ form factors from lattice QCD with domain-wall heavy quarks,” arXiv:2401.01570 [hep-lat].
  * [73] H. Jeong, “Form factors for semileptonic $B$-decays with HISQ light quarks and clover b-quarks in Fermilab interpretation,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271400/.
  * [74] A. Conigli, J. Frison, P. Fritzsch, A. Gérardin, J. Heitger, G. Herdoiza, S. Kuberski, C. Pena, H. Simma, and R. Sommer, “A strategy for B-physics observables in the continuum limit,” in 40th International Symposium on Lattice Field Theory.  12, 2023.  arXiv:2312.09811 [hep-lat].
  * [75] D. Guazzini, R. Sommer, and N. Tantalo, “Precision for $B$-meson matrix elements,” JHEP 01 (2008) 076, arXiv:0710.2229 [hep-lat].
  * [76] A. Conigli, J. Frison, P. Fritzsch, A. Gérardin, J. Heitger, G. Herdoiza, S. Kuberski, C. Pena, H. Simma, and R. Sommer, “$m_{B}$ and $f_{B^{(\star)}}$ in $2+1$ flavour QCD from a combination of continuum limit static and relativistic results,” in 40th International Symposium on Lattice Field Theory.  12, 2023.  arXiv:2312.10017 [hep-lat].
  * [77] UTfit Collaboration, M. Bona et al., “New UTfit Analysis of the Unitarity Triangle in the Cabibbo-Kobayashi-Maskawa scheme,” Rend. Lincei Sci. Fis. Nat. 34 (2023) 37–57, arXiv:2212.03894 [hep-ph].
  * [78] CKMfitter Collaboration. http://ckmfitter.in2p3.fr/.
  * [79] HFLAV Collaboration, Y. S. Amhis et al., “Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021,” Phys. Rev. D 107 no. 5, (2023) 052008, arXiv:2206.07501 [hep-ex].
  * [80] HFLAV Collaboration, “$b$-hadron lifetimes and oscillation parameters,”. https://hflav-eos.web.cern.ch/hflav-eos/osc/PDG_2023/.
  * [81] Particle Data Group Collaboration, C. J. Lin et al., “$CP$ violation in $K_{L}^{0}$ decays (Review of Particle Physics, Sec. 68),” PTEP 2022 (2022) 083C01.
  * [82] A. J. Buras, D. Guadagnoli, and G. Isidori, “On $\epsilon_{K}$ Beyond Lowest Order in the Operator Product Expansion,” Phys. Lett. B 688 (2010) 309–313, arXiv:1002.3612 [hep-ph].
  * [83] N. H. Christ, “Computing the long-distance contribution to the kaon mixing parameter $\epsilon_{K}$,” PoS LATTICE2011 (2011) 277, arXiv:1201.2065 [hep-lat].
  * [84] Z. Bai, N. H. Christ, T. Izubuchi, C. T. Sachrajda, A. Soni, and J. Yu, “$K_{L}-K_{S}$ Mass Difference from Lattice QCD,” Phys. Rev. Lett. 113 (2014) 112003, arXiv:1406.0916 [hep-lat].
  * [85] B. Wang, “Calculating $\Delta m_{K}$ with lattice QCD,” PoS LATTICE2021 (2022) 141, arXiv:2301.01387 [hep-lat].
  * [86] A. W. Jackura, R. A. Briceńo, and M. T. Hansen, “Three-pion effects in $K^{0}-\bar{K}^{0}$ mixing,” PoS LATTICE2022 (2023) 062, arXiv:2212.09951 [hep-lat].
  * [87] S. Jwa, J. Kim, S. Kim, S. Lee, W. Lee, J. Leem, J. Pak, and S. Park, “2023 Update of $\varepsilon_{K}$ with lattice QCD inputs,” in 40th International Symposium on Lattice Field Theory.  11, 2023.  arXiv:2312.02986 [hep-lat].
  * [88] J. Charles, S. Descotes-Genon, Z. Ligeti, S. Monteil, M. Papucci, K. Trabelsi, and L. Vale Silva, “New physics in $B$ meson mixing: future sensitivity and limitations,” Phys. Rev. D 102 no. 5, (2020) 056023, arXiv:2006.04824 [hep-ph].
  * [89] T. Blum et al., “$K\rightarrow\pi\pi$ $\Delta I=3/2$ decay amplitude in the continuum limit,” Phys. Rev. D 91 no. 7, (2015) 074502, arXiv:1502.00263 [hep-lat].
  * [90] RBC, UKQCD Collaboration, R. Abbott et al., “Direct CP violation and the $\Delta I=1/2$ rule in $K\to\pi\pi$ decay from the standard model,” Phys. Rev. D 102 no. 5, (2020) 054509, arXiv:2004.09440 [hep-lat].
  * [91] J. T. Tsang, “$B_{(s)}$-mixing parameters from all-domain-wall-fermion simulations,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271292/.
  * [92] M. Black, R. Harlander, F. Lange, A. Rago, A. Shindler, and O. Witzel, “Using Gradient Flow to Renormalise Matrix Elements for Meson Mixing and Lifetimes,” in 40th International Symposium on Lattice Field Theory.  10, 2023.  arXiv:2310.18059 [hep-lat].
  * [93] J. Lin, “Nonperturbative renormalization of HQET operators in position space,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/270610/.
  * [94] R. Marinelli, “Operator mixing and non-perturbative running of $\Delta F=2$ four-fermion operators,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271057/.
  * [95] M. Tomii, “New result for $\epsilon^{\prime}$ in $K\to\pi\pi$ decay using periodic boundary conditions,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/270651/.
  * [96] RBC, UKQCD Collaboration, T. Blum, P. A. Boyle, D. Hoying, T. Izubuchi, L. Jin, C. Jung, C. Kelly, C. Lehner, A. Soni, and M. Tomii, “$\Delta I=3/2$ and $\Delta I=1/2$ channels of $K\to\pi\pi$ decay at the physical point with periodic boundary conditions,” Phys. Rev. D 108 no. 9, (2023) 094517, arXiv:2306.06781 [hep-lat].
  * [97] A. Hocker, H. Lacker, S. Laplace, and F. Le Diberder, “A New approach to a global fit of the CKM matrix,” Eur. Phys. J. C 21 (2001) 225–259, arXiv:hep-ph/0104062.
  * [98] J. Charles et al., “Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ New Physics,” Phys. Rev. D 91 no. 7, (2015) 073007, arXiv:1501.05013 [hep-ph].
  * [99] BaBar Collaboration, J. P. Lees et al., “Evidence for an excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays,” Phys. Rev. Lett. 109 (2012) 101802, arXiv:1205.5442 [hep-ex].
  * [100] LHCb Collaboration, R. Aaij et al., “Observation of the decay $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\tau^{-}\overline{\nu}_{\tau}$,” Phys. Rev. Lett. 128 no. 19, (2022) 191803, arXiv:2201.03497 [hep-ex].
  * [101] M. Fedele, M. Blanke, A. Crivellin, S. Iguro, T. Kitahara, U. Nierste, and R. Watanabe, “Impact of $\Lambda_{b}\to\Lambda_{c}\tau\nu$ measurement on new physics in $b\to c\ell\nu$ transitions,” Phys. Rev. D 107 no. 5, (2023) 055005, arXiv:2211.14172 [hep-ph].
  * [102] HFLAV Collaboration, “Preliminary average of $R(D)$ and $R(D^{*})$ for Summer 2023,”. https://hflav-eos.web.cern.ch/hflav-eos/semi/summer23/html/RDsDsstar/RDRDs.html.
  * [103] Belle Collaboration, M. Huschle et al., “Measurement of the branching ratio of $\bar{B}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}\to D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle,” Phys. Rev. D 92 no. 7, (2015) 072014, arXiv:1507.03233 [hep-ex].
  * [104] Belle Collaboration, G. Caria et al., “Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^{*})$ with a semileptonic tagging method,” Phys. Rev. Lett. 124 no. 16, (2020) 161803, arXiv:1910.05864 [hep-ex].
  * [105] LHCb Collaboration, “Measurement of the ratios of branching fractions $\mathcal{R}(D^{*})$ and $\mathcal{R}(D^{0})$,” Phys. Rev. Lett. 131 (2023) 111802, arXiv:2302.02886 [hep-ex].
  * [106] Belle Collaboration, S. Hirose et al., “Measurement of the $\tau$ lepton polarization and $R(D^{*})$ in the decay $\bar{B}\to D^{*}\tau^{-}\bar{\nu}_{\tau}$,” Phys. Rev. Lett. 118 no. 21, (2017) 211801, arXiv:1612.00529 [hep-ex].
  * [107] Belle-II Collaboration, I. Adachi et al., “A test of lepton flavor universality with a measurement of $R(D^{*})$ using hadronic $B$ tagging at the Belle II experiment,” arXiv:2401.02840 [hep-ex].
  * [108] LHCb Collaboration, R. Aaij et al., “Test of lepton flavor universality using $B_{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decays with hadronic $\tau$ channels,” Phys. Rev. D 108 no. 1, (2023) 012018, arXiv:2305.01463 [hep-ex].
  * [109] LATTICE-HPQCD Collaboration, J. Harrison, C. T. H. Davies, and A. Lytle, “$R(J/\psi)$ and $B_{c}^{-}\rightarrow J/\psi\ell^{-}\bar{\nu}_{\ell}$ Lepton Flavor Universality Violating Observables from Lattice QCD,” Phys. Rev. Lett. 125 no. 22, (2020) 222003, arXiv:2007.06956 [hep-lat].
  * [110] LHCb Collaboration, R. Aaij et al., “Measurement of the ratio of branching fractions $\mathcal{B}(B_{c}^{+}\,\to\,J/\psi\tau^{+}\nu_{\tau})$/$\mathcal{B}(B_{c}^{+}\,\to\,J/\psi\mu^{+}\nu_{\mu})$,” Phys. Rev. Lett. 120 no. 12, (2018) 121801, arXiv:1711.05623 [hep-ex].
  * [111] HPQCD Collaboration, B. Chakraborty, W. G. Parrott, C. Bouchard, C. T. H. Davies, J. Koponen, and G. P. Lepage, “Improved $V_{cs}$ determination using precise lattice QCD form factors for $D\to K\ell\nu$,” Phys. Rev. D 104 no. 3, (2021) 034505, arXiv:2104.09883 [hep-lat].
  * [112] Fermilab Lattice, MILC Collaboration, A. Bazavov et al., “$D$-meson semileptonic decays to pseudoscalars from four-flavor lattice QCD,” Phys. Rev. D 107 no. 9, (2023) 094516, arXiv:2212.12648 [hep-lat].
  * [113] ETM Collaboration, V. Lubicz, L. Riggio, G. Salerno, S. Simula, and C. Tarantino, “Scalar and vector form factors of $D\to\pi(K)\ell\nu$ decays with $N_{f}=2+1+1$ twisted fermions,” Phys. Rev. D 96 no. 5, (2017) 054514, arXiv:1706.03017 [hep-lat]. [Erratum: Phys.Rev.D 99, 099902 (2019), Erratum: Phys.Rev.D 100, 079901 (2019)].
  * [114] BaBar Collaboration, B. Aubert et al., “Measurement of the hadronic form-factor in $D^{0}\to K^{-}e^{+}\nu_{e}$,” Phys. Rev. D 76 (2007) 052005, arXiv:0704.0020 [hep-ex].
  * [115] BaBar Collaboration, J. P. Lees et al., “Measurement of the $D^{0}\to\pi^{-}e^{+}\nu_{e}$ differential decay branching fraction as a function of $q^{2}$ and study of form factor parameterizations,” Phys. Rev. D 91 no. 5, (2015) 052022, arXiv:1412.5502 [hep-ex].
  * [116] CLEO Collaboration, D. Besson et al., “Improved measurements of $D$ meson semileptonic decays to $\pi$ and $K$ mesons,” Phys. Rev. D 80 (2009) 032005, arXiv:0906.2983 [hep-ex].
  * [117] BES III Collaboration, M. Ablikim et al., “Study of dynamics of $D^{0}\to K^{-}e^{+}\nu_{e}$ and $D^{0}\to\pi^{-}e^{+}\nu_{e}$ decays,” Phys. Rev. D 92 no. 7, (2015) 072012, arXiv:1508.07560 [hep-ex].
  * [118] BES III Collaboration, M. Ablikim et al., “Analysis of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ semileptonic decays,” Phys. Rev. D 96 no. 1, (2017) 012002, arXiv:1703.09084 [hep-ex].
  * [119] BES III Collaboration, M. Ablikim et al., “Measurement of the branching fraction for the semi-leptonic decay $D^{0(+)}\to\pi^{-(0)}\mu^{+}\nu_{\mu}$ and test of lepton universality,” Phys. Rev. Lett. 121 no. 17, (2018) 171803, arXiv:1802.05492 [hep-ex].
  * [120] BES III Collaboration, M. Ablikim et al., “Study of the $D^{0}\to K^{-}\mu^{+}\nu_{\mu}$ dynamics and test of lepton flavor universality with $D^{0}\to K^{-}\ell^{+}\nu_{\ell}$ decays,” Phys. Rev. Lett. 122 no. 1, (2019) 011804, arXiv:1810.03127 [hep-ex].
  * [121] BES III Collaboration, M. Ablikim et al., “First measurement of the form factors in $D^{+}_{s}\rightarrow K^{0}e^{+}\nu_{e}$ and $D^{+}_{s}\rightarrow K^{*0}e^{+}\nu_{e}$ decays,” Phys. Rev. Lett. 122 no. 6, (2019) 061801, arXiv:1811.02911 [hep-ex].
  * [122] S. Meinel, “$\Lambda_{c}\to\Lambda l^{+}\nu_{l}$ form factors and decay rates from lattice QCD with physical quark masses,” Phys. Rev. Lett. 118 no. 8, (2017) 082001, arXiv:1611.09696 [hep-lat].
  * [123] BESIII Collaboration, M. Ablikim et al., “Study of $\Lambda_{c}^{+}\rightarrow\Lambda\mu^{+}\nu_{\mu}$ and Test of Lepton Flavor Universality with $\Lambda_{c}^{+}\rightarrow\Lambda\ell^{+}\nu_{\ell}$ Decays,” Phys. Rev. D 108 no. 3, (2023) L031105, arXiv:2306.02624 [hep-ex].
  * [124] BESIII Collaboration, M. Ablikim et al., “Study of the Semileptonic Decay $\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$,” Phys. Rev. Lett. 129 no. 23, (2022) 231803, arXiv:2207.14149 [hep-ex].
  * [125] H. Bahtiyar, “$\Lambda_{c}\to\Lambda$ Form Factors in Lattice QCD,” Turk. J. Phys. 45 (2021) 4, arXiv:2107.13909 [hep-lat].
  * [126] C. Farrell and S. Meinel, “Form factors for the charm-baryon semileptonic decay $\Xi_{c}\to\Xi\ell\nu$ from domain-wall lattice QCD,” in 40th International Symposium on Lattice Field Theory.  9, 2023.  arXiv:2309.08107 [hep-lat].
  * [127] Q.-A. Zhang et al., “First lattice QCD calculation of semileptonic decays of charmed-strange baryons $\Xi_{c}$,” Chin. Phys. C 46 no. 1, (2022) 011002, arXiv:2103.07064 [hep-lat].
  * [128] D. Giusti, C. F. Kane, C. Lehner, S. Meinel, and A. Soni, “Methods for high-precision determinations of radiative-leptonic decay form factors using lattice QCD,” Phys. Rev. D 107 no. 7, (2023) 074507, arXiv:2302.01298 [hep-lat].
  * [129] D. Giusti, “Structure-dependent form factors in radiative leptonic decays of the $D_{s}$ meson with Domain Wall fermions,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/269439/.
  * [130] P. Fritzsch, J. Heitger, F. Joswig, and J. T. Kuhlmann, “Towards charm physics with stabilised Wilson fermions,” in 40th International Symposium on Lattice Field Theory.  arXiv:2401.05885 [hep-lat].
  * [131] A. Francis, P. Fritzsch, M. Lüscher, and A. Rago, “Master-field simulations of $O(a)$-improved lattice QCD: Algorithms, stability and exactness,” Comput. Phys. Commun. 255 (2020) 107355, arXiv:1911.04533 [hep-lat].
  * [132] W. Altmannshofer and F. Archilli, “Rare decays of $b$ and $c$ hadrons,” in Snowmass 2021.  arXiv:2206.11331 [hep-ph].
  * [133] B. Grinstein, M. J. Savage, and M. B. Wise, “$B\to X_{s}e^{+}e^{-}$ in the Six Quark Model,” Nucl. Phys. B 319 (1989) 271–290.
  * [134] D. van Dyk et al., “EOS: a software for flavor physics phenomenology,” Eur. Phys. J. C 82 no. 6, (2022) 569, arXiv:2111.15428 [hep-ph].
  * [135] LHCb Collaboration, R. Aaij et al., “Measurement of $CP$-Averaged Observables in the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ Decay,” Phys. Rev. Lett. 125 no. 1, (2020) 011802, arXiv:2003.04831 [hep-ex].
  * [136] S. Descotes-Genon, L. Hofer, J. Matias, and J. Virto, “On the impact of power corrections in the prediction of $B\to K^{*}\mu^{+}\mu^{-}$ observables,” JHEP 12 (2014) 125, arXiv:1407.8526 [hep-ph].
  * [137] A. Khodjamirian, T. Mannel, A. A. Pivovarov, and Y. M. Wang, “Charm-loop effect in $B\to K^{(*)}\ell^{+}\ell^{-}$ and $B\to K^{*}\gamma$,” JHEP 09 (2010) 089, arXiv:1006.4945 [hep-ph].
  * [138] LHCb Collaboration, R. Aaij et al., “Measurements of the S-wave fraction in $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays and the $B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ differential branching fraction,” JHEP 11 (2016) 047, arXiv:1606.04731 [hep-ex]. [Erratum: JHEP 04, 142 (2017)].
  * [139] R. R. Horgan, Z. Liu, S. Meinel, and M. Wingate, “Lattice QCD calculation of form factors describing the rare decays $B\to K^{*}\ell^{+}\ell^{-}$ and $B_{s}\to\phi\ell^{+}\ell^{-}$,” Phys. Rev. D 89 no. 9, (2014) 094501, arXiv:1310.3722 [hep-lat].
  * [140] R. R. Horgan, Z. Liu, S. Meinel, and M. Wingate, “Rare $B$ decays using lattice QCD form factors,” PoS LATTICE2014 (2015) 372, arXiv:1501.00367 [hep-lat].
  * [141] LHCb Collaboration, R. Aaij et al., “Branching Fraction Measurements of the Rare $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B^{0}_{s}\rightarrow f_{2}^{\prime}(1525)\mu^{+}\mu^{-}$\- Decays,” Phys. Rev. Lett. 127 no. 15, (2021) 151801, arXiv:2105.14007 [hep-ex].
  * [142] W. Altmannshofer and D. M. Straub, “New physics in $b\rightarrow s$ transitions after LHC run 1,” Eur. Phys. J. C 75 no. 8, (2015) 382, arXiv:1411.3161 [hep-ph].
  * [143] D. M. Straub, “flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond,” arXiv:1810.08132 [hep-ph].
  * [144] R. R. Horgan, Z. Liu, S. Meinel, and M. Wingate, “Calculation of $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ and $B_{s}^{0}\to\phi\mu^{+}\mu^{-}$ observables using form factors from lattice QCD,” Phys. Rev. Lett. 112 (2014) 212003, arXiv:1310.3887 [hep-ph].
  * [145] Belle Collaboration, J. T. Wei et al., “Measurement of the Differential Branching Fraction and Forward-Backward Asymmetry for $B\to K^{(*)}\ell^{+}\ell^{-}$,” Phys. Rev. Lett. 103 (2009) 171801, arXiv:0904.0770 [hep-ex].
  * [146] CDF Collaboration, T. Aaltonen et al., “Measurement of the Forward-Backward Asymmetry in the $B\to K^{(*)}\mu^{+}\mu^{-}$ Decay and First Observation of the $B^{0}_{s}\to\phi\mu^{+}\mu^{-}$ Decay,” Phys. Rev. Lett. 106 (2011) 161801, arXiv:1101.1028 [hep-ex].
  * [147] BaBar Collaboration, J. P. Lees et al., “Measurement of Branching Fractions and Rate Asymmetries in the Rare Decays $B\to K^{(*)}l^{+}l^{-}$,” Phys. Rev. D 86 (2012) 032012, arXiv:1204.3933 [hep-ex].
  * [148] LHCb Collaboration, R. Aaij et al., “Differential branching fractions and isospin asymmetries of $B\to K^{(*)}\mu^{+}\mu^{-}$ decays,” JHEP 06 (2014) 133, arXiv:1403.8044 [hep-ex].
  * [149] LHCb Collaboration, R. Aaij et al., “Measurement of lepton universality parameters in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ and $B^{0}\to K^{*0}\ell^{+}\ell^{-}$ decays,” Phys. Rev. D 108 no. 3, (2023) 032002, arXiv:2212.09153 [hep-ex].
  * [150] CMS Collaboration, “Test of lepton flavor universality in $\mathrm{B}^{\pm}\rightarrow\mathrm{K}^{\pm}\ell^{+}\ell^{-}$ decays,” tech. rep., CMS-PAS-BPH-22-005, 2023.  https://cds.cern.ch/record/2868987.
  * [151] HPQCD Collaboration, W. G. Parrott, C. Bouchard, and C. T. H. Davies, “Standard Model predictions for $B\to K\ell^{+}\ell^{-}$, $B\to K\ell_{1}^{-}\ell_{2}^{+}$ and $B\to K\nu\bar{\nu}$ using form factors from $N_{f}=2+1+1$ lattice QCD,” Phys. Rev. D 107 no. 1, (2023) 014511, arXiv:2207.13371 [hep-ph]. [Erratum: Phys.Rev.D 107, 119903 (2023)].
  * [152] LHCb Collaboration, R. Aaij et al., “Differential branching fraction and angular analysis of $\Lambda^{0}_{b}\rightarrow\Lambda\mu^{+}\mu^{-}$ decays,” JHEP 06 (2015) 115, arXiv:1503.07138 [hep-ex]. [Erratum: JHEP 09, 145 (2018)].
  * [153] T. Blake, S. Meinel, and D. van Dyk, “Bayesian Analysis of $b\to s\mu^{+}\mu^{-}$ Wilson Coefficients using the Full Angular Distribution of $\Lambda_{b}\to\Lambda(\to p\,\pi^{-})\mu^{+}\mu^{-}$ Decays,” Phys. Rev. D 101 no. 3, (2020) 035023, arXiv:1912.05811 [hep-ph].
  * [154] W. Detmold and S. Meinel, “$\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ form factors, differential branching fraction, and angular observables from lattice QCD with relativistic $b$ quarks,” Phys. Rev. D 93 no. 7, (2016) 074501, arXiv:1602.01399 [hep-lat].
  * [155] LHCb Collaboration, R. Aaij et al., “Measurement of the $\Lambda_{b}^{0}\to\Lambda(1520)\mu^{+}\mu^{-}$ Differential Branching Fraction,” Phys. Rev. Lett. 131 no. 15, (2023) 151801, arXiv:2302.08262 [hep-ex].
  * [156] S. Meinel and G. Rendon, “$\Lambda_{c}\to\Lambda^{*}(1520)$ form factors from lattice QCD and improved analysis of the $\Lambda_{b}\to\Lambda^{*}(1520)$ and $\Lambda_{b}\to\Lambda_{c}^{*}(2595,2625)$ form factors,” Phys. Rev. D 105 no. 5, (2022) 054511, arXiv:2107.13140 [hep-lat].
  * [157] Y. Amhis, M. Bordone, and M. Reboud, “Dispersive analysis of $\Lambda_{b}\to\Lambda(1520)$ local form factors,” JHEP 02 (2023) 010, arXiv:2208.08937 [hep-ph].
  * [158] HPQCD Collaboration, C. Bouchard, G. P. Lepage, C. Monahan, H. Na, and J. Shigemitsu, “Rare decay $B\to K\ell^{+}\ell^{-}$ form factors from lattice QCD,” Phys. Rev. D 88 no. 5, (2013) 054509, arXiv:1306.2384 [hep-lat]. [Erratum: Phys.Rev.D 88, 079901 (2013)].
  * [159] J. A. Bailey et al., “$B\to Kl^{+}l^{-}$ Decay Form Factors from Three-Flavor Lattice QCD,” Phys. Rev. D 93 no. 2, (2016) 025026, arXiv:1509.06235 [hep-lat].
  * [160] HPQCD Collaboration, W. G. Parrott, C. Bouchard, and C. T. H. Davies, “$B\to K$ and $D\to K$ form factors from fully relativistic lattice QCD,” Phys. Rev. D 107 no. 1, (2023) 014510, arXiv:2207.12468 [hep-lat].
  * [161] JLQCD Collaboration, K. Nakayama, T. Ishikawa, and S. Hashimoto, “Charmonium contribution to $B\to K\ell^{+}\ell^{-}$: testing the factorization approximation on the lattice,” PoS LATTICE2019 (2020) 062, arXiv:2001.10911 [hep-lat].
  * [162] B. Grinstein and D. Pirjol, “Exclusive rare $B\to K^{*}\ell^{+}\ell^{-}$ decays at low recoil: Controlling the long-distance effects,” Phys. Rev. D 70 (2004) 114005, arXiv:hep-ph/0404250.
  * [163] M. Beylich, G. Buchalla, and T. Feldmann, “Theory of $B\to K^{(*)}\ell^{+}\ell^{-}$ decays at high $q^{2}$: OPE and quark-hadron duality,” Eur. Phys. J. C 71 (2011) 1635, arXiv:1101.5118 [hep-ph].
  * [164] M. Beneke, T. Feldmann, and D. Seidel, “Systematic approach to exclusive $B\to Vl^{+}l^{-}$, $V\gamma$ decays,” Nucl. Phys. B 612 (2001) 25–58, arXiv:hep-ph/0106067.
  * [165] A. Khodjamirian, T. Mannel, and Y. M. Wang, “$B\to K\ell^{+}\ell^{-}$ decay at large hadronic recoil,” JHEP 02 (2013) 010, arXiv:1211.0234 [hep-ph].
  * [166] N. Gubernari, D. van Dyk, and J. Virto, “Non-local matrix elements in $B_{(s)}\to\\{K^{(*)},\phi\\}\ell^{+}\ell^{-}$,” JHEP 02 (2021) 088, arXiv:2011.09813 [hep-ph].
  * [167] N. Gubernari, M. Reboud, D. van Dyk, and J. Virto, “Improved theory predictions and global analysis of exclusive $b\to s\mu^{+}\mu^{-}$ processes,” JHEP 09 (2022) 133, arXiv:2206.03797 [hep-ph].
  * [168] S. Descotes-Genon, J. Matias, and J. Virto, “Understanding the $B\to K^{*}\mu^{+}\mu^{-}$ Anomaly,” Phys. Rev. D 88 (2013) 074002, arXiv:1307.5683 [hep-ph].
  * [169] D. Du, A. X. El-Khadra, S. Gottlieb, A. S. Kronfeld, J. Laiho, E. Lunghi, R. S. Van de Water, and R. Zhou, “Phenomenology of semileptonic B-meson decays with form factors from lattice QCD,” Phys. Rev. D 93 no. 3, (2016) 034005, arXiv:1510.02349 [hep-ph].
  * [170] LHCb Collaboration, R. Aaij et al., “Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett. 113 (2014) 151601, arXiv:1406.6482 [hep-ex].
  * [171] LHCb Collaboration, R. Aaij et al., “Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays,” JHEP 08 (2017) 055, arXiv:1705.05802 [hep-ex].
  * [172] LHCb Collaboration, R. Aaij et al., “Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett. 122 no. 19, (2019) 191801, arXiv:1903.09252 [hep-ex].
  * [173] LHCb Collaboration, R. Aaij et al., “Test of lepton universality with ${\Lambda}_{b}^{0}\to{pK}^{-}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-}$ decays,” JHEP 05 (2020) 040, arXiv:1912.08139 [hep-ex].
  * [174] LHCb Collaboration, R. Aaij et al., “Test of lepton universality in beauty-quark decays,” Nature Phys. 18 no. 3, (2022) 277–282, arXiv:2103.11769 [hep-ex]. [Addendum: Nature Phys. 19, (2023)].
  * [175] LHCb Collaboration, R. Aaij et al., “Tests of lepton universality using $B^{0}\to K^{0}_{S}\ell^{+}\ell^{-}$ and $B^{+}\to K^{*+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett. 128 no. 19, (2022) 191802, arXiv:2110.09501 [hep-ex].
  * [176] M. Algueró, A. Biswas, B. Capdevila, S. Descotes-Genon, J. Matias, and M. Novoa-Brunet, “To (b)e or not to (b)e: no electrons at LHCb,” Eur. Phys. J. C 83 no. 7, (2023) 648, arXiv:2304.07330 [hep-ph].
  * [177] A. Greljo, J. Salko, A. Smolkovič, and P. Stangl, “Rare b decays meet high-mass Drell-Yan,” JHEP 05 (2023) 087, arXiv:2212.10497 [hep-ph].
  * [178] E. Goudzovski et al., “Weak Decays of Strange and Light Quarks,” in Snowmass 2021.  arXiv:2209.07156 [hep-ex].
  * [179] A. J. Buras, D. Buttazzo, J. Girrbach-Noe, and R. Knegjens, “${K}^{+}\to{\pi}^{+}\nu\overline{\nu}$ and ${K}_{L}\to{\pi}^{0}\nu\overline{\nu}$ in the Standard Model: status and perspectives,” JHEP 11 (2015) 033, arXiv:1503.02693 [hep-ph].
  * [180] A. J. Buras, D. Buttazzo, and R. Knegjens, “$K\to\pi\nu\overline{\nu}$ and $\varepsilon$’/$\varepsilon$ in simplified new physics models,” JHEP 11 (2015) 166, arXiv:1507.08672 [hep-ph].
  * [181] KOTO Collaboration, J. K. Ahn et al., “Search for the $K_{L}\\!\to\\!\pi^{0}\nu\overline{\nu}$ and $K_{L}\\!\to\\!\pi^{0}X^{0}$ decays at the J-PARC KOTO experiment,” Phys. Rev. Lett. 122 no. 2, (2019) 021802, arXiv:1810.09655 [hep-ex].
  * [182] NA62 Collaboration, E. Cortina Gil et al., “Measurement of the very rare $K^{+}\to{\pi}^{+}\nu\overline{\nu}$ decay,” JHEP 06 (2021) 093, arXiv:2103.15389 [hep-ex].
  * [183] RBC, UKQCD Collaboration, N. H. Christ, X. Feng, A. Portelli, and C. T. Sachrajda, “Lattice QCD study of the rare kaon decay $K^{+}\to\pi^{+}\nu\bar{\nu}$ at a near-physical pion mass,” Phys. Rev. D 100 no. 11, (2019) 114506, arXiv:1910.10644 [hep-lat].
  * [184] RBC, UKQCD Collaboration, P. A. Boyle, F. Erben, J. M. Flynn, V. Gülpers, R. C. Hill, R. Hodgson, A. Jüttner, F. O. hÓgáin, A. Portelli, and C. T. Sachrajda, “Simulating rare kaon decays $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ using domain wall lattice QCD with physical light quark masses,” Phys. Rev. D 107 no. 1, (2023) L011503, arXiv:2202.08795 [hep-lat].
  * [185] F. Erben, V. Gülpers, M. T. Hansen, R. Hodgson, and A. Portelli, “Progress on the exploratory calculation of the rare Hyperon decay $\Sigma^{+}\to p\ell^{+}\ell^{-}$,” PoS LATTICE2022 (2023) 315, arXiv:2212.09595 [hep-lat].
  * [186] T. Blum et al., “Discovering new physics in rare kaon decays,” in Snowmass 2021.  arXiv:2203.10998 [hep-lat].
  * [187] M. Gorbahn and U. Haisch, “Charm Quark Contribution to $K_{L}\to\mu^{+}\mu^{-}$ at Next-to-Next-to-Leading,” Phys. Rev. Lett. 97 (2006) 122002, arXiv:hep-ph/0605203.
  * [188] N. H. Christ, X. Feng, L. Jin, C. Tu, and Y. Zhao, “Lattice QCD calculation of the two-photon contributions to $K_{L}\to\mu^{+}\mu^{-}$ and $\pi^{0}\to e^{+}e^{-}$ decays,” PoS LATTICE2019 (2020) 128.
  * [189] N. Christ, X. Feng, L. Jin, C. Tu, and Y. Zhao, “Lattice QCD Calculation of $\pi^{0}\to e^{+}e^{-}$ Decay,” Phys. Rev. Lett. 130 no. 19, (2023) 191901, arXiv:2208.03834 [hep-lat].
  * [190] E.-H. Chao, N. H. Christ, X. Feng, and L. Jin, “$K_{\rm L}\rightarrow\mu^{+}\mu^{-}$ from lattice QCD,” in 40th International Symposium on Lattice Field Theory.  12, 2023.  arXiv:2312.01224 [hep-lat].
  * [191] B.-L. Hoid, M. Hoferichter, and J. R. de Elvira, “Comparing phenomenological estimates of dilepton decays of pseudoscalar mesons with lattice QCD,” in 40th International Symposium on Lattice Field Theory.  12, 2023.  arXiv:2312.00520 [hep-lat].
  * [192] R. Hodgson, “Status of the exploratory calculation of the rare hyperon decay,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/270641/.
  * [193] S. Schacht and A. Soni, “Rare K decays off and on the lattice,” in 40th International Symposium on Lattice Field Theory.  12, 2023.  arXiv:2312.05527 [hep-ph].
  * [194] LHCb Collaboration, R. Aaij et al., “Observation of CP Violation in Charm Decays,” Phys. Rev. Lett. 122 no. 21, (2019) 211803, arXiv:1903.08726 [hep-ex].
  * [195] LHCb Collaboration, R. Aaij et al., “Measurement of the Time-Integrated CP Asymmetry in $D_{0}\to K^{-}K^{+}$ Decays,” Phys. Rev. Lett. 131 no. 9, (2023) 091802, arXiv:2209.03179 [hep-ex].
  * [196] A. Khodjamirian and A. A. Petrov, “Direct CP asymmetry in $D\to\pi^{-}\pi^{+}$ and $D\to K^{-}K^{+}$ in QCD-based approach,” Phys. Lett. B 774 (2017) 235–242, arXiv:1706.07780 [hep-ph].
  * [197] A. Pich, E. Solomonidi, and L. Vale Silva, “Final-state interactions in the CP asymmetries of charm-meson two-body decays,” Phys. Rev. D 108 no. 3, (2023) 036026, arXiv:2305.11951 [hep-ph].
  * [198] S. Schacht and A. Soni, “Enhancement of charm CP violation due to nearby resonances,” Phys. Lett. B 825 (2022) 136855, arXiv:2110.07619 [hep-ph].
  * [199] M. T. Hansen and S. R. Sharpe, “Multiple-channel generalization of Lellouch-Luscher formula,” Phys. Rev. D 86 (2012) 016007, arXiv:1204.0826 [hep-lat].
  * [200] M. Hansen, “Towards hadronic $D$ decays at the $SU(3)$ flavour symmetric point,” in 40th International Symposium on Lattice Field Theory.  https://indico.fnal.gov/event/57249/contributions/271297/.
  * [201] A. Di Canto and S. Meinel, “Weak Decays of $b$ and $c$ Quarks,” in Snowmass 2021.  arXiv:2208.05403 [hep-ex].