# On magnitudes of some CKM matrix elements

Zoltan Ligeti<EMAIL_ADDRESS>Ernest Orlando Lawrence Berkeley National
Laboratory,
University of California, Berkeley, CA 94720, USA

###### Abstract

This writeup follows the presentation at the Symposium, with emphasis on
topics and ideas discussed there. It is purposefully informal, not a review of
the field, and neither does it include a complete list of references. However,
I hope that readers might find some comments useful or amusing, and may
appreciate the challenges and reasons for excitement about recent progress and
future opportunities in flavor physics.

##  1 Introduction

I was asked to talk at this symposium, celebrating the 50th anniversary of the
Kobayashi–Maskawa theory Kobayashi:1973fv , about the determinations of the
magnitudes of elements of the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing
matrix Kobayashi:1973fv ; Cabibbo:1963yz . This talk only covers (some of)
those CKM elements that involve the third generation of quarks.

Given that much of the audience, as well as I, was not yet (or barely) alive
when the paper that this Symposium celebrates was written, I would like to
emphasize how stunning the timeline was. After the discovery of $CP$ violation
Christenson:1964fg  in 1964, Wolfenstein’s paper on the superweak model
Wolfenstein:1964ks  was nearly instantaneous (the PRL received dates are July
10 and August 31, 1964, respectively), whereas PTEP received the
Kobayashi–Maskawa manuscript Kobayashi:1973fv  on September 1, 1972. I am old
enough to remember people talking about excluding the superweak theory as a
goal for Belle and BaBar, and young enough to never have taken that seriously.
Imagine, those 8 years between 1964 and 1972 were just as if ATLAS
TheATLAScollaboration:2015mdt  and CMS CMS:2015dxe  had discovered a new
particle near 750 GeV in 2015, but all theory papers in the first 7 years had
turned out to be wrong, and the correct model had only been proposed in 2023.

While this talk is about magnitudes of CKM elements, it is important to
emphasize that testing the flavor sector of the standard model (SM), as a
whole, is a lot more interesting than any single measurement. It is the
combination of many measurements that gives often the most interesting
information, such as the constraints on beyond SM (BSM) contributions to
flavor-changing neutral-current processes. (Wolfenstein used to say that even
though he invented the Wolfenstein parameters Wolfenstein:1983yz , he did not
care what their values were, only whether their independent determinations
gave consistent results.) For the success of this program then, the interplay
between experimental and theoretical developments, seeking to optimize
theoretical cleanliness and experimental precision, has been crucial and also
a lot of fun.

Since a substantial part of this talk concerns semileptonic decays, it may be
amusing to note that semileptonic operators (to be precise, coefficients of
operators composed of $l\bar{l}q\bar{q}$ fields) account for 1053 (i.e., 42%)
of the 2499 parameters of the dimension-6 baryon and lepton number conserving
terms in the 3-generation SMEFT Alonso:2013hga  (558 $CP$-even and 495
$CP$-odd terms). And in the low energy effective theory below the weak scale,
there are 1944 semileptonic parameters (i.e., 54%) of the 3631 terms
Jenkins:2017jig  (1017 $CP$-even and 927 $CP$-odd).111I thank Aneesh Manohar
for help with adding (the correct) integers.

##  2 The Past

To proceed from the Kobayashi–Maskawa proposal to building asymmetric $B$
factories, and pursuing their spectacularly rich physics program, required
mixing angles and quark masses to have fortuitous values. (Technological
developments were also critical, of course.) The combination of the value of
$m_{b}$ (about half of the mass of the $\Upsilon$ resonance, discovered in
1977 E288:1977xhf ) and the smallness of $|V_{cb}|$ (discovered via the long
lifetime of $b$-flavored hadrons in 1983 Fernandez:1983az ; Lockyer:1983ev )
resulted in $b$ hadrons propagating long enough distances to be measured with
detectors developed in the 1980s and 90s. (If $|V_{cb}|$ was as large as the
Cabibbo angle, $|V_{us}|$, then it would have been impossible to make time-
dependent $CP$ violation measurements at Belle and BaBar in the early 2000s.)
Another critical ingredient was the much larger value of $m_{t}$ than
anticipated in the 1980s, enabling the ARGUS discovery of $B^{0}$ –
$\,\overline{\\!B}{}{}^{0}$ oscillation ARGUS:1987xtv . The measurement of the
ratio of $B^{0}$ decays after mixing vs. unmixed decays, $r=0.21\pm 0.08$,
implied that $m_{t}$ was much greater than the direct bound in 1987,
$m_{t}>23\,{\rm GeV}$. The comparable time scales of oscillation and decay,
$\Delta m/\Gamma=0.77$, and constraining $\Delta\Gamma\ll\Gamma$ for $B^{0}$
mixing were also important. Finally, CLEO CLEO:1989gsk  and ARGUS
ARGUS:1989eae  established a nonzero value for $|V_{ub}|$ in 1989; if it were
zero, the CKM matrix could not contain a physical $CP$ violating phase, as the
Jarskog invariant is proportional to $|V_{ub}|$. These were all crucial to
make (asymmetric) $B$ factories exciting and plausible to pursue.

The mixing of neutral mesons plays a special role in determining CKM matrix
elements and constraining BSM scenarios. This goes back to the successful
prediction of $m_{c}$ from $K^{0}$ – $\,\overline{\\!K}{}{}^{0}$ mixing
Vainshtein:1973md ; Gaillard:1974hs . Since the 1970s, $\Delta m_{K}$ and $CP$
violation in the kaon sector have provided some of the strongest constraints
on BSM scenarios, because flavor-changing neutral currents (FCNCs) are most
strongly suppressed between the 1st and 2nd generation of quarks. In the
$B^{0}$ and $B_{s}^{0}$ systems the large top mass implies that the GIM
mechanism Glashow:1970gm  does not yield strong suppressions, and FCNC decay
rates and $CP$ violating observables can be much larger than in $K^{0}$ or
$D^{0}$ meson processes. After the initial measurement of $B_{s}$ oscillation
by CDF CDF:2006imy , by now $\Delta m_{B_{s}}=(17.7656\pm 0.0057)\,{\rm
ps}^{-1}$ LHCb:2021moh  is one of the most precise CKM measurements, yielding
a $1.6\times 10^{-4}$ relative uncertainty for
$|V_{tb}V_{ts}|\,f_{B_{s}}^{2}B_{B_{s}}$. The LHCb Collaboration’s textbook
measurement of the time dependence of unmixed (“right-sign”) and mixed
(“wrong-sign“) $B_{s}$ decays is shown in Figure 1. If the uncertainty of
$f_{B_{s}}^{2}B_{B_{s}}$ from lattice QCD did not dominate, the uncertainty of
$|V_{tb}V_{ts}|$ would compete with that of $|V_{ud}|$ (the latter may
actually be underestimated Seng ).

Figure 1: Time-dependence of “right-sign” (blue) and “wrong-sign” (red)
$B_{s}$ decays LHCb:2021moh .

The evaluation of FCNC $B$ decay amplitudes at the (multi-)loop level also
started around 1986–87, including two-loop calculations of $b\to s\gamma$
Grinstein:1987vj ; Bertolini:1986th ; Deshpande:1987nr , which is important
both for its sensitivity to BSM physics, and, in the SM, to $|V_{tb}V_{ts}|$.
Remarkably, for $m_{t}\simeq 170\,{\rm GeV}$ (unknown at that time), the
prediction $\Gamma(B\to X_{s}\gamma)/\Gamma(B\to X_{c}e\bar{\nu})\approx
0.003$ Grinstein:1987vj  is very close to the current central value
Misiak:2015xwa . In the presence of unavoidable experimental phase space cuts,
the $b$-quark distribution function in the $B$ meson Bigi:1993ex ;
Neubert:1993um  becomes important to describe the photon spectrum, which is
largely BSM physics independent Kapustin:1995nr , can be extracted from the
measured decay distributions Bernlochner:2020jlt , and plays a crucial role in
the determination of $|V_{ub}|$ from inclusive decays.

The development of heavy quark symmetry Isgur:1989vq ; Isgur:1990yhj  and
heavy quark effective theory (HQET) Georgi:1990um  also started around 1990,
as well as putting the calculation of inclusive $B$ decay rates on rigorous
foundations Chay:1990da . These have played crucial roles since then, for the
determinations of $|V_{cb}|$ and $|V_{ub}|$, which are essential to constrain
BSM physics by comparing measurements dominated by tree-level processes in the
SM with loop-mediated FCNC measurements. In particular, the uncertainty of
$|V_{cb}|$ is a large part of the uncertainty of the SM predictions for
$\epsilon_{K}$, ${\cal B}(K\to\pi\nu\bar{\nu})$, ${\cal
B}(B_{d,s}\to\ell^{+}\ell^{-})$, etc. The measurement of $|V_{ub}|$ is
especially important in constraining BSM physics, because it constitutes the
dominant uncertainty in the side of the unitarity triangle (a graphical
representation of the
$V_{ud}\,V_{ub}^{*}+V_{cd}\,V_{cb}^{*}+V_{td}\,V_{tb}^{*}=0$ relation; see
Figure 3) opposite to the currently best measured angle,
$\beta(\equiv\phi_{1})$. Moreover, the same theoretical tools can be used to
analyze inclusive and exclusive FCNC decays, with complementary sensitivity to
BSM physics, such as those mediated by $b\to s\gamma$, $b\to
s\ell^{+}\ell^{-}$, and $b\to s\nu\bar{\nu}$ transitions. (A recent
illustration of how such combinations work is the semi-inclusive analysis of
$B\to X_{s}\ell^{+}\ell^{-}$ in the high-$q^{2}$ region, where the best
constraints are derived Isidori:2023unk  by comparing the data with a similar
Belle measurement of $B\to X_{u}\ell\bar{\nu}$ Belle:2021ymg , using the
method of Ref. Ligeti:2007sn .)

Since the development of HQET, the determination of $|V_{cb}|$ from exclusive
$B\to D^{(*)}\ell\bar{\nu}$ decays has relied on measuring the
$q^{2}=(p_{\ell}+p_{\nu})^{2}$ distribution near the so-called zero recoil
point, $w=1$ [$w=(m_{B}^{2}+m_{D^{(*)}}^{2}-q^{2})/(2m_{B}m_{D^{(*)}})$],
where the $D^{(*)}$ is at rest in the restframe of the decaying $B$ meson. The
rates can be schematically written as

${{\rm d}\Gamma(B\to D^{(*)}\ell\bar{\nu})\over{\rm d}w}=(\mbox{calculable
terms})\,|V_{cb}|^{2}\begin{cases}(w^{2}-1)^{1/2}\,{\cal
F}_{*}^{2}(w),&\mbox{for }B\to D^{*},\\\ (w^{2}-1)^{3/2}\,{\cal
F}^{2}(w),&\mbox{for }B\to D\,.\end{cases}$ (1)

Both ${\cal F}(w)$ and ${\cal F}_{*}(w)$ are equal to the Isgur-Wise function
in the $m_{b,c}\gg\Lambda_{\rm QCD}$ limit, and ${\cal F}_{(*)}(1)=1$ is the
basis for a model-independent determination of $|V_{cb}|$. There are
calculable corrections in powers of $\alpha_{s}(m_{c,b})$, as well as terms
suppressed by $\Lambda_{\rm QCD}/m_{c,b}$, which can only be parametrized, and
that is where hadronic uncertainties enter. Schematically,

$\displaystyle{\cal F}_{*}(1)$ $\displaystyle=$ $\displaystyle
1_{\mbox{\footnotesize(Isgur-
Wise)}}+c_{A}(\alpha_{s})+{0_{\mbox{\footnotesize(Luke)}}\over m_{c,b}}+{\cal
O}(\Lambda_{\rm QCD}^{2}/m_{c,b}^{2})\,,$ $\displaystyle{\cal F}(1)$
$\displaystyle=$ $\displaystyle 1_{\mbox{\footnotesize(Isgur-
Wise)}}+c_{V}(\alpha_{s})+{\cal O}(\Lambda_{\rm QCD}/m_{c,b})\,.$ (2)

The absence of the ${\cal O}(\Lambda_{\rm QCD}/m_{c,b})$ term for $B\to
D^{*}\ell\bar{\nu}$ at zero recoil is a consequence of Luke’s theorem
Luke:1990eg . Calculating corrections to the heavy quark limit in these decays
is a vast subject. To achieve percent level precision, lattice QCD appears to
be the only tool El-Khadra . FLAG FlavourLatticeAveragingGroupFLAG:2021npn
quotes ${\cal F}_{*}(1)=0.904\pm 0.012$ and ${\cal F}(1)=1.054\pm 0.009$,
while the experimental data are $|V_{cb}\,{\cal F}_{*}(1)|=(34.77\pm
0.36)\times 10^{-3}$ and $|V_{cb}\,{\cal F}(1)|=(41.26\pm 0.97)\times 10^{-3}$
HFLAV:2022esi . An important ingredient of the analysis which only received
full attention in recent years, after the publication of unfolded measurements
Belle:2017rcc ; Belle:2018ezy , is related to the functional form of fitting
the data and extrapolating the rate to $w=1$ (where phase space vanishes). In
the experimental determinations of $|V_{cb}|$ before 2017, the model-
independent Boyd-Grinstein-Lebed (BGL) parametrization Boyd:1995cf ;
Boyd:1995sq ; Boyd:1997kz  was supplemented by QCD sum rule results
Neubert:1992wq ; Neubert:1992pn ; Ligeti:1993hw  for the subleading ${\cal
O}(\Lambda_{\rm QCD}/m_{c,b})$ Isgur-Wise functions, to arrive at a fit
prescription with fewer parameters Caprini:1997mu  (with hard to quantify
model dependence), which all prior BaBar and Belle analyses used. Recently two
approaches were proposed to incorporate some ${\cal O}(\Lambda_{\rm
QCD}^{2}/m_{c}^{2})$ corrections Bordone:2019vic ; Bernlochner:2022ywh . The
question how to fix the number of BGL expansions parameters that are fit from
a measurement of $B\to D^{*}\ell\bar{\nu}$ distributions with a given
statistics, is subject to debate Bernlochner:2019ldg ; Gambino:2019sif ;
Simons:2023wfg , and different choices in recent experimental analyses
Belle:2023bwv ; Belle-II:2023okj  impact the extracted value of $|V_{cb}|$
significantly. Thus, in future measurements a detailed validation of the
choices made with toy Monte Carlo studies seems essential. There are
additional open questions related to (i) moderate tensions between HQET
predictions, lattice QCD results and form factor measurements; (ii) the role
of the $D^{**}$ states, etc. ZLoct23 .

For the determination of $|V_{cb}|$ from inclusive $B\to X_{c}\ell\bar{\nu}$
decay, instead of identifying all particles, final state hadrons that can be
produced by the strong interaction are summed over, subject to constraints
determined by short-distance physics, e.g., the energy of the charged lepton.
Although hadronization is nonperturbative, it occurs on much longer distance
(and time) scales than the underlying weak decay, and therefore one can use an
operator product expansion (OPE) to calculate the decay rate in a systematic
expansion. The leading order corresponds to the free $b$-quark decay rate, and
perturbative and nonperturbative corrections can systematically be accounted
for. The first few dominant nonperturbative parameters can be extracted by
fitting differential decay distributions in the same $B\to X_{c}\ell\bar{\nu}$
decays, an approach that has been pursued for over twenty years now (mainly
using fits in the so-called $1S$ Bauer:2004ve ; Bauer:2002sh  and kinetic
Alberti:2014yda ; Gambino:2004qm  mass schemes). Recently 3-loop
$\alpha_{s}^{3}$ corrections were calculated Fael:2020njb ; Fael:2020tow ;
Bordone:2021oof , $\alpha_{s}$ corrections are known up to ${\cal
O}(\Lambda_{\rm QCD}^{3}/m^{3})$ Mannel:2021zzr , and lattice QCD calculations
of inclusive rates are being developed Barone:2023tbl .

Given that determinations of $|V_{cb}|$ from exclusive and inclusive decays
have been in persistent tension (at an inconclusive, couple of sigma level),
it is possible that a detailed understanding of the composition of the
inclusive rate in terms of the sum over exclusive modes will be needed, in
order to understand the resolution of this tension (whether on the theoretical
or experimental side, or a combination). The upcoming much larger data sets
should allow this to be explored and hopefully resolved.

Determinations of $|V_{cb}|$, that are currently less precise, can also be
obtained from the measurement of $B_{s}\to D_{s}^{(*)}\mu\bar{\nu}$ decay
LHCb:2020cyw . In addition, semileptonic decays to $\tau$ leptons measured in
$B\to D^{(*)}\tau\bar{\nu}$ and related modes are also sensitive to
$|V_{cb}|$. The most precise data involving $\tau$ leptons are the
$|V_{cb}|$-independent ratios, ${\cal B}(B\to D^{(*)}\tau\bar{\nu})/{\cal
B}(B\to D^{(*)}\ell\bar{\nu})$ measured by BaBar, Belle, and LHCb for the last
ten years. If the current, approximately $3\,\sigma$ HFLAV:2022esi , hint of
lepton non-universality prevails, the determination of $|V_{cb}|$ becomes more
complicated, since one would need to know the size of the BSM contributions in
the $e$ and $\mu$ modes. It is due to heavy quark symmetry that the SM
predictions for these ratios of rates is under good theoretical control
Bernlochner:2017jka ; Bernlochner:2022ywh , even independent of lattice QCD.

##  3 The Present

There was little to say about the present at the time of this Symposium, in
February 2023, since neither LHC nor Belle was taking data, during LHC’s LS2
and the Belle II upgrade. However, ChatGPT was fairly new, and I was amused by
the poem it composed about Kobayashi, Maskawa, and Belle II, shown in Fig 2.
(I said that this was the first and last time I’d use ChatGPT in a talk, which
I hope to continue to obey.)

Figure 2: ChatGPT: write a poem about Kobayashi, Maskawa, and Belle II
(1/29/2023).

Possibly the most concise illustration of the achievements of the last two
decades is what used to be the “money plot” for BaBar and Belle, summarizing
the main constraints on the unitarity triangle, shown in Figure 3 (left plot).
These measurements of primarily $B_{u,d}$ decays (some of them further refined
by LHCb) established the Kobayashi–Maskawa mechanism as the dominant source of
$CP$ violation in flavor changing processes in the quark sector. Although this
SM CKM fit shows impressive and nontrivial consistency, the implications of
the agreement among different measurements are often overstated. The
determination of CKM parameters only using dominantly tree-level decays yields
the plot on the right in Figure 3. The constraints shown in the left plot in
Figure 3 but not on the right, all involve neutral meson mixing. Allowing BSM
contributions, FCNC processes may be altered significantly, and additional
parameters related to $CP$ and flavor violation arise, so the fits become less
constraining. The conclusion of such analyses, some of which will be discussed
below, is that ${\cal O}(20\%)$ BSM contributions to most FCNC processes are
still allowed. As a result of LHCb measurements in Run 1 and 2, the
constraints in the $B_{s}$ sector have become comparably strong to constraints
in the $B_{d}$ sector.

Figure 3: Main constraints on the unitarity triangle (left), and their subset
dominated by tree-level SM processes (right). The colored regions show 95% CL.
(From Ref. Charles:2004jd .)

##  4 The Future

In the next one to two decades, the LHC experiments Bediaga:2018lhg ;
Cerri:2018ypt , Belle II Belle-II:2018jsg , BES III BESIII:2022mxl  and their
planned and possible upgrades will increase $b$-, $c$-, and $\tau$-decay data
sets by almost two orders of magnitude. If the FCC (future circular collider)
is built, it will have an extremely rich flavor physics program FCC:2018byv ;
Grossman:2021xfq , studying $10^{12}$ $b$-flavored hadrons from the decay of
$Z$ bosons, in a clean experimental environment. An intriguing aspect of the
FCC-$ee$ flavor physics program is that in its $W^{+}W^{-}$ phase, the decay
of $10^{8}$ $W$-s will allow a determination of $|V_{cb}|$ with about 0.3%
precision Azzurri , independent of $|V_{cb}|$ measurements in $B$ decays.

The measurement of $|V_{ub}|$ from inclusive semileptonic decays belongs in my
mind largely to the future, because the existing determinations all include
some unaccounted model dependence, which we know how to eliminate. (The
determination of $|V_{ub}|$ from exclusive decays seems fully in the domain of
lattice QCD, at the relevant precision El-Khadra .) Similar to the case of
$B\to X_{s}\gamma$ discussed above, also in the analysis of inclusive $B\to
X_{u}\ell\bar{\nu}$ decays, significant experimental cuts on the phase space
are required to suppress the much larger $B\to X_{c}\ell\bar{\nu}$
backgrounds. As a result, instead of local HQET matrix elements, the
nonperturbative quantities which determine the rate are the $b$-quark
distribution functions in the $B$ meson Bigi:1993ex ; Neubert:1993um  (except
for the $q^{2}>(m_{B}-m_{D})^{2}$ cut Bauer:2000xf ). At leading order in
$\Lambda_{\rm QCD}/m_{b}$ there is only one such function, which is identical
for $B\to X_{s}\gamma$ and $B\to X_{u}\ell\bar{\nu}$, whereas at subleading
order, several different distribution functions occur. For $B\to
X_{u}\ell\bar{\nu}$, the consistent treatment of the shape function region
(where the hadronic final state is “jetty”) and the local OPE region is more
complicated than in the case of $B\to X_{s}\gamma$.

Just like the determination of $|V_{cb}|$ from inclusive decays relies on
extracting HQET matrix elements and $|V_{cb}|$ itself from the measurements of
$B\to X_{c}\ell\bar{\nu}$ absolute rates and distributions, the determination
of $|V_{ub}|$ from inclusive decays should utilize a similar strategy, fitting
$B\to X_{u}\ell\bar{\nu}$ rates and distributions (as well as the $B\to
X_{s}\gamma$ spectrum), to $b$-quark distribution functions and $|V_{ub}|$. To
develop such a self-consistent theoretical approach and fitting tool has been
the goal of SIMBA Bernlochner:2020jlt . Figure 4 shows a toy Monte Carlo study
(as an illustration, not using the state of the art theory yet), fitting the
$B\to X_{u}\ell\bar{\nu}$ charged lepton (left) and hadronic invariant mass
(right) spectra, with statistics corresponding to 75/ab Kerstin . The
sensitivity is actually underestimated, as it assumed uncertainties and
correlations similar to a BaBar full hadronic reconstruction analysis,
compared to which the Belle II hadronic tagging efficiency is already better.
This suggests that depending on the ability to constrain (linear combinations
of) subleading shape functions, an uncertainty on $|V_{ub}|$ at the few
percent level may be achievable at Belle II.

Figure 4: A (not state-of-the-art) Monte Carlo study with SIMBA, fitting the
charged lepton (left) and hadronic invariant mass (right) spectra, with
statistics corresponding to 75/ab Kerstin .

As a slight aside, the measurements of $B_{d,s}\to\mu^{+}\mu^{-}$ are usually
discussed for their BSM sensitivity, for good reasons. While ${\cal
B}(B_{s}\to\mu^{+}\mu^{-})=(2.95\pm 0.41)\times 10^{-9}$ HFLAV:2022esi  has
been measured by three LHC collaborations, the decay $B_{d}\to\mu^{+}\mu^{-}$,
which has a branching ratio around $10^{-10}$ in the SM, has not been
observed. Its uncertainty is not expected to reach 10% even by the end of HL-
LHC data taking. It is an example of a channel, whose interpretation will not
become theory limited in any planned experiment, and greater statistics would
always teach us about short-distance physics. In terms of an operator
analysis, its mass-scale sensitivity is comparable to $K\to\pi\nu\bar{\nu}$.
Amusingly, the theoretically cleanest determination of $|V_{ub}|$ I know of,
in principle, setting experimental realities aside, could come from the ratio
${\cal B}(B^{-}\to\ell\bar{\nu})/{\cal B}(B^{0}\to\mu^{+}\mu^{-})$ relying
only on perturbative QCD calculations and isospin.

###  4.1 Beyond SM sensitivity in $B$ – $\,\overline{\\!B}{}$ mixing

It has long been known that the mixing of neutral mesons is particularly
sensitive to BSM physics, and probe some of the highest scales (the smallness
of $\Delta m_{K}/m_{K}\approx 7\times 10^{-15}$ is known since the 1960s). In
a large classes of models, the dominant effect of BSM physics is to modify the
mixing amplitudes of neutral mesons, and tree-level decays are not affected.
In this case, the $3\times 3$ CKM matrix remains unitary, and the BSM effects
can be parametrized by two real parameters for each neutral meson system. The
mixing of $B_{q}^{0}$ mesons (where $q=d,s$) are simplest to analyze, as they
are dominated by short-distance physics. Writing the mixing amplitude as
$M_{12}^{q}=M_{12\,\rm(SM)}^{q}\,\big{(}1+h_{q}\,e^{2i\sigma_{q}}\big{)}$, one
has to redo the CKM fit including the effects of $h$ and $\sigma$ on the
observables. The resulting constraints on $h_{d}$ and $h_{s}$, the magnitudes
of the BSM contributions relative to the SM in $B_{d}$ and $B_{s}$ mixing,
respectively, are shown in the left plot in Figure 5, assuming independent BSM
contributions. This shows that order $(10-20)\%$ corrections to $M_{12}$ are
still allowed. (Evidence for $h_{q}\neq 0$ would rule out the SM, and the
black dot indicating the best-fit point in the left plot shows a slight pull.)
Similar conclusions apply to other neutral meson mixings, as well as many
other $\Delta F=1$ FCNC decays, such as $B\to X\gamma$, $B\to
X\ell^{+}\ell^{-}$, $B_{d,s}\to\ell^{+}\ell^{-}$, $K\to\pi\nu\bar{\nu}$, etc.
The same analysis, for a future time, using sensitivity projections for 50/ab
LHCb data and 50/ab Belle II data Charles:2020dfl  (referred there as “Phase
I”) are shown in the right plot in Figure 5. With even higher statistics
expected with the LHCb Upgrade II LHCb:2022ine  and a possible upgrade of
Belle II Forti:2022mti  (and especially FCC-$ee$), improving the determination
of $|V_{cb}|$ beyond current projections becomes crucial for this type of
analysis to continue to improve its sensitivity Charles:2020dfl . If new
developments result in theory uncertainties not becoming a limiting factor for
this type of analysis, then the statistical sensitivity could reach
Charles:2020dfl  $h_{d}\lesssim 0.02$ and $h_{s}\lesssim 0.01$ at 95% CL, with
the full LHCb Upgrade II and the possible Belle II upgrade datasets.

Figure 5: Constraints on the $h_{d}-h_{s}$ at present (left plot) and expected
future sensitivity assuming 50/ab LHCb data, and 50/ab Belle II data (right
plot). The colors indicate the confidence levels as shown, while the dashed
lines show $3\sigma$ limits. (From Ref. Charles:2020dfl .)

The correspondence between the constraints on $h_{d,s}$ and the scale of BSM
physics depends on the operator(s) that contribute. As the simplest example,
if new contributions add a term
$({C_{q}^{2}}/{\Lambda^{2}})\,(\bar{b}_{L}\gamma^{\mu}q_{L})^{2}$ to the same
operator that describes $B$ mixing in the SM, then one finds

$h_{q}\simeq\frac{|C_{q}|^{2}}{|V_{tb}^{*}\,V_{tq}|^{2}}\left(\frac{4.5\,{\rm
TeV}}{\Lambda}\right)^{2}\,.$ (3)

It is then straightforward to translate the constraints in Figure 5 to the
scale of BSM physics probed, and the summary of expected sensitivities are
shown in Table 1, for different structures. These sensitivities, even with SM-
like loop- and CKM-suppressed coefficients, are comparable to the scales
probed by the LHC. The factor of 1.5 – 2 increase in mass scale sensitivity at
Phase I compared to the present is greater than the improvement for many other
searches during the HL-LHC.

Couplings |  BSM loop |  Sensitivity for Summer 2019 [TeV] |  Phase I Sensitivity [TeV]
---|---|---|---
order |  $B_{d}$ mixing |  $B_{s}$ mixing |  $B_{d}$ mixing |  $B_{s}$ mixing
$|C_{ij}|=|V_{ti}V_{tj}^{*}|$ | tree level | 9 | 13 | 17 | 18
(CKM-like) | one loop | 0.7 | 1.0 | 1.3 | 1.4
$|C_{ij}|=1$ | tree level | $1\times 10^{3}$ | $3\times 10^{2}$ | $2\times 10^{3}$ | $4\times 10^{2}$
(no hierarchy) | one loop | 80 | 20 | $2\times 10^{2}$ | 30

Table 1: The scale of the
$({C_{q}^{2}}/{\Lambda^{2}})\,(\bar{b}_{L}\gamma^{\mu}q_{L})^{2}$ operator
probed (in TeV, at 95% CL) by $B_{d}$ and $B_{s}$ mixings at present and at
“Phase I” (described in the text) Charles:2020dfl . The impact of SM-like
hierarchy of couplings and/or loop suppression is shown.

##  5 Final Remarks

Pursuing ever more precise tests of the flavor sector remains exciting and
well motivated. If there is BSM physics below the few tens of TeV scale, which
we hope to probe directly in current and planned experiments, they cannot have
a generic flavor structure, as that is already ruled out by orders of
magnitudes. There is thus an unavoidable complementarity between direct
searches of BSM physics, and learning about their signatures (or lack of
signatures) in precision experiments. In a large class of scenarios one may
expect deviations from the SM close to the current experimental precision, so
there is significant discovery potential any time experimental sensitivities
improve substantially.

As in the past, theory will be important for the interpretation of the
measurements and for maximizing their sensitivity to BSM physics, and
experimental results will be essential inputs of theory considerations, and
triggers for new developments. In particular, the reduction of the
uncertainties of $|V_{cb}|$ and $|V_{ub}|$ are crucial. In each case, with new
measurements, the tension between inclusive and exclusive measurements will
hopefully shrink, and possibly yet unknown subtleties understood, and the
source of current tensions identified. They may lie either on the theory side
or on the experimental side, or a combination. In the case of the
determination of $|V_{ub}|$ from inclusive decays, we know how qualitatively
better measurements can be done, than those implemented so far. And the
determinations from exclusive semileptonic decays will improve with statistics
and better lattice QCD calculations. Both for $|V_{cb}|$ and $|V_{ub}|$,
measurements from two-body leptonic decays will also become available, and
will become competitive in precision, as statistics grows.

These experimental improvements need much larger data sets than available so
far, and there is a suite of measurements (which have been identified, and
will likely become further enlarged by future theory developments), in which
theoretical uncertainties will not limit probing short-distance physics.
However, I do not think anyone has seriously explored what the largest $B$
decay data sets may be, which would improve sensitivity to short-distance
physics.222I vividly remember Sanda-san at the Workshops on Higher Luminosity
$B$ Factory at Izu in 2003 Sanda , saying (as a joke? or not as a joke?) that
the question was not whether the design luminosity of the next generation $B$
factory should be $10^{35}$ or $10^{36}$, but rather if it should be $10^{53}$
or $10^{63}$. My understanding is that the sensitivity to possible signatures
of BSM physics would continue to improve with data sets well beyond all
upgrades contemplated at LHCb and Belle II, and even at the Tera-$Z$ run of
the FCC.

## Acknowledgments

I thank the organizers for the invitation to this very enjoyable Symposium,
and for their patience regarding the completion of this writeup. I thank Shoji
Hashimoto and Aneesh Manohar for helpful conversations. This work was
supported in part by the Office of High Energy Physics of the U.S. Department
of Energy under contract DE-AC02-05CH11231.

## References

  * (1) M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory of Weak Interaction,” Prog. Theor. Phys. 49 (1973) 652–657.
  * (2) N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Phys. Rev. Lett. 10 (1963) 531–533.
  * (3) J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, “Evidence for the $2\pi$ Decay of the $K_{2}^{0}$ Meson,” Phys. Rev. Lett. 13 (1964) 138–140.
  * (4) L. Wolfenstein, “Violation of CP Invariance and the Possibility of Very Weak Interactions,” Phys. Rev. Lett. 13 (1964) 562–564.
  * (5) ATLAS Collaboration, “Search for resonances decaying to photon pairs in 3.2 fb-1 of $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector,” (12, 2015)  , https://cds.cern.ch/record/2114853.
  * (6) CMS Collaboration, “Search for new physics in high mass diphoton events in proton-proton collisions at 13TeV,” (2015)  , https://cds.cern.ch/record/2114808.
  * (7) L. Wolfenstein, “Parametrization of the Kobayashi-Maskawa Matrix,” Phys. Rev. Lett. 51 (1983) 1945.
  * (8) R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, “Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology,” JHEP 04 (2014) 159, arXiv:1312.2014 [hep-ph].
  * (9) E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,” JHEP 03 (2018) 016, arXiv:1709.04486 [hep-ph].
  * (10) E288 Collaboration, S. W. Herb et al., “Observation of a Dimuon Resonance at 9.5-GeV in 400-GeV Proton-Nucleus Collisions,” Phys. Rev. Lett. 39 (1977) 252–255.
  * (11) E. Fernandez et al., “Lifetime of Particles Containing B Quarks,” Phys. Rev. Lett. 51 (1983) 1022.
  * (12) N. Lockyer et al., “Measurement of the Lifetime of Bottom Hadrons,” Phys. Rev. Lett. 51 (1983) 1316.
  * (13) ARGUS Collaboration, H. Albrecht et al., “Observation of $B^{0}$–$\,\overline{\\!B}{}{}^{0}$ mixing,” Phys. Lett. B 192 (1987) 245–252.
  * (14) CLEO Collaboration, R. Fulton et al., “Observation of B Meson Semileptonic Decays to Noncharmed Final States,” Phys. Rev. Lett. 64 (1990) 16–20.
  * (15) ARGUS Collaboration, H. Albrecht et al., “Observation of Semileptonic Charmless B Meson Decays,” Phys. Lett. B 234 (1990) 409–416.
  * (16) A. I. Vainshtein and I. B. Khriplovich, “Restrictions on masses of supercharged hadrons in the Weinberg model,” Pisma Zh. Eksp. Teor. Fiz. 18 (1973) 141–145.
  * (17) M. K. Gaillard and B. W. Lee, “Rare Decay Modes of the K-Mesons in Gauge Theories,” Phys. Rev. D 10 (1974) 897.
  * (18) S. L. Glashow, J. Iliopoulos, and L. Maiani, “Weak Interactions with Lepton-Hadron Symmetry,” Phys. Rev. D 2 (1970) 1285–1292.
  * (19) CDF Collaboration, A. Abulencia et al., “Observation of $B^{0}_{s}-\bar{B}^{0}_{s}$ Oscillations,” Phys. Rev. Lett. 97 (2006) 242003, arXiv:hep-ex/0609040.
  * (20) LHCb Collaboration, R. Aaij et al., “Precise determination of the $B_{s}^{0}$ – $\,\overline{\\!B}{}{}^{0}_{s}$ oscillation frequency,” Nature Phys. 18 no. 1, (2022) 1–5, arXiv:2104.04421 [hep-ex].
  * (21) C.-Y. Seng, “Talk at this symposium,”  2023\.  https://conference-indico.kek.jp/event/195/.
  * (22) B. Grinstein, R. P. Springer, and M. B. Wise, “Effective Hamiltonian for Weak Radiative B Meson Decay,” Phys. Lett. B 202 (1988) 138–144.
  * (23) S. Bertolini, F. Borzumati, and A. Masiero, “QCD Enhancement of Radiative b Decays,” Phys. Rev. Lett. 59 (1987) 180.
  * (24) N. G. Deshpande, P. Lo, J. Trampetic, G. Eilam, and P. Singer, “$B\to K^{*}\gamma$ and the Top Quark Mass,” Phys. Rev. Lett. 59 (1987) 183–185.
  * (25) M. Misiak et al., “Updated NNLO QCD predictions for the weak radiative B-meson decays,” Phys. Rev. Lett. 114 no. 22, (2015) 221801, arXiv:1503.01789 [hep-ph].
  * (26) I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev, and A. I. Vainshtein, “On the motion of heavy quarks inside hadrons: Universal distributions and inclusive decays,” Int. J. Mod. Phys. A 9 (1994) 2467–2504, arXiv:hep-ph/9312359.
  * (27) M. Neubert, “Analysis of the photon spectrum in inclusive $B\to X_{s}\gamma$ decays,” Phys. Rev. D 49 (1994) 4623–4633, arXiv:hep-ph/9312311.
  * (28) A. Kapustin and Z. Ligeti, “Moments of the photon spectrum in the inclusive $B\to X_{s}\gamma$ decay,” Phys. Lett. B 355 (1995) 318–324, arXiv:hep-ph/9506201.
  * (29) SIMBA Collaboration, F. U. Bernlochner, H. Lacker, Z. Ligeti, I. W. Stewart, F. J. Tackmann, and K. Tackmann, “Precision Global Determination of the $B\to X_{s}\gamma$ Decay Rate,” Phys. Rev. Lett. 127 no. 10, (2021) 102001, arXiv:2007.04320 [hep-ph].
  * (30) N. Isgur and M. B. Wise, “Weak decays of heavy mesons in the static quark approximation,” Phys. Lett. B 232 (1989) 113–117.
  * (31) N. Isgur and M. B. Wise, “Weak transition form factors between heavy mesons,” Phys. Lett. B 237 (1990) 527–530.
  * (32) H. Georgi, “An effective field theory for heavy quarks at low energies,” Phys. Lett. B 240 (1990) 447–450.
  * (33) J. Chay, H. Georgi, and B. Grinstein, “Lepton energy distributions in heavy meson decays from QCD,” Phys. Lett. B 247 (1990) 399–405.
  * (34) G. Isidori, Z. Polonsky, and A. Tinari, “Semi-inclusive $b\to s\bar{\ell}\ell$ transitions at high $q^{2}$,” Phys. Rev. D 108 no. 9, (2023) 093008, arXiv:2305.03076 [hep-ph].
  * (35) Belle Collaboration, L. Cao et al., “Measurement of Differential Branching Fractions of Inclusive ${B\to X_{u}\ell^{+}\,\nu_{\ell}}$ Decays,” Phys. Rev. Lett. 127 no. 26, (2021) 261801, arXiv:2107.13855 [hep-ex].
  * (36) Z. Ligeti and F. J. Tackmann, “Precise predictions for $B\to X_{s}l^{+}l^{-}$ in the large $q^{2}$ region,” Phys. Lett. B 653 (2007) 404–410, arXiv:0707.1694 [hep-ph].
  * (37) M. E. Luke, “Effects of subleading operators in the heavy quark effective theory,” Phys. Lett. B 252 (1990) 447–455.
  * (38) A. El-Khadra, “Talk at this symposium,”  2023\.  https://conference-indico.kek.jp/event/195/.
  * (39) 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].
  * (40) 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].
  * (41) Belle Collaboration, A. Abdesselam et al., “Precise determination of the CKM matrix element $\left|V_{cb}\right|$ with $\bar{B}^{0}\to D^{*\,+}\,\ell^{-}\,\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle,” arXiv:1702.01521 [hep-ex].
  * (42) 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)].
  * (43) C. G. Boyd, B. Grinstein, and R. F. Lebed, “Model independent extraction of $|V_{cb}|$ using dispersion relations,” Phys. Lett. B 353 (1995) 306–312, arXiv:hep-ph/9504235.
  * (44) C. G. Boyd, B. Grinstein, and R. F. Lebed, “Model independent determinations of $\bar{B}\to Dl\bar{\nu},D^{*}l\bar{\nu}$ form-factors,” Nucl. Phys. B 461 (1996) 493–511, arXiv:hep-ph/9508211.
  * (45) C. G. Boyd, B. Grinstein, and R. F. Lebed, “Precision corrections to dispersive bounds on form-factors,” Phys. Rev. D 56 (1997) 6895–6911, arXiv:hep-ph/9705252.
  * (46) M. Neubert, Z. Ligeti, and Y. Nir, “QCD sum rule analysis of the subleading Isgur-Wise form-factor $\chi_{2}(v\cdot v^{\prime})$,” Phys. Lett. B 301 (1993) 101–107, arXiv:hep-ph/9209271.
  * (47) M. Neubert, Z. Ligeti, and Y. Nir, “The Subleading Isgur-Wise form-factor $\chi_{3}(v\cdot v^{\prime})$ to order $\alpha_{s}$ in QCD sum rules,” Phys. Rev. D 47 (1993) 5060–5066, arXiv:hep-ph/9212266.
  * (48) Z. Ligeti, Y. Nir, and M. Neubert, “The Subleading Isgur-Wise form-factor $\xi_{3}(v\cdot v^{\prime})$ and its implications for the decays $\bar{B}\to D^{*}\ell\bar{\nu}$,” Phys. Rev. D 49 (1994) 1302–1309, arXiv:hep-ph/9305304.
  * (49) I. Caprini, L. Lellouch, and M. Neubert, “Dispersive bounds on the shape of anti-B —$>$ D(*) lepton anti-neutrino form-factors,” Nucl. Phys. B 530 (1998) 153–181, arXiv:hep-ph/9712417.
  * (50) M. Bordone, M. Jung, and D. van Dyk, “Theory determination of $\bar{B}\to D^{(*)}\ell^{-}\bar{\nu}$ form factors at $\mathcal{O}(1/m_{c}^{2})$,” Eur. Phys. J. C 80 no. 2, (2020) 74, arXiv:1908.09398 [hep-ph].
  * (51) F. U. Bernlochner, Z. Ligeti, M. Papucci, M. T. Prim, D. J. Robinson, and C. Xiong, “Constrained second-order power corrections in HQET: $R(D^{(*)})$, $|V_{cb}|$, and new physics,” Phys. Rev. D 106 no. 9, (2022) 096015, arXiv:2206.11281 [hep-ph].
  * (52) F. U. Bernlochner, Z. Ligeti, and D. J. Robinson, “N = 5, 6, 7, 8: Nested hypothesis tests and truncation dependence of $|V_{cb}|$,” Phys. Rev. D 100 no. 1, (2019) 013005, arXiv:1902.09553 [hep-ph].
  * (53) P. Gambino, M. Jung, and S. Schacht, “The $V_{cb}$ puzzle: An update,” Phys. Lett. B 795 (2019) 386–390, arXiv:1905.08209 [hep-ph].
  * (54) D. Simons, E. Gustafson, and Y. Meurice, “Self-consistent optimization of the $z$-Expansion for $B$ meson decays,” arXiv:2304.13045 [hep-ph].
  * (55) Belle Collaboration, M. T. Prim et al., “Measurement of differential distributions of $B\to D^{*}\ell\nu_{\ell}$ and implications on $|V_{cb}|$,” Phys. Rev. D 108 no. 1, (2023) 012002, arXiv:2301.07529 [hep-ex].
  * (56) Belle II Collaboration, I. Adachi et al., “Determination of $|V_{cb}|$ using $\,\overline{\\!B}{}{}^{0}\to D^{*+}\ell^{-}\bar{\nu}$ decays with Belle II,” Phys. Rev. D 108 no. 9, (2023) 092013, arXiv:2310.01170 [hep-ex].
  * (57) Z. Ligeti, “Talk at the Workshop on precision $|V_{cb}|$ measurements at Belle II,” 2023.  https://indico.belle2.org/event/9402/contributions/63857/attachments/25094/37246/kek23.pdf.
  * (58) C. W. Bauer, Z. Ligeti, M. Luke, A. V. Manohar, and M. Trott, “Global analysis of inclusive B decays,” Phys. Rev. D 70 (2004) 094017, arXiv:hep-ph/0408002.
  * (59) C. W. Bauer, Z. Ligeti, M. Luke, and A. V. Manohar, “B decay shape variables and the precision determination of $|V_{cb}|$ and $m_{b}$,” Phys. Rev. D 67 (2003) 054012, arXiv:hep-ph/0210027.
  * (60) A. Alberti, P. Gambino, K. J. Healey, and S. Nandi, “Precision Determination of the Cabibbo-Kobayashi-Maskawa Element $V_{cb}$,” Phys. Rev. Lett. 114 no. 6, (2015) 061802, arXiv:1411.6560 [hep-ph].
  * (61) P. Gambino and N. Uraltsev, “Moments of semileptonic B decay distributions in the 1/m(b) expansion,” Eur. Phys. J. C 34 (2004) 181–189, arXiv:hep-ph/0401063.
  * (62) M. Fael, K. Schönwald, and M. Steinhauser, “Relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of heavy quarks,” Phys. Rev. D 103 no. 1, (2021) 014005, arXiv:2011.11655 [hep-ph].
  * (63) M. Fael, K. Schönwald, and M. Steinhauser, “Third order corrections to the semileptonic $b\to c$ and the muon decays,” Phys. Rev. D 104 no. 1, (2021) 016003, arXiv:2011.13654 [hep-ph].
  * (64) M. Bordone, B. Capdevila, and P. Gambino, “Three loop calculations and inclusive $V_{cb}$,” Phys. Lett. B 822 (2021) 136679, arXiv:2107.00604 [hep-ph].
  * (65) T. Mannel, D. Moreno, and A. A. Pivovarov, “NLO QCD corrections to inclusive $b\rightarrow c\ell\bar{\nu}$ decay spectra up to $1/m_{Q}^{3}$,” Phys. Rev. D 105 no. 5, (2022) 054033, arXiv:2112.03875 [hep-ph].
  * (66) A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko, and R. Kellermann, “Approaches to inclusive semileptonic B(s)-meson decays from Lattice QCD,” JHEP 07 (2023) 145, arXiv:2305.14092 [hep-lat]. And references therein.
  * (67) 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].
  * (68) F. U. Bernlochner, Z. Ligeti, M. Papucci, and D. J. Robinson, “Combined analysis of semileptonic $B$ decays to $D$ and $D^{*}$: $R(D^{(*)})$, $|V_{cb}|$, and new physics,” Phys. Rev. D 95 no. 11, (2017) 115008, arXiv:1703.05330 [hep-ph]. [Erratum: Phys.Rev.D 97, 059902 (2018)].
  * (69) J. Charles, A. Hocker, H. Lacker, S. Laplace, F. R. Le Diberder, J. Malcles, J. Ocariz, M. Pivk, and L. Roos, “CP violation and the CKM matrix: Assessing the impact of the asymmetric $B$ factories,” Eur. Phys. J. C 41 no. 1, (2005) 1–131, arXiv:hep-ph/0406184. And updates at http://ckmfitter.in2p3.fr/.
  * (70) LHCb Collaboration, R. Aaij et al., “Physics case for an LHCb Upgrade II – Opportunities in flavour physics, and beyond, in the HL-LHC era,” arXiv:1808.08865 [hep-ex].
  * (71) A. Cerri et al., “Report from Working Group 4: Opportunities in Flavour Physics at the HL-LHC and HE-LHC,” CERN Yellow Rep. Monogr. 7 (2019) 867–1158, arXiv:1812.07638 [hep-ph].
  * (72) Belle II Collaboration, W. Altmannshofer et al., “The Belle II Physics Book,” PTEP 2019 no. 12, (2019) 123C01, arXiv:1808.10567 [hep-ex]. [Erratum: PTEP 2020, 029201 (2020)].
  * (73) BESIII Collaboration, H. B. Li et al., “Physics in the $\tau$-charm Region at BESIII,” in 2022 Snowmass Summer Study.  4, 2022.  arXiv:2204.08943 [hep-ex].
  * (74) FCC Collaboration, A. Abada et al., “FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1,” Eur. Phys. J. C 79 no. 6, (2019) 474.
  * (75) Y. Grossman and Z. Ligeti, “Theoretical challenges for flavor physics,” Eur. Phys. J. Plus 136 no. 9, (2021) 912, arXiv:2106.12168 [hep-ph].
  * (76) P. Azzurri, “Flavor tagging in $W$ decays.”. Talk at the 4th FCC Physics and Experiments Workshop, Nov. 2020, https://indico.cern.ch/event/932973/contributions/4059403/attachments/2140815/3607142/azzurriFCCeeWHF.pdf.
  * (77) C. W. Bauer, Z. Ligeti, and M. E. Luke, “A Model independent determination of $|V_{ub}|$,” Phys. Lett. B 479 (2000) 395–401, arXiv:hep-ph/0002161.
  * (78) K. Tackmann, “Extracting $|V_{ub}|$ and $B\to X_{s}\gamma$ from global fits.”. Talk at the 4th SuperB Collaboration Meeting, La Biodola (Isola d’Elba) Italy, Jun. 2012, https://agenda.infn.it/event/4880/contributions/55588/attachments/39553/46670/tackmann-superb-elba-31may12.pdf.
  * (79) 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].
  * (80) LHCb Collaboration, “Future physics potential of LHCb,”. https://inspirehep.net/literature/2071541.
  * (81) Belle II Collaboration, F. Forti, “Snowmass Whitepaper: The Belle II Detector Upgrade Program,” in Snowmass 2021.  3, 2022.  arXiv:2203.11349 [hep-ex].
  * (82) A. I. Sanda, “Talk at the 5th Workshops on Higher Luminosity $B$ Factory,” Sept., 2003.  https://belle.kek.jp/superb/workshop/2003/HL05/.