# Testing Lepton Flavor Universality with Pion, Kaon, Tau, and Beta Decays

Douglas Bryman<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of British Columbia, Vancouver, BC, Canada V6T 1Z1  TRIUMF, Vancouver, BC,
Canada V6T 2A3  Vincenzo Cirigliano<EMAIL_ADDRESS>Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545  Andreas Crivellin
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland  CERN Theory Division, CH–1211 Geneva 23,
Switzerland  Gianluca Inguglia<EMAIL_ADDRESS>Institute of High
Energy Physics, Austrian Academy of Sciences, Vienna, Austria, 1050

###### Abstract

We present an overview of searches for violation of lepton flavor universality
with focus on low energy precision probes using pions, kaons, tau leptons, and
nuclear beta decays. The current experimental results are reviewed, the
theoretical status within the context of the Standard Model is summarized, and
future prospects (both experimental and theoretical) are discussed. We review
the implications of these measurements for physics beyond the Standard Model
by performing a global model-independent fit to modified $W$ couplings to
leptons and four-fermion operators. We also discuss new physics in the context
of simplified models and review Standard Model extensions with focus on those
that can explain a possible deviation from unitarity of the Cabibbo-Kobayashi-
Maskawa quark mixing matrix.

††preprint: CERN-TH-2021-184, LA-UR-21-30608, PSI-PR-21-25, ZU-TH 54/21

###### Contents

  1. I Introduction
  2. II Standard Model Theory and Observables
    1. II.1 Light Meson decays
      1. II.1.1 Pion Decays
      2. II.1.2 Kaon Decays
    2. II.2 Beta Decays and CKM Unitarity
    3. II.3 Tau Decays
  3. III Beyond the SM (BSM) Analysis
    1. III.1 Effective Field Theory
      1. III.1.1 Modified $W\ell\nu$ couplings
      2. III.1.2 4-lepton Operators
      3. III.1.3 2-quark-2-lepton Operators
      4. III.1.4 $SU(2)_{L}$ Gauge Invariance
    2. III.2 Simplified NP models
      1. III.2.1 $W^{\prime}$ boson
      2. III.2.2 Vector-like Leptons
      3. III.2.3 Singly charged $SU(2)_{L}$ singlet scalar
      4. III.2.4 Scalar $SU(2)_{L}$ Triplet
      5. III.2.5 $SU(2)_{L}$ Neutral Vector Boson ($Z^{\prime}$):
      6. III.2.6 Leptoquark
      7. III.2.7 Charged Higgs
  4. IV Future Prospects
    1. IV.1 Pion and Kaon Experiments
    2. IV.2 Tau Experiments
    3. IV.3 Theory
  5. V Conclusions

##  I Introduction

While no particles or interactions beyond those of the Standard Model (SM)
have been observed so far, intriguing hints for the violation of lepton flavor
universality (LFU) have accumulated in recent years.111Note that interesting
hints for new scalar particles have emerged recently at the LHC [1, 2, 3].  In
particular, the measurements of the ratios of branching ratios (Br)
$R(D^{(*)})$=Br[$B\to D^{(*)}\tau\nu_{\tau}$]/Br[$B\to D^{(*)}\ell\nu_{\ell}$]
[4, 5, 6] , where $\ell=\mu,e$, and $R(K^{(*)})$=Br[$B\to
K^{(*)}\mu^{+}\mu^{-}$]$/$Br[$B\to K^{(*)}e^{+}e^{-}$] [7, 8, 9] deviate from
the SM expectation by more than $3\sigma$ [10, 11, 12, 13, 14] and $4\sigma$
[15, 16, 17, 18], respectively.222Even though the ratios $R(D)$ and $R(D^{*})$
point towards violation of $\mu-\tau$ LFU motivating new physics (NP) effects,
in the observables discussed in this review we do not include them. The NP
effects required are so large that these ratios cannot be explained by
modified $W\ell\nu$ couplings which are more stringently constrained by $\tau$
decays. Therefore, effects in 2-quark-2-lepton operators are required for
explaining $R(D)$ and $R(D^{*})$ which in general have no direct correlations
with the tests of LFUV discussed in this review, unless a flavor symmetry is
assumed. However, some of these scenarios give rise to large radiative
corrections to $\tau$ decays, such that they are excluded by low energy probes
of LFUV [19]. In addition, anomalous magnetic moments $(g-2)_{\ell}$
($\ell=e,\mu,\tau$) of charged leptons are intrinsically related to LFU
violation as they are chirality flipping quantities. Here, there is the long-
standing discrepancy in $(g-2)_{\mu}$ of $4.2\sigma$ [20, 21, 22] which can be
considered as a hint of LFU violation (LFUV), since, if compared to
$(g-2)_{e}$, the bound from the latter on flavor blind new physics (NP) is
much more stringent. In addition, there is a hint for LFUV in the difference
of the forward-backward asymmetries ($\Delta A_{\rm FB}$) in $B\to
D^{*}\mu\nu$ vs $B\to D^{*}e\nu$ [23, 24]. As another possible indication of
LFUV, CMS observed an excess in non-resonant di-electron pairs with respect to
di-muons [25]. Furthermore, the possible deficit in first-row unitarity of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix, known as the Cabibbo angle anomaly
(CAA) can also be viewed as a sign of LFUV [26, 27].

The connection of the CAA and LFUV can be seen as follows. The determination
of $V_{ud}$ from $\beta$ decays, which is most relevant for a possible
explanation of the CAA, is affected by a modified $W\mu\nu$ coupling [27].
Importantly, a modification of the $W\mu\nu$ couplings, if not compensated by
an effect in $We\nu$, would also affect, for example, the ratios of decay
rates $R^{\pi}_{e/\mu}=\frac{\Gamma(\pi\to e\nu)}{\Gamma(\pi\to\mu\nu)}$ or
$R^{\tau}_{e/\mu}=\frac{\Gamma(\tau\to
e\nu\overline{\nu})}{\Gamma(\tau\to\mu\nu\overline{\nu})}$ which provide the
best tests of LFU. In fact, recent global fits to electroweak (EW) observables
and tests of LFU show a preference for a value of $R^{\pi}_{e/\mu}$ smaller
than its SM expectation [26, 28]. Furthermore, $R(K^{*})$ can be correlated to
$R^{\pi}_{e/\mu}$ [29] and a combined explanation of the deficit in the first-
row CKM unitarity and the CMS excess in di-electrons even predicts that
$R^{\pi}_{e/\mu}$ should be smaller than its SM value [30].

These considerations provide additional motivation for us to review, summarize
and re-examine the different searches for LFUV in the charged current with
focus on pion, kaon, tau, and beta decays. In the next section we will discuss
the experimental and theoretical status of these processes. We will then
consider the impact on NP searches, first in a model independent way by a
global analysis of modified $W\ell\nu$ coupling and four-fermion operators,
and then by considering different NP models with focus on the ones that can
explain the CAA in Sec. III.333Here we will focus on models with heavy NP
components such that the effective Lagrangian only contains SM fields.
However, light but massive right-handed neutrinos [31, 32, 33, 34], Majorons
[35], as well as light Dark Matter candidates [36, 37] can also have an impact
on the tests of LFUV studied in this review; these effects are searched for in
pion, kaon, and $\tau$ decay experiments [38, 39, 40]. We then give an outlook
for future experimental and theoretical prospects in Sec. IV before we
conclude in Sec. V.

##  II Standard Model Theory and Observables

###  II.1 Light Meson decays

The ratios of the decay rates

$R^{P}_{e/\mu}=\Gamma[P\to
e\bar{\nu}_{e}(\gamma)]/\Gamma[P\to\mu\bar{\nu}_{\mu}(\gamma)]\,,$ (1)

with $P=\pi,K$, provide some of the most stringent tests of LFU of the SM
gauge interactions. In the SM, the decay rates $\Gamma[P\to
e\bar{\nu}_{e}(\gamma)]$ are helicity-suppressed due to the $V-A$ structure of
the charged current. Moreover, their ratios can be calculated with an
extraordinary precision at the $10^{-4}$ level [41, 42, 43, 44] because, to a
first approximation, the strong interaction dynamics cancel out in the ratio
$R^{P}_{e/\mu}$ and the hadronic structure dependence appears only through EW
corrections. Due to these features and the precise experimental measurements,
the ratios $R^{P}_{e/\mu}$ are very sensitive probes of all SM extensions that
induce non-universal corrections to $W$-lepton couplings as well as
$\bar{e}\nu\bar{u}d$, $\bar{e}\nu\bar{u}s$ operators, in particular, if they
generate a pseudo-scalar current or induced scalar current [45].

The most recent theoretical calculations of $R^{P}_{e/\mu}$ [43, 44] are based
on Chiral Perturbation Theory (ChPT), the low-energy effective field theory
(EFT) of QCD [46, 47, 48], generalized to include virtual photons and light
charged leptons [49]. This framework provides a controlled expansion of the
decay rates in terms of a power-counting scheme characterized by the
dimensionless ratio $Q\sim m_{\pi,K,\mu}/\Lambda_{\chi}$, with
$\Lambda_{\chi}$ $\sim 4\pi F_{\pi}$ $\sim 1.2$ ${\rm GeV}$ (where
$F_{\pi}\simeq 92.4$ MeV is the pion decay constant), and the electromagnetic
coupling $e$. In this setup one can write

$\displaystyle R^{P}_{e/\mu}$ $\displaystyle=$
$\displaystyle\bar{R}^{P}_{e/\mu}\,\Bigg{[}1+\Delta_{e^{2}Q^{0}}^{P}+\Delta_{e^{2}Q^{2}}^{P}+\Delta_{e^{2}Q^{4}}^{P}+...+\Delta^{P}_{e^{4}Q^{0}}{+...}\Bigg{]}\,,$
(2)

with

$\displaystyle\bar{R}^{P}_{e/\mu}$ $\displaystyle=$
$\displaystyle\frac{m_{e}^{2}}{m_{\mu}^{2}}\left(\frac{m_{P}^{2}-m_{e}^{2}}{m_{P}^{2}-m_{\mu}^{2}}\right)^{2}\,.$
(3)

Here we have kept all the terms needed to reach an uncertainty of $\sim
10^{-4}$ for the ratio. The leading electromagnetic corrections
$\Delta_{e^{2}Q^{0}}^{P}$ corresponds to the point-like approximation for
pions and kaons, and their expressions are well known [50]. The hadronic
structure dependence first appears through the correction
$\Delta_{e^{2}Q^{2}}^{P}\sim(\alpha/\pi)(m_{P}/\Lambda_{\chi})^{2}$, which
features both the calculable double-chiral logarithms and an a priori unknown
low-energy coupling constant (LEC), which was estimated in large-$N_{C}$ QCD
($N_{C}$ being the number of colors) [43, 44] and found to contribute
negligibly to the error budget.

####  II.1.1 Pion Decays

In the pion case ($P=\pi^{\pm}$), one usually defines the ratio to be fully
photon-inclusive such that it is infrared safe. As a consequence, one has to
include in $R^{P}_{e/\mu}$ terms arising from the structure dependent
contribution to $\pi\to\ell\bar{\nu}_{\ell}\gamma$ [51], that are formally of
$O(e^{2}Q^{4})$, but are not helicity suppressed and behave as
$\Delta^{P}_{e^{2}Q^{4}}\sim(\alpha/\pi)\,(m_{P}/\Lambda_{\chi})^{4}\,(m_{P}/m_{e})^{2}$.
Finally, at the level of uncertainty considered, one needs to include higher
order corrections in $\alpha$, namely $\Delta^{P}_{e^{4}Q^{0}}$. The leading
logarithmic correction $\Delta^{P}_{e^{4}Q^{0},LL}=(7/2)(\alpha/\pi\log
m_{\mu}/m_{e})^{2}$ has been calculated in Ref. [41] and the effect of sub-
leading contributions was estimated in Ref. [43] as $(\alpha/\pi)^{2}\log
m_{\mu}/m_{e}\sim 0.003\%$. Numerically one finds
$\Delta_{e^{2}Q^{0}}^{\pi}=-3.929\%$, $\Delta_{e^{2}Q^{2}}^{\pi}=0.053(11)\%$,
$\Delta_{e^{2}Q^{4}}^{\pi}=0.073(3)\%$, and
$\Delta_{e^{4}Q^{0}}^{(\pi)}=0.055(3)\%$, which leads to the SM expectation
444Due to a larger uncertainty estimate in $\Delta_{e^{4}Q^{0}}^{\pi}$ namely
$\Delta_{e^{4}Q^{0}}^{\pi}=0.055(10)\%$, Ref. [52] quotes a final result of
$R{(\rm SM)}^{\pi}_{e/\mu}=(1.2352\pm 0.0002)\times 10^{-4}$.

$R{(\rm SM)}^{\pi}_{e/\mu}=(1.2352\pm 0.0001)\times 10^{-4}\,.$ (4)

We reiterate here that (i) this prediction includes structure-dependent hard-
bremsstrahlung corrections to $\Gamma(\pi^{+}\rightarrow e^{+}\nu(\gamma))$
which are not helicity suppressed; and (ii) the dominant uncertainty of the SM
prediction arises from a low-energy constant in chiral perturbation theory,
followed by the non-leading logarithmic corrections of $O(\alpha^{2})$.

The most accurate measurement of $R^{\pi}_{e/\mu}$ was reported by the TRIUMF
PIENU collaboration [53],

$R({\rm Exp})^{\pi}_{e/\mu}=(1.2344\pm 0.0023(\text{stat})\pm
0.0019(\text{syst}))\times 10^{-4},$ (5)

at the 0.24% precision level. The PDG [54] average including previous
experiments done at TRIUMF [53, 55, 56] and PSI [57] is

$R({\rm\overline{Exp}})^{\pi}_{e/\mu}=(1.2327\pm 0.0023)\times 10^{-4}.$ (6)

The comparison between theory and experiment given in Eq. (4) and Eq. (6)
provides a stringent test of the $e$–$\mu$ universality of the weak
interaction. We chose to express the results in terms of the effective
couplings $A_{\ell}$ multiplying the low-energy charged current contact
interaction

$L_{\rm
CC}=A_{\ell}\bar{u}\gamma^{\mu}P_{L}d\bar{\nu}_{\ell}\gamma_{\mu}P_{L}\ell\,,$
(7)

where $P_{L}\equiv(1-\gamma_{5})/2$. In the SM at tree level the couplings are
given by $A_{\ell}=-2\sqrt{2}G_{F}V_{ud}$ and thus satisfy LFU, i.e.
$A_{\ell}/A_{\ell^{\prime}}=1$. The measurement of $R^{\pi}_{e/\mu}$ results
in

$\left(\frac{A_{\mu}}{A_{e}}\right)_{R^{\pi}_{e/\mu}}=1.0010\pm 0.0009\,,$ (8)

which is in excellent agreement with the SM expectation. A deviation from
$A_{\ell}/A_{\ell^{\prime}}=1$ can originate from various mechanisms. In the
literature it is common to interpret deviations from
$A_{\ell}/A_{\ell^{\prime}}=1$ in terms of flavor-dependent couplings
$g_{\ell}$ of the $W$-boson to the leptonic current, in which case
$A_{\ell}\propto g_{\ell}$. We discuss this scenario in detail in Sec.
III.1.1. We note that in the context of modified $W$-boson couplings LFU
tested with $R^{\pi}_{e/\mu}$ probes the couplings of a longitudinally
polarized $W$-boson whereas tests using purely leptonic reactions like
$\tau\to\ell\nu_{\tau}\nu_{\ell}$ ($\ell=e,\mu$) test the couplings of
transversely polarized $W$-boson and are thus complementary.

####  II.1.2 Kaon Decays

LFU can also be tested using the ratios

$\displaystyle R^{K}_{e/\mu}$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left[K^{+}\rightarrow
e^{+}\nu(\gamma)\right]}{\Gamma\left[K^{+}\rightarrow\mu^{+}\nu(\gamma)\right]}\,,\textrm{and}$
(9) $\displaystyle R^{K\to\pi}_{e/\mu}$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left[K\rightarrow\pi
e\nu(\gamma)\right]}{\Gamma\left[K\rightarrow\pi\mu\nu(\gamma)\right]}\,,$
(10)

where for $R^{K\to\pi}_{e/\mu}$ both neutral and charged kaons decays, e.g.
$K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ and $K^{\pm}\to\pi^{0}\ell^{\pm}\nu$, are
used.

The calculation of $R^{K}_{e/\mu}$ is similar to the one of $R^{\pi}_{e/\mu}$
described in the previous section. An important difference concerns the
definition of the infrared-safe decay rate, which requires including part of
the radiative decay mode. The radiative amplitude is the sum of the inner
bremsstrahlung ($T_{IB}$) component of $O(eQ)$ and a structure dependent
($T_{\rm SD}$) component of $O(eQ^{3})$ [51]. While the experimental
definition of $R_{e/\mu}^{(\pi)}$ is fully inclusive, the one for
$R^{K}_{e/\mu}$ includes the effect of $T_{\rm IB}$ in
$\Delta_{e^{2}Q^{0}}^{(K)}$ (dominated by soft photons) and excludes the
effect of $T_{\rm SD}$. With this definition one finds
$\Delta_{e^{2}Q^{0}}^{K}=-3.786\%$, $\Delta_{e^{2}Q^{2}}^{K}=0.135(11)\%$,
$\Delta_{e^{2}Q^{4}}^{K}=0$, and $\Delta_{e^{4}Q^{0}}^{K}=0.055(3)\%$, and the
SM expectation is [43, 44]

$R\text{(SM)}^{K}_{e/\mu}=(2.477\pm 0.001)\times 10^{-5}\,,$ (11)

where the final uncertainty accounts for higher order chiral corrections of
expected size $\Delta_{e^{2}Q^{2}}\times m_{K}^{2}/(4\pi F_{\pi})^{2}$.

The PDG [54] average of previous measurements done by the NA62 [58] and KLOE
[59] experiments is

$R(\overline{\rm Exp})^{K}_{e/\mu}=(2.488\pm 0.009)\times 10^{-5}.$ (12)

The comparison of theory and experiment given in Eqns. (11) and (12)
corresponds to a test of $e$–$\mu$ universality

$\left(\frac{A_{\mu}}{A_{e}}\right)_{R^{K}_{e/\mu}}=0.9978\pm 0.0018.$ (13)

The analogous LFU test based on the ratios $R^{K\to\pi}_{e/\mu}$ has been
discussed by the Flavianet collaboration [60]. For a given neutral or charged
initial state kaon, the Fermi constant, $V_{us}$, short-distance radiative
corrections, and the hadronic form factor at zero momentum transfer cancel out
when taking the ratio $R^{K\to\pi}_{e/\mu}$. Therefore, in the SM this ratio
is entirely determined by phase space factors and long-distance radiative
corrections [61, 62, 63, 64]. The ratios for $K_{L}$ and $K^{\pm}$ were found
to be consistent leading to the following values for $A_{\mu}/A_{e}$ [60, 65,
66]:

$\displaystyle\begin{aligned}
\left(\dfrac{A_{\mu}}{A_{e}}\right)_{R^{K_{L}\to\pi}_{e/\mu}}&=1.0022\pm
0.0024\,,\textrm{and}\\\
\left(\dfrac{A_{\mu}}{A_{e}}\right)_{R^{K^{\pm}\to\pi^{\pm}}_{e/\mu}}&=0.9995\pm
0.0026\,,\end{aligned}$ (14)

with the average for $K_{\ell 3}$ decays

$\left(\frac{A_{\mu}}{A_{e}}\right)_{R^{K\to\pi}_{e/\mu}}=1.0009\pm 0.0018\,.$
(15)

The numbers given above correspond to the recent analysis of Ref. [66], which
uses experimental input from Ref. [60] (updated in Ref. [65] with reduced
errors in the charged modes) and theoretical input on $K_{\ell 3}$ radiative
corrections from Refs. [63, 64], which incorporates a new analysis of $K_{e3}$
modes with reduced uncertainties [64].

Note that $\mu$-$e$ universality can also be determined from $B$ decays such
as ${\rm Br}[B\to D^{*}\mu\nu]/{\rm Br}[B\to D^{*}e\nu]$. Even though the
relative precision at the % level [67, 68, 69] is not competitive with the
ones obtained from kaon and pion decays, these measures of LFUV are
interesting in light of the anomalies in $R(D^{(*)})$ and $\Delta A_{FB}$ [68,
23, 24] as they test different 4-fermion operators.

###  II.2 Beta Decays and CKM Unitarity

The observables testing LFUV discussed so far involve ratios of purely
leptonic or semileptonic meson decays with an electron or muon in the final
state. While considering ratios of (semi) leptonic decay rates offers
theoretical advantages (e.g. the elements $V_{ud}$ and $V_{us}$ of the
Cabibbo-Kobayashi-Maskawa (CKM) [70, 71] matrix, part of the radiative
corrections, and hadronic matrix elements cancel), the high-precision study of
absolute semileptonic decay rates can also uncover LFUV effects. For example,
in the context of corrections to the $W\to\ell\nu_{\ell}$ vertex, the
semileptonic transition $d(s)\to ue\bar{\nu}_{e}$ ($d(s)\to
u\mu\bar{\nu}_{\mu}$) is sensitive to corrections to the muon (electron)
coupling [72, 73, 27](see Ref.[74] for a discussion within supersymmetric
models). In absolute decay rates, these BSM LFUV corrections contaminate the
extraction of the CKM elements $V_{ud}$ and $V_{us}$ from measured decay
rates. This means that beta decays and the study of CKM unitarity are
intertwined with the study of LFUV [27]. In light of this connection, we
briefly summarize the status of first-row CKM unitarity tests. We will discuss
the implications for LFUV BSM interactions in Sec. III.

Unitarity of the CKM matrix [70, 71] implies $\Delta_{\rm
CKM}\equiv|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}-1=0$, where $V_{ud}$,
$V_{us}$, $V_{ub}$ represent the mixing of up with down, strange, and beauty
quarks, respectively. In practice $|V_{ub}|^{2}<10^{-5}$ can be neglected and
CKM unitarity reduces to the original Cabibbo universality, with the
identifications $V_{ud}=\cos\theta_{C}$ and $V_{us}=\sin\theta_{C}$, where
$\theta_{C}$ is the Cabibbo angle [70]. The determination of $V_{uD}$
($D=d,s$) from various hadronic weak decays $h_{i}\to h_{f}\ell\nu_{\ell}$
($\ell=e,\mu$) relies on the following schematic formula for the decay rate
$\Gamma$,

$\Gamma=G_{F}^{2}\times|V_{uD}|^{2}\times|M_{\rm had}|^{2}\times(1+\delta_{\rm
IB}+\delta_{\rm RC})\times F_{\rm kin}\,,$ (16)

where $G_{F}$ is the Fermi constant extracted from muon decay and $F_{\rm
kin}$ is a phase space factor. Theoretical input comes in the form of (i) the
hadronic matrix elements of the weak current, $M_{\rm had}$, usually
calculated in the isospin limit of QCD (in which up and down quark masses are
equal and electromagnetic interactions are turned off); and (ii) small %-level
corrections, $\delta_{\rm IB,RC}$, due to strong isospin-breaking (IB) and
electromagnetic radiative corrections (RCs) induced by the exchange of virtual
photons and the emission of real photons, characterized by the small expansion
parameters $\epsilon_{\rm IB}\sim(m_{u}-m_{d})/\Lambda_{\rm QCD}$ and
$\epsilon_{\rm EM}\sim\alpha/\pi$, respectively, where $\alpha\sim 1/137$ is
the electromagnetic fine structure constant.

Currently, as shown in Fig. 1, the knowlegde of $V_{ud}$ is dominated by
nuclear $0^{+}\to 0^{+}$ decays where the most recent survey [75] of
experimental and theoretical input leads to $V_{ud}=0.97373(31)$. This
incorporates a reduction in the uncertainty on the so-called inner radiative
corrections [76, 77] and an increase in uncertainty due to nuclear-structure
dependent effects with input from Refs. [78, 79, 80]. 555Note that the
$V_{ud}$ value quoted in the PDG [54] does not yet reflect the increased error
in the nuclear-structure dependent radiative corrections and therefore has an
uncertainty $\delta V_{ud}=0.00014$.

Thanks to higher precision measurements of the lifetime [81] and beta
asymmetry [82] (see Ref. [83] for a recent review), neutron decay is becoming
competitive with super-allowed beta decays concerning the precision with which
$V_{ud}$ can be extracted. Using the PDG average for the neutron lifetime
(including a scale factor $S=1.6$ to account for tensions among experimental
measurements)666The Particle Data Group excludes the beam lifetime
measurements from the current ‘PDG average’, quoting $\tau_{n}=879.4(6)s$.
This would change to $\tau_{n}=879.6(8)s$ if one included the beam lifetime
measurement result. and the post-2002 PDG average777We follow here the
analysis described in the PDG review “$V_{ud}$, $V_{us}$, the Cabibbo Angle,
and CKM unitarity”[54]. A recent comprehensive analysis of beta decays can be
found in Ref. [84]. Adopting input from Ref. [84] would lead to negligible
changes in our global fit presented below. determinations of the axial
coupling $g_{A}=1.2762(5)$ [54] leads to
$V_{ud}=0.97338(33)_{\tau}(32)_{g_{A}}(10)_{RC}=0.97338(47)$, with errors
originating from the lifetime $\tau_{n}$, $g_{A}$, and radiative corrections
(RC) [54], respectively. Ongoing and planned neutron experiments aim to reduce
the uncertainty in $\tau_{n}$ and $g_{A}$ by a factor of few (see [85] and
references therein), which will put the extraction of $V_{ud}$ from neutron
decay at the same precision level as superallowed nuclear beta decays. Future
prospects of improving the extraction of $V_{ud}$ from pion beta decay are
discussed in Sec.IV.

The most precise value of $V_{us}$ is extracted from $\Gamma(K\to\pi\ell\nu)$
while $R_{A}\equiv\Gamma(K\to\mu\nu)/\Gamma(\pi\to\mu\nu)$ provides currently
the most precise determination of $V_{us}/V_{ud}$ [86]. A comprehensive
discussion of the experimental and theoretical input up to 2010 can be found
in Ref. [87] and references therein. Since then, experimental input on the
$K^{\pm}$ branching ratios and form-factor parameters has been updated, as
reviewed in Ref. [65], while the most recent theoretical input on the hadronic
matrix elements can be found in Ref. [88] and references therein. Radiative
corrections are included according to Refs, [63, 64]. With this input one
obtains [87, 66] $V_{us}=0.2231(6)$ from $K_{\ell 3}$ decays and
$V_{us}/V_{ud}=0.2313(5)$ from $R_{A}$. Recently it has been pointed out [89]
that $V_{us}/V_{ud}$ can also be obtained through the ratio of vector channel
decays
$R_{V}\equiv\Gamma(K\to\pi\ell\nu(\gamma))/\Gamma(\pi^{+}\to\pi^{0}e^{+}\nu(\gamma))$,
leading to $V_{us}/V_{ud}=0.22908(87)$, with uncertainty dominated by the pion
beta decay width, but nonetheless within a factor of two compared to the
$R_{A}$ determination. Future prospects on this front are discussed in Sect.
IV. Finally, $V_{us}$ can also be extracted from inclusive and exclusive
semileptonic decays of the $\tau$ lepton, with final state hadrons carrying
strangeness quantum numbers. This leads to a somewhat lower and less precise
value $V_{us}=0.2221(13)$ (see Ref.[54] and references therein).

Fig. 1 summarizes graphically the results on $V_{ud}$ and $V_{us}$ discussed
so far and reveals that while nuclear and neutron decay lead to a consistent
picture for $V_{ud}$, tensions exist among current determinations of $V_{us}$
($K_{\ell 3}$ vs. $K_{\ell 2}$ and $K$ vs. $\tau$ lepton). Moreover, an
overall tension with CKM unitarity is also apparent. A global fit leads to
$V_{ud}=0.97357(27)$ and $V_{us}=0.22406(34)$, with the one-sigma ellipse
represented in Fig. 1 in yellow, implying

$\Delta_{\rm CKM}=(-19.5\pm 5.3)\times 10^{-4}\,,$ (17)

a 3.7-$\sigma$ effect. Due to the tension in the input data, the $\chi^{2}$
per degree of freedom is 2.8, corresponding to a scale factor of $S=1.67$
under the assumption that there is no new physics effect. In Sec. III we will
discuss the implications of the current tensions for LFUV interactions.

Figure 1: Summary of constraints on $V_{ud}$ and $V_{us}$ (assuming the
Standard Model hypothesis) from nuclear, nucleon, meson, and $\tau$ lepton
decays. For each constraint, the one-sigma uncertainty on $V_{us}$ or $V_{ud}$
is given in parenthesis (see text for details). The one-sigma ellipse from a
global fit (with $\chi^{2}/{\rm d.o.f.}=2.8$), depicted in yellow, corresponds
to $V_{ud}=0.97357(27)$ and $V_{us}=0.22406(34)$, implying $\Delta_{\rm
CKM}=|V_{ud}|^{2}+|V_{us}|^{2}-1=(-19.5\pm 5.3)\times 10^{-4}$.

###  II.3 Tau Decays

Tests of LFU can also be obtained comparing different $\tau$ decay rates to
those of the muon or the pion (kaon). For the $\tau$ we have semileptonic as
well as purely leptonic decays at our disposal. While the former can only test
$\tau-\mu$ universality efficiently, the latter allow us to assess $\mu-e$,
$\tau-e$ and $\tau-\mu$ universality. If one defines

$\displaystyle R^{\tau}_{\mu/e}$ $\displaystyle=$ $\displaystyle\frac{\rm
Br[\tau^{-}\to\mu^{-}\bar{\nu_{\mu}}\nu_{\tau}]}{\rm Br[\tau^{-}\to
e^{-}\bar{\nu_{e}}\nu_{\tau}]}\,,$ (18) $\displaystyle
R^{\tau\pi(K)}_{\tau/\mu}$ $\displaystyle=$ $\displaystyle\frac{\rm
Br[\tau\to\pi(K)\nu_{\tau}]}{\rm Br[\pi(K)\to\mu\nu_{\mu}]}\,,$ (19)
$\displaystyle R^{\tau}_{\tau/\mu}$ $\displaystyle=$ $\displaystyle\frac{\rm
Br[\tau^{-}\to e^{-}\bar{\nu_{e}}\nu_{\tau}]}{\rm Br[\mu^{-}\to
e^{-}\bar{\nu_{e}}\nu_{\mu}]}\,,\textrm{and}$ (20) $\displaystyle
R^{\tau}_{\tau/e}$ $\displaystyle=$ $\displaystyle\frac{\rm
Br[\tau^{-}\to\mu^{-}\bar{\nu_{\mu}}\nu_{\tau}]}{\rm Br[\mu^{-}\to
e^{-}\bar{\nu_{e}}\nu_{\mu}]}\,,$ (21)

the LFU ratios can then be expressed in terms of experimentally measured rates
and theoretical input. For $\mu-e$ universality one has

$\displaystyle\left(\frac{A_{\mu}}{A_{e}}\right)_{\tau}=\sqrt{R^{\tau}_{\mu/e}\frac{f(m_{e}^{2}/m_{\tau}^{2})}{f(m_{\mu}^{2}/m_{\tau}^{2})}}\,,$
(22)

with $f(x)=-8x+8x^{3}-x^{4}-12x^{2}\log x$. The above expression receives
radiative corrections of $O(\alpha/\pi)\times(m_{\mu}/m_{\tau})^{2}$ [90],
which are hence suppressed. For $\tau-\mu$ universality one has 888 In the
case of purely leptonic decays, we write the LFU test in terms of the ratios
$A_{\ell}/A_{\ell^{\prime}}\equiv
A_{\ell\ell^{{}^{\prime\prime}}}/A_{\ell^{{}^{\prime}}\ell^{{}^{\prime\prime}}}$
with
$L=A_{\ell^{\prime}\ell}\bar{\ell}^{\prime}\gamma^{\mu}P_{L}\nu_{\ell^{\prime}}\bar{\nu}_{\ell}\gamma_{\mu}P_{L}\ell$.

$\displaystyle\left(\frac{A_{\tau}}{A_{\mu}}\right)_{h}=\frac{1-m_{\mu}^{2}/m_{h}^{2}}{1-m_{h}^{2}/m_{\tau}^{2}}\sqrt{R^{\tau
h}_{\tau/\mu}\frac{2m_{h}m_{\mu}^{2}\tau_{h}}{(1+\delta_{h})m_{\tau}^{3}\tau_{\tau}}},$
(23)

where $h=\pi,K$. An alternative method to test $\tau-\mu$ universality,
similar to the $\mu-e$ case, compares the electronic and muonic decay rates
and can be expressed as

$\displaystyle\left(\frac{A_{\tau}}{A_{\mu}}\right)_{\tau}=\sqrt{R^{\tau}_{\tau/\mu}\frac{\tau_{\mu}}{\tau_{\tau}}\frac{m_{\mu}^{2}}{m_{\tau}^{3}}(1+\delta_{W})(1+\delta_{\gamma})}\,.$
(24)

In the above equations $m_{e,\mu,\tau}$ are the masses of $e$, $\mu$, and
$\tau$, $\tau_{\tau,h}$ are the lifetimes of the particles $\tau$ and $h$, and
$\delta_{h,W,\gamma}$ are the weak and electromagnetic radiative corrections
(see Ref. [90] and references therein for details). Experimentally, these
tests have been carried out at $B$-factories where, at the nominal center-of-
mass energy of 10.58 GeV/c2, thanks to a cross section of 0.919 nb, these
machines are ”$\tau$-Factories” de facto that produce large numbers of $\tau$
pairs.

Both the BaBar and the CLEO Collaborations performed the LFU tests according
to Eq. (22) [91] and Eq. (23) [92], while only CLEO performed the measurement
according to Eq. (24). In the reaction $e^{+}e^{-}\to\tau^{+}\tau^{-}$ at a
$B$-Factory one can use the decay of the $\tau^{+}$ to tag and study the
$\tau^{-}$(and vice versa). Typically one uses either the so-called 3$\times$1
$\tau$ topology, with the decay
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu_{\tau}}$ as a tag and then a study
on the other $\tau^{-}$ is performed, or the so-called 1$\times$1 topology in
which both taus decay with one prong (lepton or hadron) and a neutrino. While
the latest BaBar measurement only focused on the 3$\times$1 topology, the
latest study from the CLEO collaborations also used the 1$\times$1.

Tests of LFU are precise measurements for which, in addition to sizable
amounts of data, one needs to control systematic effects when determining the
branching ratios. In the most recent results from BaBar, for example,
$R^{\tau}_{\mu/e}$ and $\left(\frac{A_{\mu}}{A_{e}}\right)_{\tau}$ have been
determined with a precision of $0.4\%(0.16\%_{\rm stat}\bigoplus 0.36\%_{\rm
sys})$ and $0.2\%$ [91], respectively, where the leading systematic
uncertainty (0.32$\%$) originated from particle identification; similarly,
$R^{\pi}_{\tau/\mu}$ and $\left(\frac{A_{\tau}}{A_{\mu}}\right)_{\pi}$ were
determined with a precision of $0.63\%(0.14\%_{\rm stat}\bigoplus 0.61\%_{\rm
sys})$ and $0.57\%$ where the dominant systematic source originated again from
particle identification.

The results from BaBar and CLEO, are also used to obtain the latest HFLAV
combination which includes 176 measurements and 89 constraints in $\tau$
processes [93]; for purely leptonic $\tau$ decays, these are

$\displaystyle\left(\frac{A_{\tau}}{A_{\mu}}\right)_{\tau}$ $\displaystyle=$
$\displaystyle 1.0010\pm 0.0014\,,$ (25)
$\displaystyle\left(\frac{A_{\tau}}{A_{e}}\right)_{\tau}$ $\displaystyle=$
$\displaystyle 1.0029\pm 0.0014\,,\;{\rm and}$ (26)
$\displaystyle\left(\frac{A_{\mu}}{A_{e}}\right)_{\tau}$ $\displaystyle=$
$\displaystyle 1.0018\pm 0.0014\,.$ (27)

During the preparation of this manuscript, new values of ($A_{\tau}/A_{\mu}$)h
($h=\pi,K$), where the radiative corrections were computed including the
lightest multiplets of spin-one heavy states in ChPT, were given in Ref. [94];
these new values are

$\displaystyle\left(\frac{A_{\tau}}{A_{\mu}}\right)_{\pi}$ $\displaystyle=$
$\displaystyle 0.9964\pm 0.0038\,,\;{\rm and}$ (28)
$\displaystyle\left(\frac{A_{\tau}}{A_{\mu}}\right)_{K}$ $\displaystyle=$
$\displaystyle 0.9857\pm 0.0078\,,$ (29)

with the following correlation coefficients [93]

$\begin{matrix}(A_{\tau}/A_{\mu})_{\tau}&1\\\
(A_{\tau}/A_{e})_{\tau}&0.51&1\\\ (A_{\mu}/A_{e})_{\tau}&-0.50&0.49&1\\\
(A_{\tau}/A_{\mu})_{\pi}&0.23&0.25&0.02&1\\\
(A_{\tau}/A_{\mu})_{K}&0.11&0.10&-0.01&0.06&1\\\
&(A_{\tau}/A_{\mu})_{\tau}&(A_{\tau}/A_{e})_{\tau}&(A_{\mu}/A_{e})_{\tau}&(A_{\tau}/A_{\mu})_{\pi}&(A_{\tau}/A_{\mu})_{K}\end{matrix}$
(30)

and with 100$\%$ correlation between $(A_{\tau}/A_{\mu})_{\tau}$,
$(A_{\tau}/A_{e})_{\tau}$, and $(A_{\mu}/A_{e})_{\tau}$ [93].

##  III Beyond the SM (BSM) Analysis

Let us now interpret the experimental bounds for LFUV in the charged current
in terms of constraints on NP. For this we will first study effective
operators (i.e. modified $W\ell\nu$ couplings and four-fermion operators) and
then consider simplified models which give rise to the corresponding Wilson
coefficients. In this context we will highlight a possible correlation between
the CAA and non-resonant di-lepton searches at the LHC and finally study NP
models with focus on the ones motivated by the CAA.

###  III.1 Effective Field Theory

####  III.1.1 Modified $W\ell\nu$ couplings

All observables discussed in this review are sensitive to modified $W$
couplings to leptons. In order to investigate their effects, let us therefore
use the parameterization999 In the conventions of Ref. [90] one has
$1+\epsilon_{ii}-\epsilon_{jj}=g_{i}/g_{j}$, or equivalently
$g_{i}=g_{j}(1+\epsilon_{ii}-\epsilon_{jj})$, with $i,j=e,\mu,\tau$.

$\mathcal{L}\supset-i\frac{{g_{2}}}{{\sqrt{2}}}{{\bar{\ell}}_{i}}{\gamma^{\mu}}{P_{L}}{\nu_{j}}{W_{\mu}^{-}}\left({{\delta_{ij}}+{\epsilon_{ij}}}\right)+h.c.\,,$
(31)

where $i,j=e$,$\mu$,$\tau$ and with the SM $SU(2)_{L}$ gauge coupling $g_{2}$
recovered in the limit $\epsilon_{ij}\to 0$. Here we neglected possible
effects of the PMNS matrix which drop out in the limit of vanishing neutrino
masses. Furthermore, in the following we will disregard flavor-violating
couplings ($\varepsilon_{ij}$ with $i\neq j$) since they are tightly bounded
by radiative lepton decays $\ell\to\ell^{\prime}\gamma$ and lead to effects in
LFUV observables that do not interfere with the SM and are thus suppressed.
Note that in Eq. (31) we simply parameterize the BSM effect by
$\varepsilon_{ij}$ but do not consider the $SU(2)_{L}$ gauge invariance in SM
EFT, which we will discuss later.

For the phenomenological analysis, note that all LFUV observables (encoded in
direct ratios) depend, at leading order, on differences
$\epsilon_{aa}-\epsilon_{bb}$ ($a\neq b$) while the deficit in first row CKM
unitarity, related to the determination of $V_{ud}$, is to a good
approximation only sensitive to $\epsilon_{\mu\mu}$ [27]. In order to see the
latter, the first crucial observation is that in order to extract $V_{ud}$
from beta decays, one needs the Fermi constant determined from the muon
lifetime [95]

$\frac{1}{\tau_{\mu}}=\frac{(G_{F}^{\mathcal{L}})^{2}m_{\mu}^{5}}{192\pi^{3}}(1+\Delta
q)(1+\varepsilon_{ee}+\varepsilon_{\mu\mu})^{2}\,.$ (32)

Here $G_{F}^{\mathcal{L}}$ is the Fermi constant appearing in the Lagrangian
(excluding BSM contamination) and $\Delta q$ subsumes the phase space, QED,
and electroweak radiative corrections. Therefore, the Fermi constant measured
in muon decay and extracted under the SM assumption ($G_{F}$), is related to
the one at the Lagrangian level as

$G_{F}=G_{F}^{\mathcal{L}}(1+\varepsilon_{ee}+\varepsilon_{\mu\mu})\,.$ (33)

Thus

$V_{ud}^{\beta}=V_{ud}^{\mathcal{L}}\big{(}1-\varepsilon_{\mu\mu}\big{)}\,,$
(34)

where $V_{ij}^{\mathcal{L}}$ denotes CKM matrix elements without any BSM
contamination which by definition fulfills CKM unitarity. Taking into account
that the first row and column CKM unitarity relations are very much dominated
by $V_{ud}$, being by far the biggest element of the CKM matrix, we find to a
good approximation

$\varepsilon_{\mu\mu}\approx 0.00098\pm 0.00027\,,$ (35)

reflecting the corresponding $3.7\,\sigma$ tension.

We can now reparametrize the NP effects by writing

$\epsilon_{ee}-\epsilon_{\mu\mu}\,,\qquad\epsilon_{\tau\tau}-\epsilon_{\mu\mu}\;\;{\rm
and}\;\;\epsilon_{\mu\mu}\,,$ (36)

such that differences are direct measures of LFU and are constrained by the
corresponding ratios. This allows us to perform a global fit in the
$\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$ $vs.$
$\epsilon_{ee}-\epsilon_{\mu\mu}$ plane which is uncorrelated with
$\epsilon_{\mu\mu}$ taking into account all LFU ratios discussed previously
(including correlations among them). The result in shown in Fig. 2 (left).
From this depiction, we see that while the hypothesis of LFU in the charged
current is compatible with data at the $2\sigma$ level, one observes a slight
preference for negative values of $\epsilon_{ee-\mu\mu}$.

Figure 2: Left: Global fit in the $\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$ vs.
$\epsilon_{ee}-\epsilon_{\mu\mu}$ plane, including kaon, pion, and tau decays,
quantifying LFU in the charged current. Right: Global fit in the $C^{\rm
NP}_{23}-C^{\rm NP}_{12}$ vs. $C^{\rm NP}_{13}-C^{\rm NP}_{12}$ plane from
leptonic tau and muon decays. Uncertainties are shown for one (dark color) and
two (lighter color) standard deviations.

####  III.1.2 4-lepton Operators

It is clear that 4-lepton operators only enter purely leptonic decays.
Furthermore, as (in the limit of vanishing masses of the final state leptons)
only left-handed vector operators with the same flavor structure as the SM
lead to interference with the SM in these decays, we can focus on them and
write

${\cal
L}_{4\ell}=\frac{-g_{2}^{2}}{2m_{W}^{2}}C_{fi}^{\ell\ell\nu\nu}\bar{\ell}_{f}\gamma_{\mu}P_{L}\ell_{i}\bar{\nu}_{i}\gamma^{\mu}P_{L}\nu_{f}\,,$
(37)

with $C_{fi}^{\ell\ell\nu\nu}=1+C_{fi,{\rm NP}}^{\ell\ell\nu\nu}$. The effects
of $C_{fi,{\rm NP}}^{\ell\ell\nu\nu}$ are similar to the ones of modified
$W-\ell-\nu$ couplings and we can consider the three parameters $C_{12,{\rm
NP}}^{\ell\ell\nu\nu}$, $C_{13,{\rm NP}}^{\ell\ell\nu\nu}-C_{12,{\rm
NP}}^{\ell\ell\nu\nu}$ and $C_{23,{\rm NP}}^{\ell\ell\nu\nu}-C_{12,{\rm
NP}}^{\ell\ell\nu\nu}$. However, in this case $C_{12}$ is not only determined
from the CAA but also has an impact on the global EW fit as it modifies the
determination of the Fermi constant from muon decay [96, 97]. In fact, they
turn out to prefer opposite signs

$\displaystyle\left.{C_{12,{\rm NP}}^{\ell\ell\nu\nu}}\right|_{\rm CAA}$
$\displaystyle\approx 0.00098\pm 0.00027\,,\textrm{and}$ (38)
$\displaystyle\left.{C_{12,{\rm NP}}^{\ell\ell\nu\nu}}\right|_{\rm EW}$
$\displaystyle\approx-0.00067\pm 0.00033\,.$

$C_{13,{\rm NP}}^{\ell\ell\nu\nu}-C_{12,{\rm NP}}^{\ell\ell\nu\nu}$ as well as
$C_{23,{\rm NP}}^{\ell\ell\nu\nu}-C_{12,{\rm NP}}^{\ell\ell\nu\nu}$ are
determined from the ratios of rates $\tau\to\mu\nu\nu/\tau\to e\nu\nu$,
$\tau\to\mu\nu\nu/\mu\to e\nu\nu$ and $\tau\to e\nu\nu/\mu\to e\nu\nu$, while
all ratios involving mesons remain unaffected. Therefore, we find the global
fit shown in Fig. 2 (right).

####  III.1.3 2-quark-2-lepton Operators

Concerning 2-quark-2-lepton operators, both left-handed vector operators and
scalar ones are relevant as they interfere with the SM contribution. In fact,
the latter have enhanced effects in $R^{P}_{e/\mu}$:

$\left(\dfrac{A_{\mu}}{A_{e}}\right)_{R^{P}_{e/\mu}}=\dfrac{C^{V_{L}\mu}_{fi}-C^{V_{R}\mu}_{fi}+\dfrac{m_{P}^{2}}{\left(m_{u_{f}}+m_{d_{i}}\right)m_{\mu}}\left(C_{fi}^{S_{R}\mu}-C_{fi}^{L\mu}\right)}{C^{V_{L}e}_{fi}-C^{V_{R}e}_{fi}+\dfrac{m_{P}^{2}}{\left(m_{u_{f}}+m_{d_{i}}\right)m_{e}}\left(C_{fi}^{S_{R}e}-C_{fi}^{S_{L}e}\right)}\,,$
(39)

with $u_{f}=u$ and $d_{i}=d(s)$ for $P=\pi(K)$, defined via the Lagragian

$L_{ud\ell\nu}=\dfrac{-g_{2}^{2}}{2m_{W}^{2}}V_{fi}\sum_{A=L,R}\left(C^{V_{A}\ell}_{fi}\bar{u}_{f}\gamma^{\mu}P_{A}d_{i}\bar{\ell}\gamma_{\mu}P_{L}\nu_{\ell}+C^{S_{A}\ell}_{fi}\bar{u}_{f}P_{A}d_{i}\bar{\ell}P_{L}\nu_{\ell}\right)\,,$
(40)

where $C^{V_{L}\ell}_{fi}=1+C^{V_{L}\ell}_{fi,{\rm NP}}$. All other Wilson
coefficients are zero within the SM.

From the CAA we find

$C^{V_{L}e}_{11,{\rm NP}}=-0.00098\pm 0.00027\,,$ (41)

and from LFUV ratios the bounds can be directly read off using Eqs. (8) and
(13). For the three body decays involved in $R^{K\to\pi}_{e/\mu}$, the last
term of Eq. (39) should be omitted and the sign in front of the right-handed
vector Wilson coefficient changes. Note that, in principle, constraints from
the ratios like ${\rm Br}[K\to\mu\nu]/{\rm Br}[\pi\to\mu\nu]$ (see e.g. Ref.
[98] for an overview) have to be taken into account, especially in the case of
scalar operators, even though these are not measures of LFUV.

####  III.1.4 $SU(2)_{L}$ Gauge Invariance

Interestingly, assuming NP above the EW scale which respects $SU(2)_{L}$ gauge
symmetry, any modification of the left-handed charged current also leads to a
modification of a neutral current [73]. The relevant effective operators in
explicit $SU(2)_{L}$ gauge invariant language (see Refs. [72, 99] for the
definitions of the conventions) are contained in the Lagraingain

$\displaystyle L=\dfrac{1}{\Lambda^{2}}$ $\displaystyle\left([C_{\ell
q}^{(3)}]_{ijkl}(\bar{L}_{i}\gamma^{\mu}\tau^{I}L_{j})(\bar{Q}_{k}\gamma_{\mu}\tau^{I}Q_{l})+[C_{\ell\ell}^{(3)}]_{ijkl}(\bar{L}_{i}\gamma^{\mu}\tau^{I}L_{j})(\bar{L}_{k}\gamma_{\mu}\tau^{I}L_{l})+\right.$
(42)
$\displaystyle\;\;\;\;\left.C_{\phi\ell}^{\left(1\right)ij}{\phi^{\dagger}}i\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over{D}}_{\mu}\phi\,{{\bar{L}}^{i}}{\gamma^{\mu}}{L^{j}}+C_{\phi\ell}^{\left(3\right)ij}{\phi^{\dagger}}i\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over{D}}_{\mu}^{I}\phi\,{{\bar{L}}^{i}}{\tau^{I}}{\gamma^{\mu}}{L^{j}}\right)\,.$

Making the identification with the operators discussed in the previous
subsections we find for modified $W\ell\nu$ couplings

$\epsilon_{fi}=\frac{v^{2}}{\Lambda^{2}}C_{\phi e}^{fi}\,,$ (43)

with $v=\sqrt{2}\times 174\,{\rm GeV}$ and $\Lambda$ is the mass scale of NP.
This means that at the same time $Z\to\nu\nu$ and/or $Z\to\ell\ell$ effects
appear. However, in order to avoid effects in stringently constrained $Z$
couplings to charged leptons [100], one can assume
$C^{(1)}_{\phi\ell}=-C^{(3)}_{\phi\ell}$. In this case, the constructive
effect in $Z\to\nu_{\mu}\nu_{\mu}$ can be compensated by a destructive effect
in $Z\to\nu_{e}\nu_{e}$, as only the sum of the three flavors is measured,
resulting in a very good fit to data [26].

For 4-lepton operators

$\displaystyle C_{fi,{\rm NP}}^{\ell\ell\nu\nu}$
$\displaystyle=-\dfrac{4m_{W}^{2}}{g_{2}^{2}\Lambda^{2}}[C_{\ell
q}^{(3)}]_{ffii}\,.$ (44)

Here the implications are that a purely leptonic charged current operator with
first generation leptons in general also gives rise to a neutral current with
electrons which is subject to LEP bounds from non-resonant di-lepton searches
[101].

Finally, for the 4-fermion operators, working in the down-basis, we find

$\displaystyle C_{fi,{\rm NP}}^{V_{L}\ell_{j}}$
$\displaystyle=-\dfrac{4m_{W}^{2}}{g_{2}^{2}\Lambda^{2}}[C_{\ell
q}^{(3)}]_{jjfi}\,.$ (45)

This means that for left-handed $\bar{u}d\bar{\nu}e$ operators affecting beta
decays, $\bar{u}u\bar{e}e$ and/or $\bar{d}d\bar{e}e$ operators are also
generated. This has interesting implications for an explanation of the CAA, as
it leads to an additional neutral $\bar{u}u\ell\ell$ and $\bar{d}d\ell\ell$
current after EW symmetry breaking. $[Q_{\ell q}^{(3)}]_{1111}$ therefore also
contributes to non-resonant di-electron production at the LHC, which is
tailored to search for heavy NP that is above the direct production reach
[102, 103]. The latest di-lepton results from ATLAS and CMS are presented in
Ref. [104] and Ref. [25], respectively. CMS observed a slight excess in the
di-electron cross section at high invariant lepton mass and computed the
double ratio

$R^{\rm Data}_{\mu^{+}\mu^{-}/e^{+}e^{-}}/R^{\rm
MC}_{\mu^{+}\mu^{-}/e^{+}e^{-}}\,,$ (46)

in order to reduce the uncertainties [105]. This means that they provide the
relative signal strength for muons vs. electrons, $R^{\rm
Data}_{\mu^{+}\mu^{-}/e^{+}e^{-}}$, divided by the SM expectation obtained
from Monte Carlo simulations, $R^{\rm MC}_{\mu^{+}\mu^{-}/e^{+}e^{-}}$. Taking
this into account, we find that the best fit value for the Wilson coefficient
is

$[C_{\ell q}^{(3)}]_{1111}/\Lambda^{2}\approx 1.0/(10\,{\rm TeV})^{2}\,,$ (47)

with $\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{SM}\approx-10$ and $0.3/(10\,{\rm
TeV})^{2}\lessapprox[C_{\ell q}^{(3)}]_{1111}\lessapprox 1.8/(10\,{\rm
TeV})^{2}$ . Note that this value is compatible with the corresponding ATLAS
bounds and $\left(\frac{A_{\mu}}{A_{e}}\right)_{R^{\pi}_{e/\mu}}$ in Eq. 8.
Treating the ATLAS exclusion as a hard cut, we therefore find that at $95\%$
CL

$0.6/(10{\rm TeV})^{2}\lessapprox[C_{\ell
q}^{(3)}]_{1111}/\Lambda^{2}\lessapprox 1.4/(10{\rm TeV})^{2}\,,$ (48)

which predicts

$1.0004\lessapprox\left(\frac{A_{\mu}}{A_{e}}\right)_{R^{\pi}_{e/\mu}}\lessapprox
1.0009\,,$ (49)

at the 95% CL.

###  III.2 Simplified NP models

The following NP models (see Ref. [106] for a complete categorization of tree-
level extensions of the SM) give contributions to the effective operators
discussed in the last subsection, which have a relevant impact on our
observables limiting LFUV.

####  III.2.1 $W^{\prime}$ boson

In order to be relevant for our observables, a $W^{\prime}$ bosons must be the
component of a $SU(2)_{L}$ triplet $X_{a}^{\mu}$ with hypercharge 0 such that
it can have coupling to left-handed fermions

$L_{X^{\mu}_{a}}=-g_{ji}^{\ell}X_{a}^{\mu}\bar{L}_{j}\gamma_{\mu}\frac{\tau^{a}}{2}L_{i}-g_{ji}^{q}X_{a}^{\mu}\bar{Q}_{j}\gamma_{\mu}\frac{\tau^{a}}{2}Q_{i}\,.$
(50)

These couplings can contribute to LFUV observables in several ways [28]:

  * •

Modified $W\ell\nu$ coupling via mixing with the SM $W$. In this case it
generates $C^{(3)}_{\phi\ell}$ such that $W\ell\nu$, $Z\nu\nu$ and $Z\nu\nu$
couplings are affected. As discussed in the previous subsection, this leads to
limited effects in $W\ell\nu$ due to the stringent constraints from $Z$
decays. Global fits to $C^{(3)}_{\phi\ell}$ can be found in Refs. [28, 107].

  * •

Tree-level effects in $\ell\to\ell^{\prime}\nu\nu$. In this case the Wilson
coefficient $C_{\ell\ell}^{(3)}$ is generated which results after EW symmetry
breaking in

$C_{fi}^{\ell\ell\nu\nu,{\rm
NP}}=\dfrac{g^{\ell}_{ff}}{g^{\ell}_{ii}}\dfrac{m_{W}^{2}}{m_{W^{\prime}}^{2}}\,,$
(51)

and the bounds of Sec. III.1.2 can be used.

  * •

Tree-level effects in $d\to ue\nu$. Similarly, if $W^{\prime}$ possesses
couplings to quarks and leptons, it leads to tree-level effects $d\to ue\nu$
via the Wilson coefficient $C_{qq}^{(3)}$ and the bounds from kaon and pion
decays apply.

Fro example, left-handed $W^{\prime}$ bosons appear as excitations of the SM
$W$ in [108, 109] or extra-dimensional models [110] as well as in theories
with several $SU(2)_{L}$ gauge groups [111, 112].

####  III.2.2 Vector-like Leptons

Vector-like leptons (VLL) affect $W-\ell-\nu$ coupling via their mixing with
SM leptons and are EW symmetry breaking. There are 5 representations of
vector-like leptons that can couple to SM leptons and the Higgs and mix with
the former after EW symmetry breaking. These are represented as

$\begin{array}[]{l|ccc}&SU(3)&SU(2)_{L}&U(1)_{Y}\\\ \hline\cr\ell&1&2&-1/2\\\
\mathrm{e}&1&1&-1\\\ \phi&1&2&1/2\\\ \hline\cr\mathrm{N}&1&1&0\\\
\mathrm{E}&1&1&-1\\\
\Delta_{1}=\left(\Delta_{1}^{0},\Delta_{1}^{-}\right)&1&2&-1/2\\\
\Delta_{3}=\left(\Delta_{3}^{-},\Delta_{3}^{--}\right)&1&2&-3/2\\\
\Sigma_{0}=\left(\Sigma_{0}^{+},\Sigma_{0}^{0},\Sigma_{0}^{-}\right)&1&3&0\\\
\Sigma_{1}=\left(\Sigma_{1}^{0},\Sigma_{1}^{-},\Sigma_{1}^{--}\right)&1&3&-1\end{array}$
(52)

which result in the following Wilson coefficients

$\displaystyle\frac{C_{\phi\ell}^{(1)ij}}{\Lambda^{2}}$
$\displaystyle=\frac{\lambda_{N}^{i}\lambda_{N}^{j\dagger}}{4M_{N}^{2}}-\frac{\lambda_{E}^{i}\lambda_{E}^{j\dagger}}{4M_{E}^{2}}+\frac{3}{16}\frac{\lambda_{\Sigma_{0}}^{i\dagger}\lambda_{\Sigma_{0}}^{j}}{M_{\Sigma_{0}}^{2}}-\frac{3}{16}\frac{\lambda_{\Sigma_{1}}^{i\dagger}\lambda_{\Sigma_{1}}^{j}}{M_{\Sigma_{1}}^{2}}\,,$
(53) $\displaystyle\frac{C_{\phi\ell}^{(3)ij}}{\Lambda^{2}}$
$\displaystyle=-\frac{\lambda_{N}^{i}\lambda_{N}^{j\dagger}}{4M_{N}^{2}}-\frac{\lambda_{E}^{i}\lambda_{E}^{j\dagger}}{4M_{E}^{2}}+\frac{1}{16}\frac{\lambda_{\Sigma_{0}}^{i\dagger}\lambda_{\Sigma_{0}}^{j}}{M_{\Sigma_{0}}^{2}}+\frac{1}{16}\frac{\lambda_{\Sigma_{1}}^{j\dagger}\lambda_{\Sigma_{1}}^{i}}{M_{\Sigma_{1}}^{2}}\,,$
$\displaystyle\frac{C_{\phi\mathrm{e}}^{ij}}{\Lambda^{2}}$
$\displaystyle=\frac{\lambda_{\Delta_{1}}^{i\dagger}\lambda_{\Delta_{1}}^{j}}{2M_{\Delta_{1}}^{2}}-\frac{\lambda_{\Delta_{3}}^{i\dagger}\lambda_{\Delta_{3}}^{j}}{2M_{\Delta_{3}}^{2}}\,,$

where $\lambda$ are the couplings of the VLLs to the SM leptons and the Higgs
doublet (see Ref. [28] for details). Note that only $C_{\phi\ell}^{(3)}$
generates a charged current and affects our LFUV observables while the other
coefficients enter $Z\ell\ell$ and $Z\nu\nu$ couplings entering the global EW
fit [113]. While for $C_{\phi\ell}^{(3)}$ we can apply the bounds from the
previous subsection, in order to take into account all effects, a global fit
is necessary for which we refer the reader to Ref. [28].

VLLs are predicted in many SM extensions, such as Grand Unified Theories [114,
115, 116], composite models or models with extra dimensions [117, 118, 119,
120, 121, 122, 123, 124] and are involved in the type I [125, 126] and type
III [127] seesaw mechanisms.

####  III.2.3 Singly charged $SU(2)_{L}$ singlet scalar

As it is a $SU(2)_{L}\times SU(3)_{C}$ singlet, $\phi^{+}$ has hypercharge +1
and allows only for Yukawa-type interactions with leptons. The model is
particularly interesting in the context of LFUV; due to hermicity of the
Lagrangian it has anti-symmetric (i.e. off-diagonal) couplings:

$\displaystyle\mathcal{L}=-\frac{\lambda_{ij}}{2}\,\bar{L}^{c}_{a,i}\,\varepsilon_{ab}\,L_{b,j}\,\Phi^{+}+{\rm
h.c.}\,,$ (54)

but not with quarks. Here $L$ is the left-handed $SU(2)_{L}$ lepton doublet,
$c$ stands for charge conjugation, $a$ and $b$ are $SU(2)_{L}$ indices, $i$
and $j$ are flavor indices and $\varepsilon_{ab}$ is the two-dimensional
antisymmetric tensor. Note that without loss of generality, $\lambda_{ij}$ can
be chosen to be antisymmetric in flavor space, $\lambda_{ji}=-\lambda_{ij}$,
such that $\lambda_{ii}=0$ and our free parameters are $\lambda_{12}$,
$\lambda_{13}$, and $\lambda_{23}$ giving

$C_{fi,{\rm
NP}}^{\ell\ell\nu\nu}=\frac{\left|\lambda_{fi}^{2}\right|}{g_{2}^{2}}\frac{m_{W}^{2}}{m_{\phi}^{2}}\,.$
(55)

This means that the effect of $\ell\to\ell^{\prime}\nu\overline{\nu}$ is
necessarily constructive, such that the CAA can be solved [128, 129] and also
$C_{23,{\rm NP}}^{\ell\ell\nu\nu}$ has the right sign preferred by the fit.
However, in order to not violate the bounds from $\mu\to e\gamma$,
$\lambda_{13}$ must be very close to zero [128].

Singly charged scalars have been proposed within the Babu-Zee model [130, 131]
and studied in Refs. [132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142]
as part of a larger NP spectrum, mostly with the aim of generating neutrino
masses at loop level.

####  III.2.4 Scalar $SU(2)_{L}$ Triplet

This fermion couples to a lepton doublet and a charge conjugated one as

$\displaystyle\mathcal{L_{T}}=-\frac{\kappa_{ij}}{2}\,\bar{L}^{c}_{a,i}\,\varepsilon_{ab}\tau^{I}_{cd}\,L_{d,j}\,\Phi^{T}+{\rm
h.c.}\,,$ (56)

resulting in

$C_{fi,{\rm
NP}}^{\ell\ell\nu\nu}=-\frac{\left|\kappa_{fi}^{2}\right|}{g_{2}^{2}}\frac{2m_{W}^{2}}{m_{\phi}^{2}}\,,$
(57)

such that the effect is always destructive in
$\ell\to\ell^{\prime}\nu\overline{\nu}$.

This scalar can generate neutrino masses within the type-II seesaw model [143,
144, 145, 146, 147]. While in general the tiny neutrino masses require very
small couplings $\kappa$ (for TeV scale masses), the contribution to the
Weinberg operator[148] can be suppressed, e.g. by an approximate Baryon number
symmetry, such that phenomenologically relevant effects in LFUV are possible.

####  III.2.5 $SU(2)_{L}$ Neutral Vector Boson ($Z^{\prime}$):

A $Z^{\prime}$ boson which is an $SU(2)_{L}$ singlet only interferes with the
SM amplitudes for $\ell\to\ell^{\prime}\nu\nu$ if it has couplings to lepton
doublets

$L_{Z^{\prime}}=-ig^{\ell}_{fi}\bar{L}_{f}\gamma^{\mu}L_{i}Z^{\prime}_{\mu}\,.$
(58)

Furthermore, these couplings must be flavor violating such that

$\displaystyle C_{fi,{\rm NP}}^{\ell\ell\nu\nu}$
$\displaystyle=\dfrac{2m_{W}^{2}}{g_{2}^{2}M_{Z^{\prime}}^{2}}|g^{\ell}_{fi}|^{2}\,.$
(59)

Note that the effect in $\ell\to\ell^{\prime}\nu\overline{\nu}$ is necessarily
constructive [149].

There is a huge literature on $Z^{\prime}$ bosons (see Ref. [150] for an
overview), which could again be excitations of the SM $Z$ and $\gamma$, but in
this case also originate from a gauged $U(1)$ flavor symmetry like
$L_{\mu}-L_{\tau}$ or $B-L$.

####  III.2.6 Leptoquark

There are 10 representations of leptoquarks (five scalar and five vector) with
gauge invariant couplings to quarks and leptons [151]. Out of these, six
representations generate a charged current, two a vector current, two a scalar
current and two simultaneously a vector and a scalar current. While those with
a scalar current are stringently constrained by $R^{\pi}_{e/\mu}$, for vector
currents other observables, like di-lepton pairs, low energy parity violation
or kaon decays, are, in general, more constraining. For a recent comprehensive
analysis we refer to Ref. [152, 153].

Leptoquarks arise in the Pati-Salam model [154] $SU(5)$ Grand Unified theories
[155, 156] and in the R-parity violating Mimimal Supersymmetric Model (MSSM,
see e.g. Ref. [157] for a review) and were studied in the context of LFUV in
kaon, tau and pion decays in Ref. [158, 45, 159, 160, 161].

####  III.2.7 Charged Higgs

Charged Higgs bosons have been considered in the context of leptonic meson
decays [162], in particular, $R^{K}_{\mu/e}$ in the context of the MSSM [163,
164]. Furthermore, the type-X two-Higgs-doublet model (2HDM) is constrained by
loop effects in $\tau\to\mu\nu_{\tau}\nu_{\mu}$ [165, 166, 167] relevant at
large values of $\tan\beta$.

##  IV Future Prospects

###  IV.1 Pion and Kaon Experiments

The PIENU [168, 169] and PEN [170, 171, 172] experiments aim to further
improve the precision of the $\pi\to e\nu$ branching ratio $R^{\pi}_{e/\mu}$.
However, even when these goals are realized, an experimental improvement by
more than an order of magnitude in uncertainty is warranted to confront the SM
prediction and to search for non-SM effects. A developing proposal PIONEER
[173] aims for an improvement in precision for $R^{\pi}_{e/\mu}$ by an order
of magnitude making the experimental uncertainty comparable to the theoretical
uncertainty in Eq. 4. To reach very high precision requires high statistics as
well as extensive evaluation of systematic uncertainties, backgrounds, biases,
and distortions in the data selection criteria. Like PIENU [53] and PEN [170],
PIONEER will be done using stopped pions that decay at rest. Principal
features of the experiment include a fully active Si tracking stopping target
[174] and a high resolution calorimeter, both of which contribute to
suppression of systematic effects.

PIONEER also aims to improve the precision for the branching ratio of pion
beta decay $\pi^{+}\rightarrow\pi^{0}e^{+}\nu(\gamma)$ [175]. Pion beta decay,
while providing the theoretically cleanest determination of the CKM matrix
element $V_{ud}$, is currently not competitive:
$V_{ud}=0.9739(28)_{\textrm{exp}}(1)_{\textrm{th}}$, where the experimental
uncertainty comes almost entirely from the
$\pi^{+}\rightarrow\pi^{0}e^{+}\nu(\gamma)$ branching ratio [175] (the pion
lifetime contributes ${\delta}V_{ud}=0.0001$), and the theory uncertainty has
been reduced from $({\delta}V_{ud})_{\textrm{th}}=0.0005$ [176, 177, 178] to
$({\delta}V_{ud})_{\textrm{th}}=0.0001$ via a lattice QCD calculation of the
radiative corrections [179]. As pointed out in Ref. [89] even a three-fold
improvement in precision of the pion beta decay branching ratio compared to
Ref. [175] would allow for a 0.2% accuracy in the determination of the ratio
$V_{us}/V_{ud}$ from
$R_{V}\equiv\Gamma(K\to\pi\ell\nu(\gamma))/\Gamma(\pi^{+}\to\pi^{0}e^{+}\nu(\gamma))$,
competitive with the existing determination based on
$R_{A}\equiv\Gamma(K\to\mu\nu)/\Gamma(\pi\to\mu\nu)$.

Regarding other kaon decays, the TREK [180] experiment may provide additional
information on $R^{K}_{e/\mu}$ and NA62 may make further measurements relevant
to LFUV [33, 181].

###  IV.2 Tau Experiments

The Belle II experiment is currently collecting data at and near the center-
of-mass energy of 10.58 GeV/c2 and is expected to obtain a total integrated
luminosity of $50$ ab-1, equivalent to $45$ billion $\tau-$pairs [182, 183].
Although precision measurements are ultimately dominated by the capabilities
of experiments to limit systematic factors, an improvement in the
determination of all LFU parameters in $\tau$ decays is to be expected.

The proposed future circular collider (FCC) plans to collide electrons and
positrons (FCC-ee) at different center-of-mass energies, including a 4-year
high-statistics run at the $Z$ pole [184]. Assuming an instantaneous
luminosity of 2.3$\times 10^{36}$cm-2s-1 and four interaction regions, this
would translate into 1.7$\times 10^{11}$ $\tau-$pairs produced in
$Z\to\tau^{+}\tau^{-}$ reactions available for precision studies of $\tau$
properties and polarization at the FCC-ee [185]. A rich $\tau$ physics program
would also be expected by the proposed Circular Electron Positron Collider
(CEPC), where 30 billion $\tau$ pairs could be produced at the Z pole [186].

For both the FCC-ee and CEPC, the same considerations regarding the control of
systematic effects hold as for Belle II. Although collecting high-statistics
samples of $\tau$ is a conditio sine qua non to perform precision tests of the
standard model via suppressed or forbidden processes in $\tau$ decays,
systematic effects must be understood and kept under control. All the above-
mentioned experiments will possibly reach a statistical precision at the level
of $10^{-4}-10^{-5}$ on a number of LFU parameters, given the large amount of
$\tau$ decays that will be produced and analyzed, but systematic effects will
dominate the final precision. Experiments planned in the future have the
advantage of designing the detectors to minimize potential systematic effects.

###  IV.3 Theory

While the ratios measuring LFU are, in general, theoretically very clean, the
uncertainty in the extraction of $V_{ud}$ from super-allowed beta decays is
still limited by the theoretical error. Moreover, expected experimental
improvements in the neutron lifetime and decay asymmetry will enhance the
relative impact of theoretical uncertainties in the extraction of $V_{ud}$
from neutron decay. The theoretical uncertainty in both cases arises from
electromagnetic radiative corrections. Progress in the next decade can be
expected on a number of fronts. First, at the single nucleon level, one
expects results on radiative corrections to neutron decay from lattice QCD.
The technology to perform these calculations in meson systems
($K^{+}\to\mu^{+}\nu_{\mu}$ and $\pi^{+}\to\pi^{0}e^{+}\nu_{e}$) has been
demonstrated [187, 179], but not yet applied to nucleons. Second, one can
expect progress in the analysis of few- and many-body effects in nuclear
transitions, both with dispersive techniques (see Ref. [80]) and through the
development of chiral effective field theory few-body transitions operators to
$O(G_{F}\alpha)$ coupled to first-principle many-body methods.

##  V Conclusions

In this article we reviewed the status of the searches for LFUV in the charged
current involving pions, kaons, taus, and beta decays. Averaging the values
found in Sec. II.1.1, Sec. II.1.2, and Sec. II.3 we find that the ratio of the
$W$ couplings to leptons are

$\displaystyle\frac{g_{\mu}}{g_{e}}=$
$\displaystyle\,1+\epsilon_{\mu\mu}-\epsilon_{ee}=1.0009\pm 0.0006\,,$ (60)
$\displaystyle\frac{g_{\tau}}{g_{\mu}}=$
$\displaystyle\,1+\epsilon_{\tau\tau}-\epsilon_{\mu\mu}=1.0013\pm
0.0013\,,\rm{and}$ $\displaystyle\frac{g_{\tau}}{g_{e}}=$
$\displaystyle\,1+\epsilon_{\tau\tau}-\epsilon_{ee}=1.0022\pm 0.0013\,.$

Note that these individual results for $g_{i}/g_{j}$ use values of the others
that minimize the $\chi^{2}$ of 1d fits and should thus be understood
separately i.e. no correlations between the three ratios are taken into
account. These high precision tests of LVU which agree well with the SM are
particularly interesting in light of the experimental hints for LFUV in semi-
leptonic $B$ decays, the anomalous magnetic moment of the muon, and the
Cabbibo angle anomaly.

Taking into account the current experimental and theoretical results, we
performed a combined fit to modified $W-\ell-\nu$ couplings as shown the left
plot of Fig. 2. However, the results can also be interpreted in terms of
4-lepton and 2-quark-2-lepton operators. We also reviewed the experimental and
theoretical prospects for the many observables. The proposed PIONEER
experiment aims to improve the test of $e-\mu$ universality in pion decay and
the pion beta decay determination of $V_{ud}$. Furthermore, BELLE II
measurements will lead to significant improvements in LFU tests with tau
leptons. Finally, the proposed FCC-ee and CEPC accelerators offer intriguing
possibilities for flavor physics and, in particular, for tau decays given the
expected large samples of $B$ mesons and $\tau$ leptons that could be produced
by these facilities.

###### Acknowledgements.

D.B. is supported by NSERC grant no. SAPPJ-2018-0017 (Canada). A.C. gratefully
acknowledges the support by the Swiss National Science Foundation under
Project No. PP00P21_76884. G.I. was supported by the European Research Council
under the grant agreement no. 947006 - “InterLeptons”. V.C. is supported by
the US Department of Energy through the Office of Nuclear Physics and the LDRD
program at Los Alamos National Laboratory. Los Alamos National Laboratory is
operated by Triad National Security, LLC, for the National Nuclear Security
Administration of U.S. Department of Energy

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