$|V_{q}|=O(|V_{cb}|)$, while light quark and lepton masses imply
$|\Delta_{u,d,e}|_{ij}\ll|V_{q}|$. The assumption that $V_{q}$ is the only
leading $U(2)^{5}$-breaking term in the quark sector ensures a suppression of
flavor-violating terms in higher-dimensional operators as effective as the one
implied by the hypothesis of Minimal Flavor Violation [152, 155].

An EFT constructed in terms of SM fields and arbitrary powers of the spurions
in (A.2), formally invariant under $U(2)^{5}$, is what we denote as the
minimally-broken $U(2)^{5}$ hypothesis. The flavor scaling discussed in
Section 2.1 for the leading semileptonic operators addressing the anomalies is
obtained for

$\displaystyle|V_{q}|,\leavevmode\nobreak\
|V_{\ell}|=O(10^{-1})\leavevmode\nobreak\ .$ (A.4)

This is perfectly consistent with the estimate $|V_{q}|=O(|V_{cb}|)$ from the
quark Yukawa sector and is compatible with a possible common origin for the
two leading $U(2)^{5}$-breaking terms.

### Flavor bases and diagonalization of the Yukawa couplings

We define as interaction basis the flavor basis in $U(2)^{5}$ space where
$V_{q,\ell}=|V_{q,\ell}|\times\vec{n}$, with $\vec{n}=(0,1)^{\intercal}$, and
$\Delta_{u,d,e}^{\dagger}\Delta_{u,d,e}$ is diagonal. Without loss of
generality, the Yukawa matrices can be brought to this basis via appropriate
field redefinitions, yielding

$\displaystyle
Y_{u}=|y_{t}|\begin{pmatrix}U_{q}^{\dagger}O_{u}^{\intercal}\,\hat{\Delta}_{u}&|V_{q}|\,|x_{t}|\,e^{i\phi_{q}}\,\vec{n}\\\
0&1\end{pmatrix}\,,\qquad
Y_{d}=|y_{b}|\begin{pmatrix}\;\;\;U_{q}^{\dagger}\hat{\Delta}_{d}&|V_{q}|\,|x_{b}|\,e^{i\phi_{q}}\,\vec{n}\\\
\;\;\;0&1\end{pmatrix}\,,$ $\displaystyle
Y_{e}=|y_{\tau}|\begin{pmatrix}\;\;O_{e}^{\intercal}\,\hat{\Delta}_{e}\;\;&|V_{\ell}|\,|x_{\tau}|\,\vec{n}\\\
\;\;0&1\end{pmatrix}\,,$ (A.5)

where $\hat{\Delta}_{u,d,e}$ are $2\times 2$ diagonal positive matrices,
$O_{u,e}$ are $2\times 2$ orthogonal matrices, and $U_{q}$ is of the form

$\displaystyle U_{q}=\begin{pmatrix}c_{d}&s_{d}\,e^{i\alpha_{d}}\\\
-s_{d}\,e^{-i\alpha_{d}}&c_{d}\end{pmatrix}\,,\qquad
s_{d}\equiv\sin\theta_{d}\,,\quad c_{d}\equiv\cos\theta_{d}\,.$ (A.6)

The unitary matrices that diagonalize the Yukawa matrices above, defined as
$L_{f}^{\dagger}Y_{f}R_{f}={\rm diag}(Y_{f})$ for $f=u,d,e$, assume the form
[60]

$\displaystyle\begin{aligned}
L_{d}&\approx\begin{pmatrix}c_{d}&-s_{d}\,e^{i\alpha_{d}}&0\\\
s_{d}\,e^{-i\alpha_{d}}&c_{d}&s_{b}\\\
-s_{d}\,s_{b}\,e^{-i(\alpha_{d}+\phi_{q})}&-c_{d}\,s_{b}\,e^{-i\phi_{q}}&e^{-i\phi_{q}}\end{pmatrix}\,,\qquad\qquad
L_{e}\approx\begin{pmatrix}c_{e}&-s_{e}&0\\\ s_{e}&c_{e}&s_{\tau}\\\
-s_{e}s_{\tau}&-c_{e}s_{\tau}&1\end{pmatrix}\,,\\\\[5.0pt]
R_{d}&\approx\begin{pmatrix}1&0&0\\\ 0&1&\frac{m_{s}}{m_{b}}\,s_{b}\\\
0&-\frac{m_{s}}{m_{b}}\,s_{b}\,e^{-i\phi_{q}}&e^{-i\phi_{q}}\end{pmatrix},\;\;R_{u}\approx\begin{pmatrix}1&0&0\\\
0&1&\frac{m_{c}}{m_{t}}\,s_{t}\\\
0&-\frac{m_{c}}{m_{t}}\,s_{t}\,e^{-i\phi_{q}}&e^{-i\phi_{q}}\end{pmatrix},\;\;R_{e}\approx\begin{pmatrix}1&0&0\\\
0&1&\frac{m_{\mu}}{m_{\tau}}\,s_{\tau}\\\
0&-\frac{m_{\mu}}{m_{\tau}}\,s_{\tau}&1\end{pmatrix}\,,\end{aligned}$ (A.7)

with $L_{u}=L_{d}\,V_{\rm CKM}^{\dagger}$ and $s_{t}=s_{b}-V_{cb}$. The
parameters $s_{d}$ and $\alpha_{d}$ are not free and can be expressed in terms
of CKM parameters:

$\displaystyle s_{d}/c_{d}$ $\displaystyle=|V_{td}/V_{ts}|\,,$
$\displaystyle\alpha_{d}$ $\displaystyle=\arg(V_{td}^{*}/V_{ts}^{*})\,.$ (A.8)

The fact that the $2\times 2$ upper block of $L_{d}$ is entirely determined in
terms of CKM elements is the origin of the relation in (2.16) and the
condition $\beta_{L}^{d\tau}/\beta_{L}^{s\tau}=V^{*}_{td}/V^{*}_{ts}$ among
the $U_{1}$ couplings.

The flavor-mixing parameters (spurion structures) which are left unconstrained
(i.e. cannot be determined in terms of SM parameters) are:

  * •

quark sector: 2–3 mixing angle $s_{b}/c_{b}=|x_{b}|\,|V_{q}|$ and CP-violating
phase $\phi_{q}$;

  * •

lepton sector: 2–3 mixing angle $s_{\tau}/c_{\tau}=|x_{\tau}|\,|V_{\ell}|$ and
1–2 mixing angle $s_{e}$ (encoded in $O_{e}$).

The approximate down-alignment in the 2–3 sector, necessary to satisfy the
tight bounds from $B_{s}$ mixing when addressing the $B$ anomalies, is
achieved for $|x_{b}|\ll 1$. In the exact down-alignment limit $|x_{b}|\to 0$
(or equivalently $s_{b}\to 0$), the phase $\phi_{q}$ becomes unphysical.

##  Appendix B Theoretical expressions for flavor observables

Here we collect the expressions for the relevant flavor observables used in
our analysis in terms of Wilson coefficients and quark masses defined at the
low scale $\mu_{b}=m_{b}$.

###  B.1 $b\to c(u)\tau\bar{\nu}$ transitions

The low-energy effective Lagrangian describing $b\to u_{i}\tau\bar{\nu}$
transitions (with $i=1,2$) can be written as

$\displaystyle\mathcal{L}_{b\to
u_{i}\tau\bar{\nu}}=-\frac{4G_{F}}{\sqrt{2}}\,V_{ib}\,\Big{[}\big{(}1+C_{V_{L}}^{i}\big{)}\,\mathcal{O}_{V_{L}}^{i}+C_{V_{R}}^{i}\,\mathcal{O}_{V_{R}}^{i}+C_{S_{L}}^{i}\,\mathcal{O}_{S_{L}}^{i}+C_{S_{R}}^{i}\,\mathcal{O}_{S_{R}}^{i}+C_{T}^{i}\,\mathcal{O}_{T}^{i}\Big{]}\,,$
(B.1)

with operators

$\displaystyle\begin{aligned}
\mathcal{O}_{V_{L}}^{i}&=(\bar{\tau}_{L}\gamma_{\mu}{\nu}_{L})(\bar{u}^{i}_{L}\gamma^{\mu}b_{L})\,,\qquad\qquad&\mathcal{O}_{V_{R}}^{i}&=(\bar{\tau}_{L}\gamma_{\mu}\nu_{L})(\bar{u}^{i}_{R}\gamma^{\mu}b_{R})\,,\\\
\mathcal{O}_{S_{L}}^{i}&=(\bar{\tau}_{R}\,\nu_{L})(\bar{u}^{i}_{R}\,b_{L})\,,&\mathcal{O}_{S_{R}}^{i}&=(\bar{\tau}_{R}\,\nu_{L})(\bar{u}^{i}_{L}\,b_{R})\,,\\\
\mathcal{O}_{T}^{i}\leavevmode\nobreak\
&=(\bar{\tau}_{R}\sigma_{\mu\nu}\nu_{L})(\bar{u}^{i}_{R}\sigma^{\mu\nu}b_{L})\,.\end{aligned}$
(B.2)

In our case, we only generate NP contributions to the operators
$\mathcal{O}_{V_{L}}$ and $\mathcal{O}_{S_{R}}$, with coefficients given by

$\displaystyle C_{V_{L}}^{i}\approx C_{LL}^{u_{i}}\,,\qquad
C_{S_{R}}^{i}\approx-2\,\eta_{S}\,C_{LR}^{u_{i}}\,,$ (B.3)

where the coefficients $C_{LL}^{u_{i}}$ and $C_{LR}^{u_{i}}$ have been defined
in (2.14), and the evolution factor for the scalar current evaluates to
$\eta_{S}\approx 1.7$ for $\Lambda=2$ TeV.

The relevant observables in this category are the LFU ratios $R_{D}$ and
$R_{D^{\ast}}$, defined as

$\displaystyle R_{D^{(\ast)}}$
$\displaystyle=\frac{\mathcal{B}\left(\bar{B}\to
D^{(\ast)}\tau\bar{\nu}\right)}{\mathcal{B}\left(\bar{B}\to
D^{(\ast)}\ell\bar{\nu}\right)}\,,\qquad\ell=e,\mu\,,$ (B.4)

and the leptonic decays $B^{-}\to\tau\bar{\nu}$ and
$B_{c}^{-}\to\tau\bar{\nu}$. The expressions for these observables in terms of
the Wilson coefficients in (B.1) read

$\displaystyle R_{D}$
$\displaystyle=R_{D}^{\mathrm{SM}}\left[\left|1+C_{V_{L}}^{c}\right|^{2}+1.5\,\mathrm{Re}\left\\{\left(1+C_{V_{L}}^{c}\right)C_{S_{R}}^{c\ast}\right\\}+1.03\left|C_{S}^{c}\right|^{2}\right],$
(B.5) $\displaystyle R_{D^{\ast}}$
$\displaystyle=R_{D^{\ast}}^{\mathrm{SM}}\left[\left|1+C_{V_{L}}^{c}\right|^{2}+0.12\,\mathrm{Re}\left\\{\left(1+C_{V_{L}}^{c}\right)C_{S_{R}}^{c\ast}\right\\}+0.04\left|C_{S_{R}}^{c}\right|^{2}\right],$

and

$\displaystyle\mathcal{B}(B_{i}^{-}\to\tau\bar{\nu})$
$\displaystyle=\mathcal{B}(B_{i}^{-}\to\tau\bar{\nu})_{\mathrm{SM}}\left|1+C_{V_{L}}^{i}+C_{S_{R}}^{i}\,\frac{m_{B_{i}}^{2}}{m_{\tau}(m_{b}+m_{i})}\right|^{2},$
(B.6)

where $B^{-}\equiv B_{u}^{-}$ is understood.

###  B.2 $b\to s(d)\ell^{-}\ell^{+}$ transitions

The low-energy effective Lagrangian describing $b\to
d_{i}\ell_{\alpha}^{-}\ell_{\beta}^{+}$ transitions (with $i=1,2$) is given by

$\displaystyle\mathcal{L}_{b\to d_{i}\ell_{\alpha}^{-}\ell_{\beta}^{+}}$
$\displaystyle\supset\frac{4G_{F}}{\sqrt{2}}\,V_{ti}^{*}V_{tb}\sum_{k}\,\mathcal{C}_{k}^{i\alpha\beta}\,\mathcal{O}_{k}^{i\alpha\beta}\,,$
(B.7)

with operators

$\displaystyle\begin{aligned}
\mathcal{O}_{9}^{i\alpha\beta}&=\frac{\alpha}{4\pi}\left(\bar{d}_{L}^{i}\gamma_{\mu}b_{L}\right)\left(\bar{\ell}_{\alpha}\gamma^{\mu}\ell_{\beta}\right),&\mathcal{O}_{S}^{i\alpha\beta}&=\frac{\alpha}{4\pi}(\bar{d}_{L}^{i}b_{R})(\bar{\ell}_{\alpha}\ell_{\beta})\,,\\\\[5.0pt]
\mathcal{O}_{10}^{i\alpha\beta}&=\frac{\alpha}{4\pi}\left(\bar{d}_{L}^{i}\gamma_{\mu}b_{L}\right)\left(\bar{\ell}_{\alpha}\gamma^{\mu}\gamma_{5}\,\ell_{\beta}\right)\,,&\mathcal{O}_{P}^{i\alpha\beta}&=\frac{\alpha}{4\pi}(\bar{d}_{L}^{i}b_{R})(\bar{\ell}_{\alpha}\gamma_{5}\,\ell_{\beta})\,.\end{aligned}$
(B.8)

Unless specified, the Wilson coefficients ${\mathcal{C}}_{k}^{i\alpha\beta}$
include both the SM and the NP contributions, i.e.

$\displaystyle{\mathcal{C}}_{k}^{i\alpha\beta}={\mathcal{C}}_{k,\mathrm{SM}}^{i\alpha\beta}+{\mathcal{C}}_{k,\mathrm{NP}}^{i\alpha\beta}\,.$
(B.9)

At tree level, the NP contributions can be matched to the Lagrangian in (2.2)
via the relations

$\displaystyle\begin{aligned}
{\mathcal{C}}_{9,\rm{NP}}^{i\alpha\beta}&=-{\mathcal{C}}_{10,\rm{NP}}^{i\alpha\beta}=-\frac{2\pi}{\alpha
V_{ts}^{*}V_{tb}}\,{\mathcal{C}}^{i3\beta\alpha}_{LL}\,,\\\
{\mathcal{C}}_{S}^{i\alpha\beta}&=-{\mathcal{C}}_{P}^{i\alpha\beta}=\frac{4\pi}{\alpha
V_{ts}^{*}V_{tb}}\,{\mathcal{C}}^{i3\beta\alpha}_{LR}\,.\end{aligned}$ (B.10)

Whenever possible, we simplify the notation by omitting the superscript $i$
for $b\to s$ transitions. Also, for lepton-flavor conserving operators we use
the superscript $\ell_{\alpha}$ instead of $\ell_{\alpha}\ell_{\alpha}$.

####  $R_{K}$ and $R_{K^{\ast}}$

The LFU ratios $R_{K}$ and $R_{K^{\ast}}$ are defined as

$\displaystyle R_{K^{(\ast)}}^{\rm[q^{2}_{\rm{min}},q^{2}_{\rm{max}}]}$
$\displaystyle=\frac{\int_{q^{2}_{\rm{min}}}^{q^{2}_{\rm{max}}}dq^{2}\frac{d\mathcal{B}}{dq^{2}}(B\to
K^{(*)}\mu^{+}\mu^{-})}{\int_{q^{2}_{\rm{min}}}^{q^{2}_{\rm{max}}}dq^{2}\frac{d\mathcal{B}}{dq^{2}}(B\to
K^{(*)}e^{+}e^{-})}\,.$ (B.11)

Their expressions in terms of the Wilson coefficients
$\mathcal{C}_{9,\mathrm{NP}}^{\mu}$ and $\mathcal{C}_{10,\mathrm{NP}}^{e}$
read

$\displaystyle\begin{aligned}
R_{K}^{\rm[1.1,6]}&=R_{K,\mathrm{SM}}^{[1.1,6]}\,\frac{1+0.24\,{\rm
Re}({\mathcal{C}}_{9,\mathrm{NP}}^{\mu})-0.26\,{\rm
Re}({\mathcal{C}}_{10,\mathrm{NP}}^{\mu})+0.03\,\big{(}|{\mathcal{C}}_{9,\mathrm{NP}}^{\mu}|^{2}+|{\mathcal{C}}_{10,\mathrm{NP}}^{\mu}|^{2}\big{)}}{1+0.24\,{\rm
Re}({\mathcal{C}}_{9,\mathrm{NP}}^{e})-0.26\,{\rm
Re}({\mathcal{C}}_{10,\mathrm{NP}}^{e})+0.03\,\big{(}|{\mathcal{C}}_{9,\mathrm{NP}}^{e}|^{2}+|{\mathcal{C}}_{10,\mathrm{NP}}^{e}|^{2}\big{)}}\,,\\\
R_{K^{\ast}}^{[1.1,6]}&=R_{K^{\ast},\mathrm{SM}}^{[1.1,6]}\,\frac{1+0.18\,{\rm
Re}({\mathcal{C}}_{9,\mathrm{NP}}^{\mu})-0.29\,{\rm
Re}({\mathcal{C}}_{10,\mathrm{NP}}^{\mu})+0.03\,\big{(}|{\mathcal{C}}_{9,\mathrm{NP}}^{\mu}|^{2}+|{\mathcal{C}}_{10,\mathrm{NP}}^{\mu}|^{2}\big{)}}{1+0.18\,{\rm
Re}({\mathcal{C}}_{9,\mathrm{NP}}^{e})-0.29\,{\rm
Re}({\mathcal{C}}_{10,\mathrm{NP}}^{e})+0.03\,\big{(}|{\mathcal{C}}_{9,\mathrm{NP}}^{e}|^{2}+|{\mathcal{C}}_{10,\mathrm{NP}}^{e}|^{2}\big{)}}\,,\\\
R_{K^{\ast}}^{[0.045,1.1]}&=R_{K^{\ast},\mathrm{SM}}^{[0.045,1.1]}\,\frac{1+0.040\,{\rm
Re}({\mathcal{C}}_{9,\mathrm{NP}}^{\mu})-0.080\,{\rm
Re}({\mathcal{C}}_{10,\mathrm{NP}}^{\mu})+0.010\,\big{(}|{\mathcal{C}}_{9,\mathrm{NP}}^{\mu}|^{2}+|{\mathcal{C}}_{10,\mathrm{NP}}^{\mu}|^{2}\big{)}}{1+0.037\,{\rm
Re}({\mathcal{C}}_{9,\mathrm{NP}}^{e})-0.075\,{\rm
Re}({\mathcal{C}}_{10,\mathrm{NP}}^{e})+0.009\,\big{(}|{\mathcal{C}}_{9,\mathrm{NP}}^{e}|^{2}+|{\mathcal{C}}_{10,\mathrm{NP}}^{e}|^{2}\big{)}}\,,\end{aligned}$
(B.12)

where the superscripts refer to the bins in $q^{2}\,[\mathrm{GeV}^{2}]$. As
can be seen from these expressions, $R_{K}^{\rm[1.1,6]}$ and
$R_{K^{\ast}}^{[1.1,6]}$ are insensitive to possible lepton-universal NP
contributions. A subleading sensitivity to universal corrections is present in
the low-$q^{2}$ bin of $R_{K^{\ast}}$, due to $m_{\ell}$-dependent kinematical
effects close to threshold (recall that $R_{K^{\ast}}^{[0.045,1.1]}\neq 1$
already in the SM). To accurately describe this effect, the expansion
coefficients are given to higher accuracy. The latter are estimated taking
into account also the non-universal QED corrections estimated in [64].

#### Leptonic and semileptonic $B$ decays

The theoretical expressions for the branching fractions of the relevant
leptonic and semileptonic $B$ decays are

$\displaystyle\begin{aligned}
\mathcal{B}(B_{i}\to\ell^{+}\ell^{-})&=\mathcal{B}(B_{i}\to\ell^{+}\ell^{-})_{\mathrm{SM}}\left\\{\left|1+\frac{{\mathcal{C}}_{10,\mathrm{NP}}^{i\ell}}{{\mathcal{C}}_{10,\mathrm{SM}}}+\frac{{\mathcal{C}}_{P}^{i\ell}}{C_{10,\mathrm{SM}}}\,\frac{m_{B_{i}}^{2}}{2m_{\ell}(m_{b}+m_{i})}\right|^{2}\right.\\\
&\qquad\qquad\qquad\qquad\qquad\left.+\left(1-\frac{4m_{\ell}^{2}}{m_{B_{i}}^{2}}\right)\left|\frac{{\mathcal{C}}_{S}^{i\ell}}{{\mathcal{C}}_{10,\mathrm{SM}}}\,\frac{m_{B_{i}}^{2}}{2m_{\ell}(m_{b}+m_{i})}\right|^{2}\right\\},\end{aligned}$
(B.13) $\displaystyle\mathcal{B}(B\to K\tau^{+}\tau^{-})$
$\displaystyle=10^{-9}\left(2.2\,\left|{\mathcal{C}}_{9}^{\tau}\right|^{2}+6.0\,\left|{\mathcal{C}}_{10}^{\tau}\right|^{2}+8.3\,\left|{\mathcal{C}}_{S}^{\tau}\right|^{2}+8.9\,\left|{\mathcal{C}}_{P}^{\tau}\right|^{2}\right.$
$\displaystyle\left.\quad+4.8\,\mathrm{Re}\left\\{{\mathcal{C}}_{S}^{\tau}\,{\mathcal{C}}_{9}^{\tau\ast}\right\\}+5.9\,\mathrm{Re}\left\\{{\mathcal{C}}_{P}^{\tau}\,{\mathcal{C}}_{10}^{\tau\ast}\right\\}\right)\,,$
(B.14) $\displaystyle\mathcal{B}(B^{+}\to K^{+}\tau^{+}\mu^{-})$
$\displaystyle=10^{-9}\left(9.6\,\left|{\mathcal{C}}_{9}^{\tau\mu}\right|^{2}+10\,\left|{\mathcal{C}}_{10}^{\tau\mu}\right|^{2}+13.58\,\left|{\mathcal{C}}_{S}^{\tau\mu}\right|^{2}+14.54\,\left|{\mathcal{C}}_{P}^{\tau\mu}\right|^{2}\right.$
$\displaystyle\left.\quad+12.41\,\mathrm{Re}\left\\{{\mathcal{C}}_{S}^{\tau\mu}\,{\mathcal{C}}_{9}^{\tau\mu\ast}\right\\}+15.20\,\mathrm{Re}\left\\{{\mathcal{C}}_{P}^{\tau\mu}\,{\mathcal{C}}_{10}^{\tau\mu\ast}\right\\}\right),$
(B.15) $\displaystyle\mathcal{B}(B_{s}\to\tau^{-}\mu^{+})$
$\displaystyle=\frac{\tau_{B_{s}}G_{F}^{2}f_{B_{s}}^{2}}{16\pi}\,m_{\tau}^{2}\,m_{B_{s}}\left(1-\frac{m_{\tau}^{2}}{m_{B_{s}}^{2}}\right)^{2}\frac{\alpha^{2}}{4\pi^{2}}\,\left|V_{tb}V_{ts}^{\ast}\right|^{2}$
(B.16)
$\displaystyle\quad\times\left\\{\left|{\mathcal{C}}_{10}^{\tau\mu}+\frac{m_{B_{s}}^{2}}{2m_{\tau}(m_{b}+m_{s})}\,{\mathcal{C}}_{P}^{\tau\mu}\right|^{2}+\left|{\mathcal{C}}_{9}^{\tau\mu}+\frac{m_{B_{s}}^{2}}{2m_{\tau}(m_{b}+m_{s})}\,{\mathcal{C}}_{S}^{\tau\mu}\right|^{2}\right\\}.$

The hadronic coefficients in (B.2) and (B.2) have been computed using inputs
from lattice QCD as compiled in [156].

###  B.3 $\tau$ decays

#### Universality tests in $\tau$ decays

$\tau$ leptonic decay rates provide a powerful test of lepton-flavor
universality via the ratios

$\displaystyle\begin{aligned}
\left(\frac{g_{\tau}}{g_{\mu(e)}}\right)_{\ell}&=\left[\frac{\mathcal{B}(\tau\to
e(\mu)\nu\bar{\nu})/\mathcal{B}(\tau\to e(\mu)\nu\bar{\nu})_{\rm
SM}}{\mathcal{B}(\mu\to e\nu\bar{\nu})/\mathcal{B}(\mu\to e\nu\bar{\nu})_{\rm
SM}}\right]^{\frac{1}{2}},\\\
\left(\frac{g_{\tau}}{g_{\mu}}\right)_{\pi}&=\left[\frac{\mathcal{B}(\tau\to\pi\nu)/\mathcal{B}(\tau\to\pi\nu)_{\rm
SM}}{\mathcal{B}(\pi\to\mu\bar{\nu})/\mathcal{B}(\pi\to\mu\bar{\nu})_{\rm
SM}}\right]^{\frac{1}{2}},\\\
\left(\frac{g_{\tau}}{g_{\mu}}\right)_{K}&=\left[\frac{\mathcal{B}(\tau\to
K\nu)/\mathcal{B}(\tau\to K\nu)_{\rm
SM}}{\mathcal{B}(K\to\mu\bar{\nu})/\mathcal{B}(K\to\mu\bar{\nu})_{\rm
SM}}\right]^{\frac{1}{2}}.\end{aligned}$ (B.17)

In terms of the high-energy Wilson coefficients, they are given by

$\displaystyle\left(\frac{g_{\tau}}{g_{\mu}}\right)_{\ell,\pi,K}\approx\left(\frac{g_{\tau}}{g_{e}}\right)_{\ell}\approx
1-0.079\,C_{LL}^{33\tau\tau}(\Lambda)\,,$ (B.18)

where the running is computed assuming $\Lambda=2\,\mathrm{TeV}$.

#### LFV $\tau$ decays

In our setup, the radiative decay $\tau\to\mu\gamma$ is conveniently described
via the effective Lagrangian

$\displaystyle\mathcal{L}_{\tau\to\mu\gamma}=-\frac{2}{v^{2}}\left[{\mathcal{C}}_{e\gamma}^{\mu\tau}{\mathcal{O}}_{e\gamma}^{\mu\tau}+{\mathcal{C}}_{LR}^{33\tau\mu}{\mathcal{O}}_{LR}^{33\tau\mu}+\mathrm{h.c.}\right],$
(B.19)

where ${\mathcal{O}}_{LR}^{33\tau\mu}$ has been defined in (2.1), and
$O_{e\gamma}^{\beta\alpha}=e\,(\bar{\ell}_{L}^{\beta}\sigma^{\mu\nu}e_{R}^{\alpha})HF_{\mu\nu}$.
The running of the Wilson coefficients from $\Lambda=2\leavevmode\nobreak\
{\rm TeV}$ to $m_{\tau}$ is given by

$\displaystyle\begin{aligned}
{\mathcal{C}}^{33\tau\mu}_{LR}(m_{\tau})&\approx\,\eta_{S}\,{\mathcal{C}}^{33\tau\mu}_{LR}(\Lambda)\,,\\\
{\mathcal{C}}_{e\gamma}^{\alpha\beta}(m_{\tau})&\approx
0.92\,{\mathcal{C}}_{e\gamma}^{\mu\tau}(\Lambda)\,,\end{aligned}$ (B.20)

and the Wilson coefficients at the high scale in the $U_{1}$ model are defined
in (3.4) and (4.18), respectively. Starting from (B.19), we find the decay
amplitude (with $q=p-p^{\prime}$)

$\displaystyle
i\mathcal{A}(\tau\to\mu\gamma)=-\frac{4e}{v^{2}}\,\bar{u}(p^{\prime})\,\sigma^{\alpha\nu}q_{\nu}\,\big{(}\mathcal{A}_{R}P_{R}+\mathcal{A}_{L}P_{L}\big{)}\,u(p)\,\varepsilon_{\alpha}^{\ast}(q)\,,$
(B.21)

with

$\displaystyle\begin{aligned}
\mathcal{A}_{R}&=\frac{v}{\sqrt{2}}\,\,{\mathcal{C}}_{e\gamma}^{\mu\tau}(m_{\tau})+\frac{m_{b}}{16\pi^{2}}\,\,{\mathcal{C}}^{33\tau\mu}_{LR}(m_{\tau})\,,\\\
\mathcal{A}_{L}&=\frac{v}{\sqrt{2}}\,\,{\mathcal{C}}_{e\gamma}^{\tau\mu\ast}(m_{\tau})\,.\end{aligned}$
(B.22)

Neglecting the muon mass, the branching ratio is given by

$\displaystyle\mathcal{B}(\tau\to\mu\gamma)$ $\displaystyle=\frac{8\hskip
0.85358ptG_{F}^{2}\hskip 0.85358pt\alpha\hskip
0.85358ptm_{\tau}^{3}}{\Gamma_{\tau}}\left(\left|\mathcal{A}_{R}\right|^{2}+\left|\mathcal{A}_{L}\right|^{2}\right).$
(B.23)

For the decay $\tau\to\mu\phi$, we find

$\displaystyle\mathcal{B}(\tau\to\mu\phi)$
$\displaystyle=\frac{1}{\Gamma_{\tau}}\frac{G_{F}^{2}f_{\phi}^{2}}{16\pi}\,m_{\tau}^{3}\left(1-\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\right)^{2}\left(1+2\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\right)\,\big{|}C_{LL}^{22\tau\mu}\big{|}^{2}\,,$
(B.24)

where $f_{\phi}\approx 225\leavevmode\nobreak\ {\rm MeV}$.

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