$\displaystyle=-\delta_{kl}\frac{\upsilon^{3}m_{d_{k}}}{\sqrt{2}m_{h}^{2}\Lambda^{2}}\,\mathcal{C}^{\ast}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}\,,$ $\displaystyle C^{(\ell)}_{\begin{subarray}{c}S_{RL}\\\
ijkk\end{subarray}}$
$\displaystyle=-\frac{v^{2}}{\Lambda^{2}}\frac{vm_{\ell_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}S_{LR}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=-\frac{v^{2}}{\Lambda^{2}}\frac{vm_{\ell_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}^{\ast}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}\,,$ $\displaystyle C^{(\ell)}_{\begin{subarray}{c}S_{RR}\\\
ijkk\end{subarray}}$
$\displaystyle=-\frac{v^{2}}{\Lambda^{2}}\frac{vm_{\ell_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}S_{LL}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=-\frac{v^{2}}{\Lambda^{2}}\frac{vm_{\ell_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}^{\ast}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}\,.$

### Tensor Operators

$\displaystyle C^{(u)}_{\begin{subarray}{c}T_{R}\\\ ijkl\end{subarray}}$
$\displaystyle=-\frac{\upsilon^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}^{\prime(3)}\,,$ $\displaystyle
C_{\begin{subarray}{c}T_{L}\\\ ijkl\end{subarray}}^{(u)}$
$\displaystyle=-\frac{\upsilon^{2}}{\Lambda^{2}}[\mathcal{C}_{\begin{subarray}{c}lequ\\\
jilk\end{subarray}}^{\prime(3)}]^{\ast}\,,$ $\displaystyle
C_{\begin{subarray}{c}T_{L}\\\ ijkl\end{subarray}}^{(d)}$
$\displaystyle=C_{\begin{subarray}{c}T_{R}\\\ ijkl\end{subarray}}^{(d)}=0\,,$
(103) $\displaystyle C_{\begin{subarray}{c}T_{L}\\\
ijkk\end{subarray}}^{(\ell)}$ $\displaystyle=C_{\begin{subarray}{c}T_{R}\\\
ijkk\end{subarray}}^{(\ell)}=0\,.$

### Leptonic Dipole Operators

$\displaystyle C_{\begin{subarray}{c}D_{R}\\\ ij\end{subarray}}^{(\ell)}$
$\displaystyle=\frac{\upsilon^{3}}{\sqrt{2}m_{\ell}\Lambda^{2}}\Big{(}-\sin{\theta_{W}}\,\mathcal{C}_{\begin{subarray}{c}eW\\\
ij\end{subarray}}+\cos{\theta_{W}}\,\mathcal{C}_{\begin{subarray}{c}eB\\\
ij\end{subarray}}\Big{)}\,,$ (104) $\displaystyle
C_{\begin{subarray}{c}D_{L}\\\ ij\end{subarray}}^{(\ell)}$
$\displaystyle=\frac{\upsilon^{3}}{\sqrt{2}m_{\ell}\Lambda^{2}}\Big{(}-\sin{\theta_{W}}\,\mathcal{C}_{\begin{subarray}{c}eW\\\
ji\end{subarray}}^{\ast}+\cos\theta_{W}\,\mathcal{C}_{\begin{subarray}{c}eB\\\
ji\end{subarray}}^{\ast}\Big{)}\,.$

##  Appendix D Semileptonic differential formulas

###  D.1 $P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$

The expression for the differential distribution of $P\to
P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$ can be expressed in full generality as

$\displaystyle\dfrac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}}=a(q^{2})+b(q^{2})\cos\theta_{\ell}+c(q^{2})\,\cos^{2}\theta_{\ell}\,,$
(105)

where $\theta_{\ell}$ stands for the angle between $\ell_{i}^{-}$ and the
$P^{\prime}$ meson line-of-flight, in the rest frame of the lepton-pair rest
frame, and $q^{2}$ is the dilepton invariant mass. The kinematical
coefficients $a(q^{2})$, $b(q^{2})$ and $c(q^{2})$ can be expressed in terms
of effective coefficients defined in Eq. (45)–(48) as follows,

$\displaystyle a(q^{2})$
$\displaystyle=\mathcal{N}_{P}\frac{\lambda_{\ell}^{1/2}}{4q^{2}}\bigg{[}|h_{0}^{V}(q^{2})|^{2}+|h_{0}^{A}(q^{2})|^{2}+\frac{(m_{\ell_{i}}-m_{\ell_{j}})^{2}}{q^{2}}|h_{t}^{VS}(q^{2})|^{2}+\frac{(m_{\ell_{i}}+m_{\ell_{j}})^{2}}{q^{2}}|h_{t}^{AP}(q^{2})|^{2}\bigg{]}\,,$
$\displaystyle b(q^{2})$
$\displaystyle=\mathcal{N}_{P}\frac{\lambda_{\ell}^{1/2}}{2q^{2}}(m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2})\,\text{Re}\Big{[}h_{0}^{V}(q^{2})\big{(}h_{t}^{VS}(q^{2})\big{)}^{\ast}+h_{0}^{A}(q^{2})h_{t}^{AP}(q^{2})^{\ast}\Big{]}\,,$
$\displaystyle c(q^{2})$
$\displaystyle=-\mathcal{N}_{P}\frac{\lambda_{\ell}}{4q^{4}}\Big{(}|h_{0}^{V}(q^{2})|^{2}+|h_{0}^{A}(q^{2})|^{2}\Big{)}\,,$
(106)

where the normalization is defined by,

$\displaystyle\mathcal{N}_{P}=\dfrac{\tau_{P}G_{F}^{2}\lambda_{P}^{1/2}\lambda_{\ell}^{1/2}}{512\pi^{3}m_{P}^{3}}\,,$
(107)

and we define $\lambda_{P}=\lambda(m_{P}^{2},m_{P^{\prime}}^{2},q^{2})$ and
$\lambda_{\ell}=\lambda(m_{\ell_{i}}^{2},m_{\ell_{j}}^{2},q^{2})$, with
$\lambda(a^{2},b^{2},c^{2})=(a^{2}-(b-c)^{2})(a^{2}-(b+c)^{2})$. The
transversity amplitudes for a decay based on the $d_{k}\to
d_{l}\ell_{i}^{-}\ell_{j}^{+}$ transition is given by the following equations,

$\displaystyle h_{0}^{V}(q^{2})$
$\displaystyle=\frac{C^{(q)}_{VV}}{\sqrt{q^{2}}}f_{+}(q^{2})\lambda_{P}^{1/2}\,,$
(108) $\displaystyle h_{0}^{A}(q^{2})$
$\displaystyle=\frac{C^{(q)}_{AV}}{\sqrt{q^{2}}}f_{+}(q^{2})\lambda_{P}^{1/2}\,,$
(109) $\displaystyle h_{t}^{VS}(q^{2})$
$\displaystyle=\Big{(}C^{(q)}_{VV}+\frac{C^{(q)}_{SS}}{m_{\ell_{i}}-m_{\ell_{j}}}\frac{q^{2}}{m_{q_{k}}-m_{q_{l}}}\Big{)}\frac{m_{P}^{2}-m_{P^{\prime}}^{2}}{\sqrt{q^{2}}}f_{0}(q^{2})\,,$
(110) $\displaystyle h_{t}^{AP}(q^{2})$
$\displaystyle=\Big{(}C^{(q)}_{AV}+\frac{C^{(q)}_{PS}}{m_{\ell_{i}}+m_{\ell_{j}}}\frac{q^{2}}{m_{q_{k}}-m_{q_{l}}}\Big{)}\frac{m_{P}^{2}-m_{P^{\prime}}^{2}}{\sqrt{q^{2}}}f_{0}(q^{2})\,.$
(111)

where flavor indices should be replaced following Eq. (56). The form-factors
in the above equations are defined as usual,

$\displaystyle\langle
P^{\prime}(k)|\bar{q}_{l}\gamma^{\mu}q_{k}|P(p)\rangle=\Big{[}(p+k)^{\mu}-\dfrac{(m_{P}^{2}-m_{P^{\prime}}^{2})}{q^{2}}q^{\mu}\Big{]}f_{+}(q^{2})+\dfrac{m_{P}^{2}-m_{P^{\prime}}^{2}}{q^{2}}q^{\mu}f_{0}(q^{2})$
(112)

where $f_{+}$ ($f_{0}$) stands for the vector (scalar) form-factors. Note, in
particular, that the above expressions are in agreement with the ones obtained
for the charged current decays $d\to u\ell\bar{\nu}$ in the limit where
$m_{\ell}\to 0$, as provided e.g. in Ref. [102]. 161616Notice that we have
neglected tensor operators in the above formulas since they represent sub-
leading corrections to the $d_{k}\to d_{l}\ell_{i}^{-}\ell_{j}^{+}$ transition
in the SM EFT. See Ref. [103] for a calculation of the branching fractions
including these operators.

###  D.2 $P\to V\ell_{i}^{-}\ell_{j}^{+}$

The expressions for the angular distributions of $P\to
V\ell_{i}^{-}\ell_{j}^{+}$ have been derived in Ref. [42]. For completeness,
we collect these expressions in this Appendix,

$\displaystyle\dfrac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}\mathrm{d}\cos\theta_{\ell}\mathrm{d}\cos\theta_{K}\mathrm{d}\phi}=\dfrac{9}{32\pi}I(q^{2},\theta_{\ell},\theta_{K},\phi)\,,$
(113)

with

$\displaystyle I(q^{2},\theta_{\ell},\theta_{K},\phi)$ $\displaystyle\equiv
I_{1}^{s}(q^{2})\sin^{2}\theta_{K}+I_{1}^{c}(q^{2})\cos^{2}\theta_{K}+[I_{2}^{s}(q^{2})\sin^{2}\theta_{K}+I_{2}^{c}(q^{2})\,\cos^{2}\theta_{K}]\cos
2\theta_{\ell}$
$\displaystyle+I_{3}(q^{2})\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\phi+I_{4}(q^{2})\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\phi$
$\displaystyle+I_{5}(q^{2})\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi+[I_{6}^{s}(q^{2})\sin^{2}\theta_{K}+I_{6}^{c}(q^{2})\,\cos^{2}\theta_{K}]\cos\theta_{\ell}$
$\displaystyle+I_{7}(q^{2})\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi+I_{8}(q^{2})\sin 2\theta_{K}\sin
2\theta_{\ell}\sin\phi$
$\displaystyle+I_{9}(q^{2})\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\phi\,,$ (114)

where we adopt the same angular conventions used for
$\overline{B}\to\overline{K}^{\ast 0}(\to K\pi)\ell_{i}^{-}\ell_{j}^{+}$ in
Ref. [42]. These angular coefficients can be expressed in terms of effective
coefficients defined in Eq. (45)–(48) [42],

$\displaystyle I_{1}^{s}(q^{2})$
$\displaystyle=\Big{[}|A_{\perp}^{L}|^{2}+|A_{\parallel}^{L}|^{2}+(L\to
R)\Big{]}\frac{\lambda_{\ell}+2[q^{4}-(m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2})^{2}]}{4q^{4}}$
$\displaystyle\hskip
85.35826pt+\frac{4m_{\ell_{i}}m_{\ell_{j}}}{q^{2}}\,\mathrm{Re}\left(A_{\parallel}^{L}A_{\parallel}^{R\ast}+A_{\perp}^{L}A_{\perp}^{R\ast}\right)\,,$
(115) $\displaystyle I_{1}^{c}(q^{2})$
$\displaystyle=\bigl{[}|A_{0}^{L}|^{2}+|A_{0}^{R}|^{2}\bigr{]}\frac{q^{4}-(m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2})^{2}}{q^{4}}+\frac{8m_{\ell_{i}}m_{\ell_{j}}}{q^{2}}\mathrm{Re}(A_{0}^{L}A_{0}^{R\ast}-A_{t}^{L}A_{t}^{R\ast})$
$\displaystyle\hskip
85.35826pt-2\frac{(m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2})^{2}-q^{2}(m_{\ell_{i}}^{2}+m_{\ell_{j}}^{2})}{q^{4}}\bigl{(}|A_{t}^{L}|^{2}+|A_{t}^{R}|^{2}\bigr{)}\,,$
(116) $\displaystyle I_{2}^{s}(q^{2})$
$\displaystyle=\frac{\lambda_{\ell}}{4q^{4}}\,[|A_{\perp}^{L}|^{2}+|A_{\parallel}^{L}|^{2}+(L\to
R)]\,,$ (117) $\displaystyle I_{2}^{c}(q^{2})$
$\displaystyle=-\frac{\lambda_{\ell}}{q^{4}}\,(|A_{0}^{L}|^{2}+|A_{0}^{R}|^{2})\,,$
(118) $\displaystyle I_{3}(q^{2})$
$\displaystyle=\frac{\lambda_{\ell}}{2q^{4}}\,[|A_{\perp}^{L}|^{2}-|A_{\parallel}^{L}|^{2}+(L\to
R)]\,,$ (119) $\displaystyle I_{4}(q^{2})$
$\displaystyle=-\frac{\lambda_{\ell}}{\sqrt{2}q^{4}}\,[\mathrm{Re}(A_{\parallel}^{L}A_{0}^{L\ast})+(L\to
R)]\,,$ (120) $\displaystyle I_{5}(q^{2})$
$\displaystyle=\frac{\sqrt{2}\lambda_{\ell}^{1/2}}{q^{2}}\Big{[}\mathrm{Re}(A_{0}^{L}A_{\perp}^{L\ast}-(L\to
R))-\frac{m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2}}{q^{2}}\mathrm{Re}(A_{t}^{L}A_{\parallel}^{L\ast}+(L\to
R))\Big{]}\,,$ (121) $\displaystyle I_{6}^{s}(q^{2})$
$\displaystyle=-\frac{2\lambda_{\ell}^{1/2}}{q^{2}}[\mathrm{Re}(A_{\parallel}^{L}A_{\perp}^{L\ast}-(L\to
R))]\,,$ (122) $\displaystyle I_{6}^{c}(q^{2})$
$\displaystyle=-\frac{4\lambda_{\ell}^{1/2}}{q^{2}}\frac{m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2}}{q^{2}}\mathrm{Re}(A_{0}^{L}A_{t}^{L\ast}+(L\to
R))\,,$ (123) $\displaystyle I_{7}(q^{2})$
$\displaystyle=-\frac{\sqrt{2}\lambda_{\ell}^{1/2}}{q^{2}}\Big{[}\mathrm{Im}(A_{0}^{L}A_{\parallel}^{L\ast}-(L\to
R))+\frac{m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2}}{q^{2}}\,\mathrm{Im}(A_{\perp}^{L}A_{t}^{L\ast}+(L\to
R))\Big{]}\,,$ (124) $\displaystyle I_{8}(q^{2})$
$\displaystyle=\frac{\lambda_{\ell}}{\sqrt{2}q^{4}}\,\mathrm{Im}(A_{0}^{L}A_{\perp}^{L\ast}+(L\to
R)),$ (125) $\displaystyle I_{9}(q^{2})$
$\displaystyle=-\frac{\lambda_{\ell}}{q^{4}}\,\mathrm{Im}(A_{\perp}^{L}A_{\parallel}^{L\ast}+A_{\perp}^{R}A_{\parallel}^{R\ast})\,,$
(126)

where the transversity amplitudes are defined by

$\displaystyle A_{\perp}^{L(R)}$
$\displaystyle=\mathcal{N}_{V}\sqrt{2}\lambda_{V}^{1/2}\dfrac{V(q^{2})}{m_{P}+m_{V}}\big{(}C_{VV}^{(q)}\mp
C_{VA}^{(q)}\big{)}\,,$ (127) $\displaystyle A_{\parallel}^{L(R)}$
$\displaystyle=\mathcal{N}_{V}\sqrt{2}(m_{P}+m_{V})A_{1}(q^{2})\big{(}C_{AV}^{(q)}\mp
C_{AA}^{(q)}\big{)}\,,$ (128) $\displaystyle A_{0}^{L(R)}$
$\displaystyle=\dfrac{\mathcal{N}_{V}(m_{P}+m_{V})}{2m_{V}\sqrt{q^{2}}}\bigg{[}(m_{P}^{2}-m_{V}^{2}-q^{2})A_{1}(q^{2})-\dfrac{\lambda_{V}A_{2}(q^{2})}{m_{P}+m_{V}}\bigg{]}\big{(}C_{AV}^{(q)}\mp
C_{AA}^{(q)}\big{)}\,,$ (129) $\displaystyle A_{t}^{L(R)}$
$\displaystyle=\mathcal{N}_{V}\dfrac{\lambda_{P}^{1/2}}{\sqrt{q^{2}}}A_{0}(q^{2})\bigg{[}C_{AV}^{(q)}\mp
C_{AA}^{(q)}-\dfrac{q^{2}}{m_{q_{k}}+m_{q_{l}}}\bigg{(}\dfrac{C_{PS}^{(q)}}{m_{\ell_{i}}-m_{\ell_{j}}}\mp\dfrac{C_{PP}^{(q)}}{m_{\ell_{i}}+m_{\ell_{j}}}\bigg{)}\bigg{]}\,,$
(130)

where, for simplicity, flavor indices are omitted. The square of the
normalization reads

$\displaystyle\mathcal{N}_{V}^{\,2}\equiv\dfrac{\tau_{P}\,G_{F}^{2}\lambda_{V}^{1/2}\lambda_{\ell}^{1/2}}{3\times
2^{10}\pi^{3}m_{P}^{3}}\,,$ (131)

where $\lambda_{V}=\lambda(m_{P}^{2},m_{V}^{2},q^{2})$ and
$\lambda_{\ell}=\lambda(m_{\ell_{i}}^{2},m_{\ell_{j}}^{2},q^{2})$, and the
$P\to V$ form-factors are defined as follows [104],

$\displaystyle\langle V(k)|\bar{q}_{l}\gamma^{\mu}(1-\gamma_{5})q_{k}$
$\displaystyle|P(p)\rangle=\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{\ast\nu}p^{\rho}k^{\sigma}\dfrac{2V(q^{2})}{m_{P}+m_{V}}-i\varepsilon_{\mu}^{\ast}(m_{P}+m_{V})A_{1}(q^{2})$
(132) $\displaystyle+i(p+k)_{\mu}(\varepsilon^{\ast}\cdot
q)\dfrac{A_{2}(q^{2})}{m_{P}+m_{V}}+iq_{\mu}(\varepsilon^{\ast}\cdot
q)\dfrac{2m_{V}}{q^{2}}[A_{3}(q^{2})-A_{0}(q^{2})]\,,$

where $\varepsilon^{\mu}$ is the polarization vector of the $V$-meson, and the
form-factor $A_{3}(q^{2})$ is related to $A_{1,2}(q^{2})$ via
$2m_{V}A_{3}(q^{2})=(m_{P}+m_{V})A_{1}(q^{2})-(m_{P}-m_{V})A_{2}(q^{2})$.
Notice, once again, that we have not included the contributions from tensor
operators in the above expressions.

##  Appendix E SM Yukawa running

In this Appendix, we briefly discuss the impact of the SM Yukawa
renormalization, which can impact semileptonic observables through loop-
induced non-diagonal elements that are not necessarily present in the
ultraviolet. More specifically, the Yukawa RGEs read [105]

$\displaystyle
16\pi^{2}\dfrac{\mathrm{d}\,y_{d}}{\mathrm{d}\log\mu}\simeq\dfrac{3}{2}\big{(}y_{d}\,y_{d}^{\dagger}\,y_{d}-y_{d}\,y_{u}^{\dagger}\,y_{u}\big{)}+3\,\mathrm{Tr}\big{[}y_{u}^{\dagger}\,y_{d}+y_{d}^{\dagger}\,y_{d}\big{]}y_{d}-8\,g_{3}^{2}y_{d}\,,$
(133)

where the up- and down-type quark Yukawas are denoted by $y_{u}$ and $y_{d}$,
respectively, and we have neglected the lepton Yukawas and the electroweak
gauge couplings, cf. Appendix A for our notation. Keeping only the top-quark
Yukawa contribution, it is straightforward to show that off-diagonal
contributions are induced via the Yukawa running,

$\displaystyle\qquad y_{\begin{subarray}{c}d\\\
ij\end{subarray}}(\mu=m_{t})\simeq\dfrac{3V_{ti}V_{tj}^{\ast}\,y_{b}y_{t}^{2}}{32\pi^{2}}\log\bigg{(}\dfrac{\Lambda}{m_{t}}\bigg{)}\,,\qquad\qquad\quad(i\neq
j)\,,$ (134)

where $y_{b}$ and $y_{t}$ are the physical top- and bottom-quark Yukawas to
first approximation. The contributions in Eq. (134) must be reabsorbed when
diagonalizing the SM Yukawas at $\mu\simeq\mu_{\mathrm{ew}}$ via the following
rotation of the left-handed down-type quarks,

$U_{d_{L}}\simeq{\mathbb{1}_{3\times
3}}+\frac{3y_{t}^{2}}{32\pi^{2}}\log\bigg{(}\frac{\Lambda}{m_{t}}\bigg{)}\begin{pmatrix}\begin{array}[]{ccc}0&0&V_{td}^{\ast}V_{tb}\\\
0&0&V_{ts}^{\ast}V_{tb}\\\ -V_{td}V_{tb}^{\ast}&-V_{ts}V_{tb}^{\ast}&0\\\
\end{array}\end{pmatrix}\;,$ (135)

whereas the other fermions are not affected by the running to first
approximation. These effects are included in our analysis, but we find that
they are sub-leading for the effective scenarios and the observables that we
consider in Sec. 5. Explicit examples where these effects could be larger have
been discussed e.g. in Ref. [106].

##  Appendix F Tree-level mediators

In this Appendix, we collect the Lagrangian of each tree-level mediator
considered in Sec. 5.2, as well as the matching to the SM EFT semileptonic
Wilson coefficients at tree level. For simplicity, we omit $SU(3)_{c}$ and
$SU(2)_{L}$ indices in the expressions below.

  * $\bullet$

$\Phi^{\prime}\sim(\mathbf{1},\,\mathbf{2},\,\frac{1}{2})$ : 171717For
simplicity, we assume that $\Phi^{\prime}$ does not acquire a vacuum
expectation value and does not mix with the SM Higgs.

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{\Phi^{\prime}}\supset-(y_{\Phi^{\prime}}^{d})_{ij}\,\Phi^{\prime\dagger}\bar{d}_{i}q_{j}-(y_{\Phi^{\prime}}^{u})_{ij}\,\widetilde{\Phi}^{\prime\dagger}\bar{u}_{i}q_{j}-(y_{\Phi^{\prime}}^{e})_{ij}\,\Phi^{\prime\dagger}\bar{e}_{i}l_{j}+\mathrm{h.c.}$
(136)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(y_{\Phi}^{u})_{kl}(y_{\Phi}^{e})_{ji}^{\ast}}{M_{\Phi}^{2}}\,,\qquad\quad\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ledq\\\
ijkl\end{subarray}}=\frac{(y_{\Phi}^{d})_{kl}(y_{\Phi}^{e})_{ji}^{\ast}}{M_{\Phi}^{2}}\,.$
(137)

  * $\bullet$

$Z^{\prime}\sim(\mathbf{1},\,\mathbf{1},\,0)$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{Z^{\prime}}\supset\sum_{\psi=l,q,e,u,d}(g_{Z^{\prime}}^{\psi})_{ij}\,\bar{\psi}_{i}$
$\displaystyle\gamma_{\mu}\psi_{j}\,Z^{\prime\mu}\,.$ (138)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(g_{Z^{\prime}}^{l})_{ij}(g_{Z^{\prime}}^{q})_{kl}^{\ast}}{M_{Z^{\prime}}^{2}}\,,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eq\\\
klij\end{subarray}}$
$\displaystyle=-\frac{(g_{Z^{\prime}}^{e})_{ij}(g_{Z^{\prime}}^{q})_{kl}^{\ast}}{M_{Z^{\prime}}^{2}}\,,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lu\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(g_{Z^{\prime}}^{l})_{ij}(g_{Z^{\prime}}^{u})_{kl}^{\ast}}{M_{Z^{\prime}}^{2}}\,,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eu\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(g_{Z^{\prime}}^{e})_{ij}(g_{Z^{\prime}}^{u})_{kl}^{\ast}}{M_{Z^{\prime}}^{2}}\,,$
(139) $\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ld\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(g_{Z^{\prime}}^{l})_{ij}(g_{Z^{\prime}}^{d})_{kl}^{\ast}}{M_{Z^{\prime}}^{2}}\,,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ed\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(g_{Z^{\prime}}^{e})_{ij}(g_{Z^{\prime}}^{d})_{kl}^{\ast}}{M_{Z^{\prime}}^{2}}\,.$

  * $\bullet$

$V\sim(\mathbf{1},\,\mathbf{3},\,0)$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{V}\supset(g_{V}^{q})_{ij}\,(\bar{q}_{i}\tau^{I}\gamma^{\mu}q_{j})\,V^{I}_{\mu}+(g_{V}^{l})_{ij}\,(\bar{l}_{i}\tau^{I}\gamma^{\mu}\l_{j})\,V^{I}_{\mu}\,.$
(140)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(g_{V}^{l})_{ij}(g_{V}^{q})_{kl}}{M_{V}^{2}}\;.$ (141)

  * $\bullet$

$S_{1}\sim(\mathbf{\bar{3}},\,\mathbf{1},\,\frac{1}{3})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{S_{1}}\supset(y^{L}_{S_{1}})_{ij}\,\bar{q}_{i}^{c}i\tau_{2}l_{j}S_{1}+(y^{R}_{S_{1}})_{ij}\,\bar{u}_{i}^{c}e_{j}S_{1}+\text{h.c.}$
(142)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(y^{L}_{S_{1}})_{lj}(y^{L}_{S_{1}})^{\ast}_{ki}}{4M_{S_{1}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(y^{L}_{S_{1}})_{lj}(y^{L}_{S_{1}})^{\ast}_{ki}}{4M_{S_{1}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(y^{R}_{S_{1}})_{lj}(y^{L}_{S_{1}})^{\ast}_{ki}}{2M_{S_{1}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(y^{R}_{S_{1}})_{lj}(y^{L}_{S_{1}})^{\ast}_{ki}}{8M_{S_{1}}^{2}}\;,$
(143) $\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eu\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(y^{R}_{S_{1}})_{lj}(y^{R}_{S_{1}})^{\ast}_{ki}}{2M_{S_{1}}^{2}}\;.$

  * $\bullet$

$\widetilde{S}_{1}\sim(\mathbf{\bar{3}},\,\mathbf{1},\,\frac{4}{3})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{\widetilde{S}_{1}}\supset(y_{\widetilde{S}_{1}}^{R})_{ij}\,\bar{d_{i}^{c}}e_{j}\,\widetilde{S}_{1}+\text{h.c.}$
(144)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ed\\\
ijkl\end{subarray}}=\frac{(y_{\widetilde{S}_{1}}^{R})_{lj}(y_{\widetilde{S}_{1}}^{R})^{\ast}_{ki}}{2M_{\widetilde{S}_{1}}^{2}}\;.$
(145)

  * $\bullet$

$S_{3}\sim(\mathbf{\bar{3}},\,\mathbf{3},\,\frac{1}{3})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{S_{3}}\supset$
$\displaystyle(y^{L}_{S_{3}})_{ij}\,\big{(}\bar{q}_{i}^{c}i\tau_{2}\tau^{I}l_{j}\big{)}\,S_{3}^{I}+\text{h.c.}$
(146)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{3(y^{L}_{S_{3}})_{lj}(y^{L}_{S_{3}})^{\ast}_{ki}}{4M_{S_{3}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}=\frac{(y^{L}_{S_{3}})_{lj}(y^{L}_{S_{3}})^{\ast}_{ki}}{4M_{S_{3}}^{2}}\;.$
(147)

  * $\bullet$

$R_{2}\sim(\mathbf{3},\,\mathbf{2},\,\frac{7}{6})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{R_{2}}\supset-(y_{R_{2}}^{L})_{ij}\,$
$\displaystyle\bar{u}_{i}R_{2}i\tau_{2}l_{j}-(y_{R_{2}}^{R})_{ij}\,\bar{q}_{i}e_{j}R_{2}+\text{h.c.}$
(148)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lu\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(y_{R_{2}}^{L})_{kj}(y_{R_{2}}^{L})_{li}^{\ast}}{2M_{R_{2}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(y_{R_{2}}^{R})_{kj}(y_{R_{2}}^{R})_{li}^{\ast}}{2M_{R_{2}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(y_{R_{2}}^{R})_{kj}(y_{R_{2}}^{L})_{li}^{\ast}}{2M_{R_{2}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(y_{R_{2}}^{R})_{kj}(y_{R_{2}}^{L})_{li}^{\ast}}{8M_{R_{2}}^{2}}\;.$
(149)

  * $\bullet$

$\widetilde{R}_{2}\sim(\mathbf{3},\,\mathbf{2},\,\frac{1}{6})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{\widetilde{R}_{2}}\supset-(y^{L}_{\widetilde{R}_{2}})_{ij}\,\bar{d_{i}}\widetilde{R}_{2}i\tau_{2}l_{j}+\text{h.c.}$
(150)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ld\\\
ijkl\end{subarray}}=-\frac{(y^{L}_{\widetilde{R}_{2}})_{kj}(y^{L}_{\widetilde{R}_{2}})^{\ast}_{li}}{2M^{2}_{\widetilde{R}_{2}}}\;.$
(151)

  * $\bullet$

$U_{1}\sim(\mathbf{3},\,\mathbf{1},\,\frac{2}{3})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{U_{1}}\supset(x_{U_{1}}^{L})_{ij}\,\bar{q}_{i}\gamma^{\mu}l_{j}\,U_{1\mu}+(x_{U_{1}}^{R})_{ij}\,\bar{d}_{i}\gamma^{\mu}e_{j}\,U_{1\mu}+\text{h.c.}$
(152)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(x_{U_{1}}^{L})_{kj}(x_{U_{1}}^{L})_{li}^{\ast}}{2M_{U_{1}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(x_{U_{1}}^{L})_{kj}(x_{U_{1}}^{L})_{li}^{\ast}}{2M_{U_{1}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ed\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{(x_{U_{1}}^{R})_{kj}(x_{U_{1}}^{R})_{li}^{\ast}}{M_{U_{1}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ledq\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{2(x_{U_{1}}^{R})_{kj}(x_{U_{1}}^{L})_{li}^{\ast}}{M_{U_{1}}^{2}}\;.$
(153)

  * $\bullet$

$\widetilde{U}_{1}\sim(\mathbf{3},\,\mathbf{1},\,\frac{5}{3})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{\widetilde{U}_{1}}\supset(x_{\widetilde{U}_{1}}^{R})_{ij}\,\bar{u}_{i}\gamma_{\mu}e_{j}\,\widetilde{U}^{\mu}_{1}+\text{h.c.}$
(154)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eu\\\
ijkl\end{subarray}}=-\frac{(x_{\widetilde{U}_{1}}^{R})_{kj}(x_{\widetilde{U}_{1}}^{R})^{\ast}_{li}}{M_{\widetilde{U}_{1}}^{2}}\;.$
(155)

  * $\bullet$

$V_{2}\sim(\mathbf{\bar{3}},\,\mathbf{2},\,\frac{5}{6})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{V_{2}}\supset(x_{V_{2}}^{L})_{ij}\,\bar{d}^{c}_{i}\gamma_{\mu}V^{\mu}_{2}i\tau_{2}l_{j}+(x_{V_{2}}^{R})_{ij}\,\bar{q}^{c}_{i}\gamma_{\mu}V^{\mu}_{2}i\tau_{2}e_{j}+\text{h.c.}$
(156)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eq\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(x_{V_{2}}^{R})_{lj}(x_{V_{2}}^{R})^{\ast}_{ki}}{M_{V_{2}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ld\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(x_{V_{2}}^{L})_{lj}(x_{V_{2}}^{L})^{\ast}_{ki}}{M_{V_{2}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ledq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{2(x_{V_{2}}^{R})_{lj}(x_{V_{2}}^{L})^{\ast}_{ki}}{M_{V_{2}}^{2}}\;.$
(157)

  * $\bullet$

$\widetilde{V}_{2}\sim(\mathbf{\bar{3}},\,\mathbf{2},\,-\frac{1}{6})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{\widetilde{V}_{2}}\supset(x_{\widetilde{V}_{2}}^{L})_{ij}\,\bar{u}^{c}_{i}\gamma_{\mu}\widetilde{V}^{\mu}_{2}i\tau_{2}l_{j}+\text{h.c.}$
(158)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lu\\\
ijkl\end{subarray}}=\frac{(x_{\widetilde{V}_{2}}^{L})_{lj}(x_{\widetilde{V}_{2}}^{L})^{\ast}_{ki}}{M_{\widetilde{V}_{2}}^{2}}\;.$
(159)

  * $\bullet$

$U_{3}\sim(\mathbf{3},\,\mathbf{3},\,\frac{2}{3})$ :

    * -

UV Lagrangian:

$\displaystyle\mathcal{L}_{U_{3}}\supset$
$\displaystyle(x_{U_{3}})_{ij}\,\big{(}\bar{q}_{i}\gamma^{\mu}\tau^{I}l_{j}\big{)}U_{3\mu}^{I}+\text{h.c.}$
(160)

    * -

SM EFT coefficients:

$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(1)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=-\frac{3(x_{U_{3}})_{kj}(x_{U_{3}})_{li}^{\ast}}{2M_{U_{3}}^{2}}\;,$
$\displaystyle\frac{1}{\Lambda^{2}}\mathcal{C}^{(3)}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{(x_{U_{3}})_{kj}(x_{U_{3}})_{li}^{\ast}}{2M_{U_{3}}^{2}}\;.$
(161)

## References

  * [1] S. Weinberg, Phys. Rev. Lett. 43 (1979), 1566-1570
  * [2] S. T. Petcov, Sov. J. Nucl. Phys. 25 (1977), 340 [erratum: Sov. J. Nucl. Phys. 25 (1977), 698; erratum: Yad. Fiz. 25 (1977), 1336] JINR-E2-10176; A. Ilakovac and A. Pilaftsis, Nucl. Phys. B 437 (1995), 491 [arXiv:hep-ph/9403398 [hep-ph]].
  * [3] L. Calibbi and G. Signorelli, Riv. Nuovo Cim. 41 (2018) no.2, 71-174 [arXiv:1709.00294 [hep-ph]].
  * [4] A. M. Baldini et al. [MEG II], Eur. Phys. J. C 78 (2018) no.5, 380 [arXiv:1801.04688 [physics.ins-det]].
  * [5] L. Bartoszek et al. [Mu2e], [arXiv:1501.05241 [physics.ins-det]].
  * [6] A. Blondel, A. Bravar, M. Pohl, S. Bachmann, N. Berger, M. Kiehn, A. Schoning, D. Wiedner, B. Windelband and P. Eckert, et al. [arXiv:1301.6113 [physics.ins-det]].
  * [7] R. Abramishvili et al. [COMET], PTEP 2020 (2020) no.3, 033C01 [arXiv:1812.09018 [physics.ins-det]].
  * [8] E. Kou et al. [Belle-II], PTEP 2019 (2019) no.12, 123C01 [erratum: PTEP 2020 (2020) no.2, 029201] [arXiv:1808.10567 [hep-ex]].
  * [9] R. Aaij et al. [LHCb], JHEP 03 (2018), 078 [arXiv:1710.04111 [hep-ex]]; R. Aaij et al. [LHCb], Phys. Rev. Lett. 123 (2019) no.24, 241802 [arXiv:1909.01010 [hep-ex]]; R. Aaij et al. [LHCb], JHEP 06 (2023), 073 [arXiv:2207.04005 [hep-ex]]; R. Aaij et al. [LHCb], JHEP 06 (2023), 143 [arXiv:2209.09846 [hep-ex]].
  * [10] L. Nayak et al. [Belle], JHEP 08 (2023), 178 [arXiv:2301.10989 [hep-ex]]; S. Watanuki et al. [Belle], Phys. Rev. Lett. 130 (2023) no.26, 261802 [arXiv:2212.04128 [hep-ex]].
  * [11] E. Cortina Gil et al. [NA62], JINST 12 (2017) no.05, P05025 [arXiv:1703.08501 [physics.ins-det]].
  * [12] T. Yamanaka [KOTO], PTEP 2012 (2012), 02B006
  * [13] M. Ablikim et al. [BESIII], Chin. Phys. C 44 (2020) no.4, 040001 [arXiv:1912.05983 [hep-ex]].
  * [14] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, JHEP 10 (2010), 085 [arXiv:1008.4884 [hep-ph]]; W. Buchmuller and D. Wyler, Nucl. Phys. B 268 (1986), 621-653
  * [15] Y. Kuno and Y. Okada, Rev. Mod. Phys. 73 (2001), 151-202 [arXiv:hep-ph/9909265 [hep-ph]].
  * [16] A. Crivellin, S. Davidson, G. M. Pruna and A. Signer, JHEP 05 (2017), 117 [arXiv:1702.03020 [hep-ph]].
  * [17] J. Elias Miro, C. Fernandez, M. A. Gumus and A. Pomarol, JHEP 06 (2022), 126 [arXiv:2112.12131 [hep-ph]].
  * [18] G. M. Pruna and A. Signer, JHEP 10 (2014), 014 [arXiv:1408.3565 [hep-ph]]; S. Davidson, Y. Kuno and M. Yamanaka, Phys. Lett. B 790 (2019), 380-388 [arXiv:1810.01884 [hep-ph]]; S. Davidson, JHEP 02 (2021), 172 [arXiv:2010.00317 [hep-ph]]; S. Davidson and B. Echenard, Eur. Phys. J. C 82 (2022) no.9, 836 [arXiv:2204.00564 [hep-ph]].
  * [19] E. E. Jenkins, A. V. Manohar and P. Stoffer, JHEP 01 (2018), 084 [arXiv:1711.05270 [hep-ph]].
  * [20] E. E. Jenkins, A. V. Manohar and M. Trott, JHEP 10 (2013), 087 [arXiv:1308.2627 [hep-ph]]; E. E. Jenkins, A. V. Manohar and M. Trott, JHEP 01 (2014), 035 [arXiv:1310.4838 [hep-ph]]; R. Alonso, E. E. Jenkins, A. V. Manohar and M. Trott, JHEP 04 (2014), 159 [arXiv:1312.2014 [hep-ph]].
  * [21] S. Davidson and A. Saporta, Phys. Rev. D 99 (2019) no.1, 015032 [arXiv:1807.10288 [hep-ph]]; J. Kumar, JHEP 01 (2022), 107 [arXiv:2107.13005 [hep-ph]]; L. Calibbi, T. Li, X. Marcano and M. A. Schmidt, Phys. Rev. D 106 (2022) no.11, 115039 [arXiv:2207.10913 [hep-ph]].
  * [22] F. Feruglio, P. Paradisi and A. Pattori, Phys. Rev. Lett. 118 (2017) no.1, 011801 [arXiv:1606.00524 [hep-ph]]; F. Feruglio, P. Paradisi and A. Pattori, JHEP 09 (2017), 061 [arXiv:1705.00929 [hep-ph]]; C. Cornella, F. Feruglio and P. Paradisi, JHEP 11 (2018), 012 [arXiv:1803.00945 [hep-ph]].
  * [23] S. Davidson, Eur. Phys. J. C 76 (2016) no.7, 370 [arXiv:1601.07166 [hep-ph]].
  * [24] F. Fortuna, A. Ibarra, X. Marcano, M. Marín and P. Roig, Phys. Rev. D 107 (2023) no.1, 015027 [arXiv:2210.05703 [hep-ph]].
  * [25] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. B 78 (1978), 443-446; J. Hisano, K. Ishiwata and N. Nagata, Phys. Rev. D 82 (2010), 115007 [arXiv:1007.2601 [hep-ph]].
  * [26] B. Bhattacharya, R. Morgan, J. Osborne and A. A. Petrov, Phys. Lett. B 785 (2018), 165-170 [arXiv:1802.06082 [hep-ph]]; Y. Cai and M. A. Schmidt, JHEP 02 (2016), 176 [arXiv:1510.02486 [hep-ph]]; Y. Cai, M. A. Schmidt and G. Valencia, JHEP 05 (2018), 143 [arXiv:1802.09822 [hep-ph]]; V. Cirigliano, K. Fuyuto, C. Lee, E. Mereghetti and B. Yan, JHEP 03 (2021), 256 [arXiv:2102.06176 [hep-ph]]; A. A. Petrov and D. V. Zhuridov, Phys. Rev. D 89 (2014) no.3, 033005 [arXiv:1308.6561 [hep-ph]].
  * [27] M. González-Alonso, J. Martin Camalich and K. Mimouni, Phys. Lett. B 772 (2017), 777-785 [arXiv:1706.00410 [hep-ph]].
  * [28] A. Crivellin, S. Najjari and J. Rosiek, JHEP 04 (2014), 167 doi:10.1007/JHEP04(2014)167 [arXiv:1312.0634 [hep-ph]].
  * [29] F. Feruglio, P. Paradisi and O. Sumensari, JHEP 11 (2018), 191 [arXiv:1806.10155 [hep-ph]].
  * [30] P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01
  * [31] M. Ardu and S. Davidson, JHEP 08 (2021), 002 [arXiv:2103.07212 [hep-ph]];
  * [32] M. Ciuchini, E. Franco, L. Reina and L. Silvestrini, Nucl. Phys. B 421 (1994), 41-64 [arXiv:hep-ph/9311357 [hep-ph]].
  * [33] E. E. Jenkins, A. V. Manohar and P. Stoffer, JHEP 03 (2018), 016 [arXiv:1709.04486 [hep-ph]].
  * [34] R. Alonso, B. Grinstein and J. Martin Camalich, Phys. Rev. Lett. 113 (2014), 241802 [arXiv:1407.7044 [hep-ph]].
  * [35] M. Ardu, S. Davidson and M. Gorbahn, Phys. Rev. D 105 (2022) no.9, 096040 [arXiv:2202.09246 [hep-ph]].
  * [36] A. Angelescu, D. Bečirević, D. A. Faroughy, F. Jaffredo and O. Sumensari, Phys. Rev. D 104 (2021) no.5, 055017 [arXiv:2103.12504 [hep-ph]]; D. Bečirević, F. Jaffredo, J. P. Pinheiro and O. Sumensari, _in preparation_.
  * [37] M. Bordone, M. Rahimi and K. K. Vos, Eur. Phys. J. C 81 (2021) no.8, 756 [arXiv:2106.05192 [hep-ph]].
  * [38] R. Aaij et al. [LHCb], JHEP 06 (2021), 044 [arXiv:2011.00217 [hep-ex]].
  * [39] B. Aubert et al. [BaBar], Phys. Rev. Lett. 99 (2007), 051801 [arXiv:hep-ex/0703018 [hep-ex]].
  * [40] J. Li and Z. You, [arXiv:2308.03405 [hep-ex]].
  * [41] Y. Aoki et al. [Flavour Lattice Averaging Group (FLAG)], Eur. Phys. J. C 82 (2022) no.10, 869 [arXiv:2111.09849 [hep-lat]].
  * [42] D. Bečirević, O. Sumensari and R. Zukanovich Funchal, Eur. Phys. J. C 76 (2016) no.3, 134 [arXiv:1602.00881 [hep-ph]].
  * [43] M. Borsato, V. V. Gligorov, D. Guadagnoli, D. Martinez Santos and O. Sumensari, Phys. Rev. D 99 (2019) no.5, 055017 [arXiv:1808.02006 [hep-ph]].
  * [44] A. Angelescu, D. A. Faroughy and O. Sumensari, Eur. Phys. J. C 80 (2020) no.7, 641 [arXiv:2002.05684 [hep-ph]].
  * [45] S. Descotes-Genon, D. A. Faroughy, I. Plakias and O. Sumensari, Eur. Phys. J. C 83 (2023) no.8, 753 [arXiv:2303.07521 [hep-ph]].
  * [46] G. C. Donald et al. [HPQCD], Phys. Rev. D 90 (2014) no.7, 074506 [arXiv:1311.6669 [hep-lat]].
  * [47] D. Becirevic and F. Sanfilippo, JHEP 01 (2013), 028 [arXiv:1206.1445 [hep-lat]].
  * [48] B. Colquhoun, R. J. Dowdall, C. T. H. Davies, K. Hornbostel and G. P. Lepage, Phys. Rev. D 91 (2015) no.7, 074514 [arXiv:1408.5768 [hep-lat]].
  * [49] D. Bečirević, G. Duplančić, B. Klajn, B. Melić and F. Sanfilippo, Nucl. Phys. B 883 (2014), 306-327 [arXiv:1312.2858 [hep-ph]].
  * [50] R. Lewis and R. M. Woloshyn, Phys. Rev. D 85 (2012), 114509 [arXiv:1204.4675 [hep-lat]].
  * [51] C. Allton et al. [RBC-UKQCD], Phys. Rev. D 78 (2008), 114509 [arXiv:0804.0473 [hep-lat]].
  * [52] N. Carrasco, P. Lami, V. Lubicz, L. Riggio, S. Simula and C. Tarantino, Phys. Rev. D 93 (2016) no.11, 114512 [arXiv:1602.04113 [hep-lat]]; A. Bazavov, C. Bernard, C. Bouchard, C. DeTar, D. Du, A. X. El-Khadra, J. Foley, E. D. Freeland, E. Gámiz and S. Gottlieb, et al. Phys. Rev. Lett. 112 (2014) no.11, 112001 [arXiv:1312.1228 [hep-ph]].
  * [53] J. A. Bailey, A. Bazavov, C. Bernard, C. M. Bouchard, C. DeTar, D. Du, A. X. El-Khadra, J. Foley, E. D. Freeland and E. Gámiz, et al. Phys. Rev. D 93 (2016) no.2, 025026 [arXiv:1509.06235 [hep-lat]]; W. G. Parrott et al. [(HPQCD collaboration)§ and HPQCD], Phys. Rev. D 107 (2023) no.1, 014510 [arXiv:2207.12468 [hep-lat]].
  * [54] D. Bečirević, G. Piazza and O. Sumensari, Eur. Phys. J. C 83 (2023) no.3, 252 [arXiv:2301.06990 [hep-ph]].
  * [55] Y. S. Amhis et al. [HFLAV], Phys. Rev. D 107 (2023) no.5, 052008 [arXiv:2206.07501 [hep-ex]].
  * [56] J. A. Bailey et al. [Fermilab Lattice and MILC], Phys. Rev. D 92 (2015) no.1, 014024 [arXiv:1503.07839 [hep-lat]]; J. M. Flynn, T. Izubuchi, T. Kawanai, C. Lehner, A. Soni, R. S. Van de Water and O. Witzel, Phys. Rev. D 91 (2015) no.7, 074510 [arXiv:1501.05373 [hep-lat]]; B. Colquhoun et al. [JLQCD], Phys. Rev. D 106 (2022) no.5, 054502 [arXiv:2203.04938 [hep-lat]].
  * [57] S. Descotes-Genon and J. Virto, JHEP 04 (2015), 045 [erratum: JHEP 07 (2015), 049] [arXiv:1502.05509 [hep-ph]]; S. Descotes-Genon, I. Plakias and O. Sumensari, JHEP 02 (2023), 096 [arXiv:2210.11995 [hep-ph]].
  * [58] V. Lubicz et al. [ETM], Phys. Rev. D 96 (2017) no.5, 054514 [erratum: Phys. Rev. D 99 (2019) no.9, 099902; erratum: Phys. Rev. D 100 (2019) no.7, 079901] [arXiv:1706.03017 [hep-lat]]; V. Lubicz et al. [ETM], Phys. Rev. D 98 (2018) no.1, 014516 [arXiv:1803.04807 [hep-lat]].
  * [59] A. Bazavov et al. [Fermilab Lattice and MILC], Phys. Rev. D 107 (2023) no.9, 094516 [arXiv:2212.12648 [hep-lat]].
  * [60] A. Bharucha, D. M. Straub and R. Zwicky, JHEP 08 (2016), 098 [arXiv:1503.05534 [hep-ph]].
  * [61] N. Gubernari, A. Kokulu and D. van Dyk, JHEP 01 (2019), 150 [arXiv:1811.00983 [hep-ph]]; S. Descotes-Genon, A. Khodjamirian and J. Virto, JHEP 12 (2019), 083 [arXiv:1908.02267 [hep-ph]].
  * [62] D. Black, T. Han, H. J. He and M. Sher, Phys. Rev. D 66 (2002), 053002 [arXiv:hep-ph/0206056 [hep-ph]]; A. Brignole and A. Rossi, Nucl. Phys. B 701 (2004), 3-53 [arXiv:hep-ph/0404211 [hep-ph]]; E. Arganda, M. J. Herrero and J. Portoles, JHEP 06 (2008), 079 [arXiv:0803.2039 [hep-ph]].
  * [63] M. Beneke and M. Neubert, Nucl. Phys. B 651 (2003), 225-248 [arXiv:hep-ph/0210085 [hep-ph]].
  * [64] D. J. Gross, S. B. Treiman and F. Wilczek, Phys. Rev. D 19 (1979), 2188
  * [65] T. Feldmann, P. Kroll and B. Stech, Phys. Rev. D 58 (1998), 114006 [arXiv:hep-ph/9802409 [hep-ph]]; T. Feldmann, Int. J. Mod. Phys. A 15 (2000), 159-207 [arXiv:hep-ph/9907491 [hep-ph]].
  * [66] R. Brüser, T. Feldmann, B. O. Lange, T. Mannel and S. Turczyk, JHEP 10 (2015), 082 [arXiv:1506.07786 [hep-ph]]; B. M. Dassinger, T. Feldmann, T. Mannel and S. Turczyk, JHEP 10 (2007), 039 [arXiv:0707.0988 [hep-ph]]; A. Matsuzaki and A. I. Sanda, Phys. Rev. D 77 (2008), 073003 [arXiv:0711.0792 [hep-ph]]; M. Giffels, J. Kallarackal, M. Kramer, B. O’Leary and A. Stahl, Phys. Rev. D 77 (2008), 073010 [arXiv:0802.0049 [hep-ph]]; A. Celis, V. Cirigliano and E. Passemar, Phys. Rev. D 89 (2014) no.9, 095014 [arXiv:1403.5781 [hep-ph]]; W. Dekens, E. E. Jenkins, A. V. Manohar and P. Stoffer, JHEP 01 (2019), 088 [arXiv:1810.05675 [hep-ph]]; G. Isidori, J. Pagès and F. Wilsch, JHEP 03 (2022), 011 [arXiv:2111.13724 [hep-ph]].
  * [67] R. Kitano, M. Koike and Y. Okada, Phys. Rev. D 66 (2002), 096002 [erratum: Phys. Rev. D 76 (2007), 059902] [arXiv:hep-ph/0203110 [hep-ph]]; Y. Kuno and Y. Okada, Phys. Rev. Lett. 77 (1996), 434-438 [arXiv:hep-ph/9604296 [hep-ph]]; V. Cirigliano, R. Kitano, Y. Okada and P. Tuzon, Phys. Rev. D 80 (2009), 013002 [arXiv:0904.0957 [hep-ph]]; V. Cirigliano, S. Davidson and Y. Kuno, Phys. Lett. B 771 (2017), 242-246 [arXiv:1703.02057 [hep-ph]]; S. Davidson, Y. Kuno and A. Saporta, Eur. Phys. J. C 78 (2018) no.2, 109 [arXiv:1710.06787 [hep-ph]].
  * [68] A. Crivellin, M. Hoferichter and M. Procura, Phys. Rev. D 89 (2014), 093024 [arXiv:1404.7134 [hep-ph]]; M. Hoferichter, J. Ruiz de Elvira, B. Kubis and U. G. Meißner, Phys. Rev. Lett. 115 (2015), 092301 [arXiv:1506.04142 [hep-ph]]; A. Crivellin, M. Hoferichter and M. Procura, Phys. Rev. D 89 (2014), 054021 [arXiv:1312.4951 [hep-ph]]; P. Junnarkar and A. Walker-Loud, Phys. Rev. D 87 (2013), 114510 [arXiv:1301.1114 [hep-lat]].
  * [69] J. Heeck, R. Szafron and Y. Uesaka, Nucl. Phys. B 980 (2022), 115833 [arXiv:2203.00702 [hep-ph]].
  * [70] T. Suzuki, D. F. Measday and J. P. Roalsvig, Phys. Rev. C 35 (1987), 2212
  * [71] W. H. Bertl et al. [SINDRUM II], Eur. Phys. J. C 47 (2006), 337-346
  * [72] G. Aad et al. [ATLAS], Phys. Rev. D 108 (2023), 032015 [arXiv:2204.10783 [hep-ex]].
  * [73] A. Hayrapetyan et al. [CMS], Phys. Rev. D 108 (2023) no.7, 072004 [arXiv:2305.18106 [hep-ex]].
  * [74] G. Aad et al. [ATLAS], JHEP 07 (2023), 166 [arXiv:2302.05225 [hep-ex]].
  * [75] A. Tumasyan et al. [CMS], JHEP 06 (2022), 082 [arXiv:2201.07859 [hep-ex]].
  * [76] C. A. Gottardo [ATLAS], [arXiv:1809.09048 [hep-ex]].
  * [77] [ATLAS], ATLAS-CONF-2023-001.
  * [78] L. Calibbi, X. Marcano and J. Roy, Eur. Phys. J. C 81 (2021) no.12, 1054 [arXiv:2107.10273 [hep-ph]].
  * [79] T. Han and D. Marfatia, Phys. Rev. Lett. 86 (2001), 1442-1445 [arXiv:hep-ph/0008141 [hep-ph]]. A. Goudelis, O. Lebedev and J. h. Park, Phys. Lett. B 707 (2012), 369-374 [arXiv:1111.1715 [hep-ph]]; G. Blankenburg, J. Ellis and G. Isidori, Phys. Lett. B 712 (2012), 386-390 [arXiv:1202.5704 [hep-ph]]; R. Harnik, J. Kopp and J. Zupan, JHEP 03 (2013), 026 [arXiv:1209.1397 [hep-ph]].
  * [80] P. Paradisi, JHEP 02 (2006), 050 [arXiv:hep-ph/0508054 [hep-ph]]; S. Kanemura, T. Ota and K. Tsumura, Phys. Rev. D 73 (2006), 016006 [arXiv:hep-ph/0505191 [hep-ph]]. S. Davidson and G. J. Grenier, Phys. Rev. D 81 (2010), 095016 [arXiv:1001.0434 [hep-ph]].
  * [81] S. Davidson, M. L. Mangano, S. Perries and V. Sordini, Eur. Phys. J. C 75 (2015) no.9, 450 [arXiv:1507.07163 [hep-ph]].
  * [82] M. Farina, G. Panico, D. Pappadopulo, J. T. Ruderman, R. Torre and A. Wulzer, Phys. Lett. B 772 (2017), 210-215 [arXiv:1609.08157 [hep-ph]].
  * [83] L. Allwicher, D. A. Faroughy, F. Jaffredo, O. Sumensari and F. Wilsch, JHEP 03 (2023), 064 [arXiv:2207.10714 [hep-ph]].
  * [84] L. Allwicher, D. A. Faroughy, F. Jaffredo, O. Sumensari and F. Wilsch, high-$p_{T}$ Drell-Yan tails beyond the standard model,” Comput. Phys. Commun. 289 (2023), 108749 [arXiv:2207.10756 [hep-ph]].
  * [85] [CMS], CMS-PAS-EXO-19-014.
  * [86] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645 (2002), 155-187 [arXiv:hep-ph/0207036 [hep-ph]].
  * [87] R. Barbieri, G. Isidori, J. Jones-Perez, P. Lodone and D. M. Straub, Eur. Phys. J. C 71 (2011), 1725 [arXiv:1105.2296 [hep-ph]].
  * [88] D. A. Faroughy, G. Isidori, F. Wilsch and K. Yamamoto, JHEP 08 (2020), 166 [arXiv:2005.05366 [hep-ph]]; A. Greljo, A. Palavrić and A. E. Thomsen, JHEP 10 (2022), 010 [arXiv:2203.09561 [hep-ph]].
  * [89] D. Bečirević, N. Košnik, O. Sumensari and R. Zukanovich Funchal, JHEP 11 (2016), 035 [arXiv:1608.07583 [hep-ph]].
  * [90] A. Angelescu, D. Bečirević, D. A. Faroughy and O. Sumensari, JHEP 10 (2018), 183 [arXiv:1808.08179 [hep-ph]].
  * [91] M. Bordone, C. Cornella, J. Fuentes-Martín and G. Isidori, JHEP 10 (2018), 148 [arXiv:1805.09328 [hep-ph]]; C. Cornella, D. A. Faroughy, J. Fuentes-Martin, G. Isidori and M. Neubert, JHEP 08 (2021), 050 [arXiv:2103.16558 [hep-ph]].
  * [92] D. M. Straub, [arXiv:1810.08132 [hep-ph]].
  * [93] J. Aebischer, J. Kumar and D. M. Straub, Eur. Phys. J. C 78 (2018) no.12, 1026 [arXiv:1804.05033 [hep-ph]].
  * [94] J. de Blas, J. C. Criado, M. Perez-Victoria and J. Santiago, JHEP 03 (2018), 109 [arXiv:1711.10391 [hep-ph]].
  * [95] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012), 1-102 [arXiv:1106.0034 [hep-ph]].
  * [96] A. Crivellin, A. Kokulu and C. Greub, Phys. Rev. D 87 (2013) no.9, 094031 [arXiv:1303.5877 [hep-ph]].
  * [97] D. Pappadopulo, A. Thamm, R. Torre and A. Wulzer, JHEP 09 (2014), 060 [arXiv:1402.4431 [hep-ph]].
  * [98] I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik and N. Košnik, Phys. Rept. 641 (2016), 1-68 [arXiv:1603.04993 [hep-ph]].
  * [99] J. Fuentes-Martín, G. Isidori, M. König and N. Selimović, Phys. Rev. D 101 (2020) no.3, 035024 [arXiv:1910.13474 [hep-ph]]; J. Fuentes-Martín, G. Isidori, M. König and N. Selimović, Phys. Rev. D 102 (2020) no.3, 035021 [arXiv:2006.16250 [hep-ph]]; J. Fuentes-Martín, G. Isidori, M. König and N. Selimović, Phys. Rev. D 102 (2020), 115015 [arXiv:2009.11296 [hep-ph]].
  * [100] R. Mandal and A. Pich, JHEP 12 (2019), 089 [arXiv:1908.11155 [hep-ph]].
  * [101] G. Panico, A. Pomarol and M. Riembau, JHEP 04 (2019), 090 [arXiv:1810.09413 [hep-ph]].
  * [102] D. Bečirević, F. Jaffredo, A. Peñuelas and O. Sumensari, JHEP 05 (2021), 175 [arXiv:2012.09872 [hep-ph]].
  * [103] J. Gratrex, M. Hopfer and R. Zwicky, Phys. Rev. D 93 (2016) no.5, 054008 [arXiv:1506.03970 [hep-ph]].
  * [104] W. Altmannshofer, P. Ball, A. Bharucha, A. J. Buras, D. M. Straub and M. Wick, JHEP 01 (2009), 019 [arXiv:0811.1214 [hep-ph]].
  * [105] M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 222 (1983), 83-103; M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 236 (1984), 221-232; M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 249 (1985), 70-92
  * [106] R. Coy, M. Frigerio, F. Mescia and O. Sumensari, Eur. Phys. J. C 80 (2020) no.1, 52 [arXiv:1909.08567 [hep-ph]]; J. Aebischer and J. Kumar, JHEP 09 (2020), 187 [arXiv:2005.12283 [hep-ph]].