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P3H-24-013, TTP24-003

# Flavor Phenomenology of Light Dark Vectors

Jordi Folch Eguren<EMAIL_ADDRESS>Department of Physics, TU
Dortmund, D-44221 Dortmund, Germany Sophie Klingel  Institute for Theoretical
Particle Physics, KIT, 76128 Karlsruhe, Germany  Emmanuel Stamou
<EMAIL_ADDRESS>Department of Physics, TU Dortmund, D-44221
Dortmund, Germany
Mustafa Tabet<EMAIL_ADDRESS>Department of Physics, TU Dortmund,
D-44221 Dortmund, Germany Robert Ziegler<EMAIL_ADDRESS>Institute for
Theoretical Particle Physics, KIT, 76128 Karlsruhe, Germany  Institute for
Theoretical Physics, Heidelberg University, 69120 Heidelberg, Germany

###### Abstract

Light dark matter with flavor-violating couplings to fermions may be copiously
produced in the laboratory as missing energy from decays of SM particles. Here
we study the effective Lagrangian of a light dark vector with generic dipole
or vector couplings. We calculate the resulting two-body decay rates of
mesons, baryons and leptons as a function of the dark vector mass and show
that existing experimental limits probe UV scales as large as
$10^{12}\,{\mathrm{G}eV}$. We also derive the general RGEs in order to
constrain the flavor-universal UV scenario, where all flavor violation arises
radiatively proportional to the CKM matrix.

##  1 Introduction

In recent years light new particles interacting very weakly with the Standard
Model (SM) have gained increased interest. The so far negative results on
searches for heavy particles above the electroweak scale at the LHC and high-
intensity experiments have increased the interest in less explored scenarios,
with additional degrees of freedom beyond the SM with masses at sub-GeV
scales. Such particles can be motivated by dynamics addressing the Strong CP
Problem (in case of the QCD axion) or the origin of neutrino masses (in case
of sterile neutrinos), but probably the main motivation is the possibility
that such light particles could be connected to the origin of particle dark
matter (DM) [1].

In this context a popular scenario is the dark photon [2, 3], which is either
itself DM or is the only mediator (“Vector Portal”) between the SM and a
hidden “dark sector”, which contains one or several DM particles [4, 5], see
Ref. [6] for a review. The term “dark photon” usually refers to a light vector
particle coupled to the SM only via kinetic mixing or dipole operators and
that is often taken as the only new degree of freedom. Instead, the term
$Z^{\prime}$ is typically reserved for the vast model space of theories of
gauged $U(1)^{\prime}$ extensions of the SM, where also a complete Higgs
sector for $U(1)^{\prime}$ breaking is explicitly present, besides additional
matter needed for anomaly cancellation, see, e.g., Ref. [7] for a
classification. While the $Z^{\prime}$ vector boson is often taken to be
heavy, with a mass much above the electroweak (EW) scale, this particle can
also be much lighter. The resulting coupling patterns are often related to the
underlying UV symmetries, see, e.g., Refs. [8, 9, 10, 11], and can leave
imprints in low-energy phenomenology/anomalies in current data, e.g., in
$(g-2)_{\mu}$ [12] or in low-energy QCD [13]. Beyond perturbative models,
light vector particles can also be in the spectrum of light resonances of low-
energy, dark strongly coupled sectors, see, e.g., Ref. [14]. To encompass all
these cases, we employ in the current work the term “light dark vector” (LDV),
which is a massive vector boson with mass much below the EW scale, and
sufficiently suppressed couplings to SM particles such that it is stable on
collider scales. For the purpose of low-energy phenomenology we leave its UV
origin unspecified.

While constraints on light particles have been extensively studied in the
context of colliders, beam-dump experiments, astrophysics, and cosmology,
their phenomenology at precision flavor experiments has so far received less
attention (see Ref. [15, 16] for early studies). Even if flavor-violating
couplings may be considered more model-dependent than flavor-diagonal
couplings, they can provide for an efficient production of light invisible
particles from decays of SM leptons, mesons or baryons. Interestingly, direct
searches at laboratory experiments for such two-body decays with missing
energy have the potential to probe enormously large scales, as the relevant
Lagrangian interactions can be dimension-five, instead of dimension-six as in
the case of heavy New Physics. For example, in models with sufficiently light
invisible bosons like the QCD axion, precision flavor experiments are
sensitive to scales as large as $10^{12}\,{\mathrm{G}eV}$ from
$K\to\pi+{\mathrm{i}nvis.}$ searches at NA62 [17], $10^{10}\,{\mathrm{G}eV}$
from $\mu\to e+{\mathrm{i}nvis.}$ searches at MEG-II [18, 19], Mu3e [20], Mu2e
or COMET [21], and $10^{8}\,{\mathrm{G}eV}$ for $b\to d/s$ transitions at
Belle II [22].111For the flavor phenomenology of the QCD axion and light
invisible axion-like particles see Refs. [23, 16, 24, 22, 18, 25].

The aim of the current work is to systematically study the flavor
phenomenology of light dark vector particles (LDVs), both in the quark and the
lepton sectors. We restrict the discussion to invisible particles, since after
all the main (only) motivation for these particles is the observed DM
abundance, and we have in mind scenarios where either the LDV is itself stable
on cosmological scales or promptly decays to stable DM particles. This
analysis includes scenarios where the LDVs are just sufficiently long-lived to
appear as missing energy. This is particularly justified for vector particles
lighter than the electron, as their decay into two photons is forbidden by the
Landau–Yang theorem [26, 27]. As we shall discuss, the resulting limits on
flavor-violating interactions can be as strong as in the axion case, which is
not unexpected due to the Goldstone-boson equivalence theorem. In light of
past and ongoing experimental searches, it is thus important to systematically
study the phenomenological differences between light dark scalars and vectors
originating from their distinct helicity and coupling structure.

Earlier works have focused on the case of flavor-violating dipole couplings of
a massless dark photon in $\mu\to e$ and $s\to d$ transitions [28, 29, 30, 6,
31, 32], or considered general interactions and masses, but using only the
available experimental limits on three-body decays to neutrinos to study
limits from $s\to d$ and $b\to s$ transitions [16]. Here instead we consider
the case of a light vector particle with generic mass and either dipole or
minimal couplings to SM fermions. We work within the framework of a general
effective-field-theory (EFT) approach and consider all possible quark flavor-
violating transitions except those involving the top quark (where constraints
are very weak), and all possible lepton flavor-violating (LFV) transitions. We
also discuss the decays of polarized leptons, which play an important role in
separating signal from SM background. We derive bounds in the general
parameter plane of light-vector mass and the appropriate flavor-changing
coupling by comparing theoretical predictions for the decay rates to the
experimental bounds from various flavor factories, such as NA62 [33, 34],
BaBar [35, 36], CLEO [37], Belle II [38, 39], BES III [40], and TWIST [41].
Whenever not available (as in the case of, e.g., $B\to
K/K^{*}/\pi+{\mathrm{i}nvis.}$ or $D\to\pi+{\mathrm{i}nvis.}$ decays), we
derive model-independent limits on the two-body decay rate as a function of
the invisible particle mass by recasting experimental data on the three-body
decay with two invisible neutrinos. Finally, we also discuss the scenario
where the light vector has only flavor-universal couplings to SM fermions in
the UV, so that all quark flavor-changing effects in the IR are induced
radiatively by the Cabibbo–Kobayashi–Maskawa (CKM) matrix, satisfying the
paradigm of Minimal Flavor-Violation (MFV) [42, 43]. For this analysis we
derive the relevant renormalization-group equations (RGEs) for both dipole and
minimal couplings, and use our results to convert limits on the flavor-
changing interactions into limits on flavor-diagonal couplings.

This work is organized as follows. In Section 2 we define our basic setup by
providing the effective Lagrangian for dipole and minimal (vector)
interactions of the LDV. The resulting phenomenology is studied in the
subsequent sections, separately for the quark (Section 3) and lepton (Section
4) sectors, where we present our main results, the model-independent bounds on
generic flavor-violating LDV couplings as a functions of its mass. In Section
5 we use these constraints to derive bounds on flavor-universal UV couplings
with either dipole or vector interactions from RG-induced flavor violation. We
conclude in Section 6. Many technical details are deferred to appendices:
Appendix C contains the details and results of our recast of two-body flavor-
violating decays with missing energy for generic masses of the invisible
particles (extending the analysis for a massless invisible particle in Ref.
[22]). Appendix D contains the bounds on flavor-violating couplings in the
chiral ${\mathrm{L}}/{\mathrm{R}}$ basis (as opposed to the V/A basis in
Section 3 and 4). The complete set of RGEs relevant for Section 5 is given in
Appendix B, and Appendix E contains the full expressions of two-body decay
rates of mesons, baryons, and polarized leptons, for a generic mass for the
light vector. We have also collected the hadronic matrix elements entering the
numerical analysis in Appendix E.1. Finally, Appendix A contains a discussion
of the EFT description of flavor-violating vector couplings and their possible
UV origin.

##  2 Setup

We extend the SM by a new, neutral, massive vector boson $V^{\prime}_{\mu}$
with a small mass $m_{V^{\prime}}$, which arises either by spontaneous
symmetry breaking of, e.g., a $U(1)^{\prime}$ gauge symmetry or by the
Stueckelberg mechanism [44, 45, 46]. Here we focus on the case where this mass
is much below the electroweak scale, and the light dark vector (LDV) is either
stable on collider scales or decays into stable invisible particles.

The most general interactions of the LDV with the SM fermions can be
parametrized using an EFT approach, by considering the most general operators
that respect the unbroken part of the SM gauge group, $SU(3)_{c}\times
U(1)_{\mathrm{e}m}$. Here we focus on flavor-violating interactions written
without loss of generality in the fermion-mass basis. We can further assume
that a possible kinetic mixing between the photon and the LDV, i.e.,
$\propto\epsilon A^{\mu}V^{\prime}_{\mu}$, has been diagonalized such that
$V^{\prime}_{\mu}$ is also in the mass-eigenstate basis. This diagonalization
can be performed equally well for a massless $V^{\prime}_{\mu}$ (cf. Ref.
[29]), and the difference with respect to the massive case is merely that for
massless vectors there remains an unphysical ambiguity in the choice of “mass-
eigenstate” basis, due to the presence of an unbroken $SO(2)$ symmetry of the
free Lagrangian. Thus our setup applies equally well to the “massless dark
photon” considered in Ref. [29] in the limit of $m_{V^{\prime}}\to 0$.

Below the EW scale the lowest dimensional interactions of the LDV are
described by two classes of operators: dipole and vector interactions.
Firstly, we consider flavor-violating, dimension-five dipole interactions of
the form

$\displaystyle\mathscr{L}_{{\mathrm{D}}}$
$\displaystyle=-\frac{1}{4}V^{\prime}_{\mu\nu}V^{\prime\mu\nu}+\frac{m_{V^{\prime}}^{2}}{2}V^{\prime}_{\mu}V^{\prime\mu}+\frac{1}{\Lambda}V^{\prime}_{\mu\nu}\,\overline{f}_{i}\sigma^{\mu\nu}\left(\mathbb{C}^{{\mathrm{D}}}_{ij}+i\mathbb{C}^{{\mathrm{D}}5}_{ij}\gamma_{5}\right)f_{j}\,,$
(2.1)

where
$V^{\prime}_{\mu\nu}=\partial_{\mu}V^{\prime}_{\nu}-\partial_{\nu}V^{\prime}_{\mu}$
is the LDV field strength,
$\sigma^{\mu\nu}=\tfrac{i}{2}[\gamma^{\nu},\gamma^{\nu}]$, and $i\neq j$
denote SM quark or lepton flavors. $\Lambda$ is the UV-completion scale of the
associated dipole couplings $\mathbb{C}^{{\mathrm{D}}}_{ij}$ and
$\mathbb{C}^{{\mathrm{D}}5}_{ij}$, which are hermitian matrices in flavor
space,
$\big{(}\mathbb{C}^{{\mathrm{D}}}_{ij}\big{)}^{*}=\mathbb{C}^{{\mathrm{D}}}_{ji}$
and
$\big{(}\mathbb{C}^{{\mathrm{D}}5}_{ij}\big{)}^{*}=\mathbb{C}^{{\mathrm{D}}5}_{ji}$.

Secondly, we consider flavor-violating couplings of the LDV to SM vector and
axial-vector currents. Naively these are dimension-four interactions below the
EW scale. However, such flavor-violating couplings violate $U(1)^{\prime}$
gauge invariance (flavor-violating currents are not conserved), and thus must
be proportional to some power of the $U(1)^{\prime}$-breaking order parameter,
which we take as the vacuum expectation value (VEV) in the dark sector.
Therefore, the flavor-violating vector couplings are actually dimension-five
or higher, depending on the underlying UV model. In perturbative UV
completions the lowest possible scaling is proportional to a single power of
the dark VEV, which upon including the dark gauge coupling becomes the LDV
mass $m_{V^{\prime}}$. Normalizing by some UV scale $\Lambda$, the flavor-
violating vector interactions are

$\mathscr{L}_{{\mathrm{V}}}=-\frac{1}{4}V^{\prime}_{\mu\nu}V^{\prime\mu\nu}+\frac{m_{V^{\prime}}^{2}}{2}V^{\prime}_{\mu}V^{\prime\mu}+\frac{m_{V^{\prime}}}{\Lambda}V^{\prime}_{\mu}\,\overline{f}_{i}\gamma^{\mu}\left(\mathbb{C}^{{\mathrm{V}}}_{ij}+\mathbb{C}^{{\mathrm{V}}5}_{ij}\gamma_{5}\right)f_{j}\,,$
(2.2)

where again $i\neq j$ denote SM quark or lepton flavors and the vector
couplings $\mathbb{C}^{{\mathrm{V}}}_{ij}$ and
$\mathbb{C}^{{\mathrm{V}}5}_{ij}$ are hermitian matrices in flavor space,
$\big{(}\mathbb{C}^{{\mathrm{V}}}_{ij}\big{)}^{*}=\mathbb{C}^{{\mathrm{V}}}_{ji}$
and
$\big{(}\mathbb{C}^{{\mathrm{V}}5}_{ij}\big{)}^{*}=\mathbb{C}^{{\mathrm{V}}5}_{ji}$.

By choosing a scaling that is linear in $m_{V^{\prime}}/\Lambda$, we ensure
that the growth of amplitudes with longitudinally polarized LDVs in initial
and/or final states $\propto E/m_{V^{\prime}}$ as $m_{V^{\prime}}\to 0$ is
cancelled by the $m_{V^{\prime}}$ dependence in the interaction. This leads to
finite amplitudes in the $m_{V^{\prime}}\to 0$ limit (see Refs. [16, 47, 48,
49, 50] for related discussions), which are just the amplitudes with the
corresponding Goldstone bosons as initial/final states. An explicit example
for a UV model that provides this linear scaling is provided by
Froggatt–Nielsen type models [51], discussed in Appendix A. However, the
linear scaling with $m_{V^{\prime}}$ is only one possibility. For example, in
UV models in which SM fermions do not carry $U(1)^{\prime}$ charges the
scaling can be quadratic in the dark VEV, as the coefficients involve
additional powers of the $U(1)^{\prime}$ breaking scale $v^{\prime}$, $\propto
m_{V^{\prime}}v^{\prime}/\Lambda^{2}$. An explicit realization of this
scenario is also discussed in Appendix A.

The interactions in Eq. (2.1) and (2.2) can also be written in the chiral
basis, which is more suited to match explicit UV models. In this basis

$\displaystyle\mathscr{L}_{{\mathrm{D}}}$
$\displaystyle=\frac{1}{\Lambda}V^{\prime}_{\mu\nu}\,\overline{f}_{i}\sigma^{\mu\nu}\left(\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{ij}P_{{\mathrm{L}}}+\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}P_{{\mathrm{R}}}\right)f_{j}\,,$
(2.3) $\displaystyle\mathscr{L}_{{\mathrm{V}}}$
$\displaystyle=\frac{m_{V^{\prime}}}{\Lambda}V^{\prime}_{\mu}\,\bar{f}_{i}\gamma^{\mu}\left(\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ij}P_{{\mathrm{L}}}+\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ij}P_{{\mathrm{R}}}\right)f_{j}\,,$

where
$\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{ij}=(\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ji})^{*}$,
$\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ij}=(\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ji})^{*}$,
$\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ij}=(\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ji})^{*}$
and the relations between the “${\mathrm{V}}/{\mathrm{A}}$” and the
“${\mathrm{L}}/{\mathrm{R}}$” bases are

$\displaystyle\mathbb{C}^{{\mathrm{D}}}_{ij}$
$\displaystyle=\frac{1}{2}\left(\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{ij}+\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}\right)=\frac{1}{2}\left(\big{(}\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ji}\big{)}^{*}+\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}\right)\,,$
$\displaystyle\mathbb{C}^{{\mathrm{V}}}_{ij}$
$\displaystyle=\frac{1}{2}\left(\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ij}+\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ij}\right)\,,$
(2.4) $\displaystyle\mathbb{C}^{{\mathrm{D}}5}_{ij}$
$\displaystyle=\frac{i}{2}\left(\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{ij}-\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}\right)=\frac{i}{2}\left(\big{(}\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ji}\big{)}^{*}-\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}\right)\,,$
$\displaystyle\mathbb{C}^{{\mathrm{V}}5}_{ij}$
$\displaystyle=\frac{1}{2}\left(\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ij}-\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ij}\right)\,.$
(2.5)

Above the EW scale the operators must be expressed in a manifestly
$SU(2)_{{\mathrm{L}}}\times U(1)_{Y}$ invariant manner. For
$\mathscr{L}_{{\mathrm{V}}}$ this is directly the case after embedding the
left- and right-handed fermions in the corresponding $SU(2)_{{\mathrm{L}}}$
doublets and singlets, respectively. Instead, the dipole operators in
$\mathscr{L}_{{\mathrm{D}}}$ require an additional Higgs insertion, making
them dimension-six operators

$\displaystyle\mathscr{L}_{{\mathrm{D}}6}$
$\displaystyle=\frac{1}{\Lambda_{6}^{2}}V^{\prime}_{\mu\nu}\left(\overline{F}_{i}HC^{{\mathrm{D}}}_{ij}\sigma^{\mu\nu}P_{{\mathrm{R}}}f_{j}+\text{h.c.}\right)\,,$
(2.6)

with $F_{i}$ and $f_{j}$ denoting here $SU(2)_{{\mathrm{L}}}$ doublets and
singlets, respectively, and $H\to\widetilde{H}$, depending on the fermion
sector and the hypercharge conventions. The matching to
$\mathscr{L}_{{\mathrm{D}}}$ is provided by identifying
$\Lambda_{6}=\sqrt{v\Lambda}$, where $v=174\,{\mathrm{G}eV}$ is the Higgs VEV.

In the following we derive bounds on the flavor-violating couplings in Eq.
(2.1) and (2.2) from hadronic and leptonic decays with missing energy in the
final state. This discussion is unaffected by other possible interactions of
the LDV with SM fields, in particular flavor-diagonal couplings, as long as
these couplings are sufficiently small to ensure that the LDV is invisible on
collider scales.

For massive LDVs neither the flavor-violating dipole (Eq. (2.1)) nor the
vector (Eq. (2.2)) interactions are UV complete. The UV completion depends on
the origin of the mass for the LDV and the corresponding (highly model-
dependent) radial mode required for the unitarity of the theory. In turn this
implies that unless the complete dark Higgs sector of the theory is specified,
there exist perturbative unitarity constraints on the couplings of the LDV,
similar to the unitarity constraints from $WW\to WW$ scattering in the Higgs-
less SM. We briefly note that, as long as the flavor bounds are applicable,
i.e., LDV masses in the kinematically allowed region, unitarity of $2\to 2$
scattering poses constraints on the corresponding couplings that are weaker
than those limits by order of magnitudes. We thus refrain from elaborating
upon these constraints in the current work. For the case of unitarity bounds
on massless fermions with flavor-diagonal couplings coupled to transversely
polarized vectors see, e.g., Ref. [52]. The more general case including
massive fermions with flavor-violating couplings to LDVs will be presented in
Ref. [53].

##  3 Quark Phenomenology of Light Dark Vectors

Figure 1: Illustrative Feynman diagrams with a flavor-violating $q_{j}\to
q_{i}$ transition in two-body decays of type $P\to P^{\prime}+V^{\prime}$,
$P\to\mathcal{V}+V^{\prime}$, and $B\to B^{\prime}+V^{\prime}$, in the left,
middle, and right panel, respectively.

In this section we derive bounds on the flavor-violating couplings
$\mathbb{C}^{{\mathrm{D}}(5)}_{ij}$ in Eq. (2.1) and
$\mathbb{C}^{{\mathrm{V}}(5)}_{ij}$ in Eq. (2.2) for the quark-flavor
transitions: $s\to d$, $b\to s$, $b\to d$, and $c\to u$. We employ the
following three types of two-body decays containing the LDV as an invisible
final state222Three-body decays and neutral meson mixing typically give weaker
constraints, e.g., for example LHCb constraints on $B_{(s)}\to\mu\mu a$ cannot
compete with Belle II limits [54].

  * •

$P\to P^{\prime}+V^{\prime}$: pseudoscalar meson to pseudoscalar meson and
LDV,

  * •

$P\to\mathcal{V}+V^{\prime}\,\,$: pseudoscalar meson to vector meson and LDV,

  * •

$B\to B^{\prime}+V^{\prime}$: baryon to baryon and LDV.

Figure 1 shows representative Feynman diagrams for the three types of decays.

Appendix E contains the analytical expressions for the corresponding decays
rates (including the dependence on $m_{V}^{\prime}$); the relevant form
factors are collected in Appendix E.1. Comparing the decay rates to the
experimental upper limits on the branching ratios, we set upper bounds on the
couplings in the ${\mathrm{V}}/{\mathrm{A}}$ basis333 In Appendix D we show
the bounds in the ${\mathrm{L}}/{\mathrm{R}}$ basis. of Eq. (2.1) and Eq.
(2.2), i.e. on the set $\\{\mathbb{C}^{{\mathrm{D}}}_{ij}$,
$\mathbb{C}^{{\mathrm{D}}5}_{ij}$, $\mathbb{C}^{{\mathrm{V}}}_{ij}$,
$\mathbb{C}^{{\mathrm{V}}5}_{ij}\\}$. The limits are determined as a function
of the LDV mass, with range $0\leq
m_{V^{\prime}}^{2}\leq(m_{\mathrm{I}}-m_{\mathrm{F}})^{2}\equiv
m_{V^{\prime}\text{max}}^{2}$ depending on the masses of the initial,
$m_{\mathrm{I}}$, and final, $m_{\mathrm{F}}$, states of the decay at hand.
Crucially, the form factors depend on the LDV mass and it is, therefore,
essential to consider the full form-factor parametrization for an accurate
analysis.

The available theoretical and experimental information is summarized in Table
1, where we collect the references for the form factors and relevant
experimental limits. Often the experimental collaborations do not provide
limits on two-body decays with missing energy. Yet, in some cases there is
enough information to extract this bound from available data. We indicate this
case by a subindex “$r$” in the last column of the table, and either use
existing recasts in the literature or perform our own recast, e.g., to find a
bound on $B\to\pi/K/K^{*}+{\mathrm{i}nvis.}$ from BaBar data on the
corresponding three-body decays [35, 36], see Appendix C for details.

Concretely we use our recast for $B\to K^{(*)}+{\mathrm{i}nvis.}$ only for LDV
masses above $3\,{\mathrm{G}eV}$. Note that we can recast only the
experimental results of the BaBar collaboration and cannot use the newer Belle
measurements, since the Belle collaboration does not provide the event count
as a (binned) function of the missing-momentum distribution. We use existing
recasts for $B\to\rho+{\mathrm{i}nvis.}$ decays from LEP [55, 56], $B\to
K+{\mathrm{i}nvis.}$ decays from Belle II [57, 39] (this recast is limited to
masses below $m_{V^{\prime}}=3\,{\mathrm{G}eV}$), $B\to
K^{*}+{\mathrm{i}nvis.}$ decays from BaBar [57, 36] (below
$m_{V^{\prime}}=3\,{\mathrm{G}eV}$). For invisible baryon decays for which
there is no analysis, we derive limits using the total lifetime from the PDG
[58] after subtracting all observed channels as in Ref. [22].

For the bound based on $D\to\pi+{\mathrm{i}nvis.}$ decays we use the result of
Ref. [22] for $m_{V^{\prime}}\approx 0$, obtained from recasting CLEO data on
$D\to(\tau\to\pi\nu)\nu$ [37]. We also perform a recast of these data for LDV
masses up to $m_{V^{\prime}}\approx 0.5\,{\mathrm{G}eV}$ (which is the upper
range of the CLEO data set), assuming the efficiency in all bins to be the
same as for $m_{V^{\prime}}\approx 0$. Note that recasting BES III data [59]
on $D\to\pi\nu\overline{\nu}$ gives weaker constraints [22], although this
result does not use the full experimental information. It would be interesting
if BES III would provide an explicit two-body recast of their full data set.
The collaboration actually does this for the case of two-body hyperon decays
$\Lambda_{c}\to p+{\mathrm{i}nvis.}$, albeit only for “massless” invisible
particles. Their signal region in fact covers invisible masses up to
$316\,\,{\mathrm{M}eV}$, and leads to limits that are much stronger than the
ones obtained by saturating the total $\Lambda_{c}$ lifetime [22]. As a
conservative limit, to be replaced by a dedicated experimental analysis, we
multiply their limit for the massless case by a factor 1/2 (since close to the
endpoint of the signal region half of the signal events are lost due to energy
resolution). We use the resulting bound ${\mathrm{B}R}(\Lambda_{c}\to
pV^{\prime})<1.6\times 10^{-4}$ for LDV masses up to $316\,\,{\mathrm{M}eV}$,
and take lifetime limits above $316\,\,{\mathrm{M}eV}$. We notice that a
search for the decay $D\to\pi+X$ would not suffer from two-body SM backgrounds
in contrast to hyperon decays, where $\Lambda_{c}\to p+\gamma$ contributes to
the signal of a massless $X$, if the photon is missed.

Quark Transition |  Hadronic Process |  Form Factors |  Experimental Limit
---|---|---|---
$s\to d$ |  $K^{+}\to\pi^{+}+V^{\prime}$ |  [60, 61] |  NA62 [34, 33, 17]
|  $\Sigma^{+}\to p+V^{\prime}$ |  [32, 62, 63, 64] |  BES III [65], Lifetimer[58, 22]
|  $\Xi^{-}\to\Sigma^{-}+V^{\prime}$ |  [32, 62, 63, 64] |  Lifetimer[58, 22]
|  $\Xi^{0}\to\Sigma^{0}+V^{\prime}$ |  [32, 62, 63, 64] |  Lifetimer[58, 22]
|  $\Xi^{0}\to\Lambda+V^{\prime}$ |  [32, 62, 63, 64] |  Lifetimer[58, 22]
|  $\Lambda\to n+V^{\prime}$ |  [32, 62, 63, 64] |  Lifetimer[58, 22]
|  $B^{+}\to K^{+}+V^{\prime}$ |  [66, 66] |  BaBarr [36], Belle IIr [57, 39]
$b\to s$ |  $B\to K^{\ast}+V^{\prime}$ |  [66, 66] |  BaBarr [36, 57]
|  $\Lambda_{b}\to\Lambda+V^{\prime}$ |  [67, 67] |  Lifetimer[58, 22]
|  $B^{+}\to\pi^{+}+V^{\prime}$ |  [68, 66] |  BaBarr [35]
$b\to d$ |  $B\to\rho+V^{\prime}$ |  [66, 66] |  LEPr [55, 56]
|  $\Lambda_{b}\to n+V^{\prime}$ |  [67, 69] |  Lifetimer [58, 22]
$c\to u$ |  $D^{+}\to\pi^{+}+V^{\prime}$ |  [70, 71] |  CLEOr [37, 22]
|  $\Lambda_{c}\to p+V^{\prime}$ |  [72, 72] |  BES III [40], Lifetimer [58, 22]
Table 1:  Overview of considered hadron decays with invisibles in the final
state. The first column shows the underlying quark-flavor transition, the
second the specific hadronic process. The relevant vector and dipole form
factors are taken from the references in the third column. The last column
contains the references for the experimental upper limits on the respective
branching ratios. A subindex “$r$” indicates that a recast of experimental
data was needed, see text and Appendix C for details.

To set constraints on the couplings $\\{\mathbb{C}^{{\mathrm{D}}}_{ij}$,
$\mathbb{C}^{{\mathrm{D}}5}_{ij}$, $\mathbb{C}^{{\mathrm{V}}}_{ij}$,
$\mathbb{C}^{{\mathrm{V}}5}_{ij}\\}$ we consider dipole
($\mathscr{L}_{{\mathrm{D}}}$) and vector interactions
($\mathscr{L}_{{\mathrm{V}}}$) separately, and turn on a single coupling at a
time. We use the theory predictions in Appendix E together with the form
factors in Table 1 (see also Appendix E.1) to calculate the decay rates as a
function of the couplings and the LDV mass. The rates are then compared to the
experimental limits to obtain the bounds in the mass–coupling plane. We
include statistical and systematic uncertainties as follows. For the theory
predictions we only use the systematic uncertainties associated with hadronic
form factors (these are the most relevant ones), while the treatment of
uncertainties of experimental limits depend on their nature: for decays where
the experimental collaborations provide two-body interpretations (or a theory
recast exists), we add the experimental and form-factor uncertainties in
quadrature. In the case where we performed our own two-body recast (as
described in Appendix C) we treat theory uncertainties as Gaussian
uncertainties smearing the expectation values of the underlying Poisson
probability distribution functions.

Our results are summarized in Figures 2 and 3 in which we show the lower
bounds on the effective inverse coupling $\Lambda/\mathbb{C}_{ij}$ for given
LDV mass $m_{V^{\prime}}$. The plots are organized according to the underlying
flavor transition, i.e, $s\to d$, $b\to s$, $b\to d$, and $c\to u$ and we
separate dipole $\\{\mathbb{C}^{{\mathrm{D}}}_{ij}$,
$\mathbb{C}^{{\mathrm{D}}5}_{ij}\\}$ (Figure 2) and vector couplings
$\\{\mathbb{C}^{{\mathrm{V}}}_{ij}$, $\mathbb{C}^{{\mathrm{V}}5}_{ij}\\}$
(Figure 3). Each plot shows the bound on a single coupling for a given quark-
flavor transition, with each line corresponding to a particular hadronic
decay, excluding the region below. Note that $P\to P^{\prime}+V^{\prime}$
decays are only sensitive to $\\{\mathbb{C}^{{\mathrm{D}}}_{ij}$ and
$\mathbb{C}^{{\mathrm{V}}}_{ij}\\}$ couplings, which follows from parity
conservation of the strong interactions and the Lorentz structure of the form
factors (see Appendix E.1). Also note that dipole operators are dimension-six
above the electroweak scale, so in fact the actual UV scale probed is
$\Lambda_{6}=\sqrt{v\Lambda}$ in all transitions.

###  3.1 Dark Dipole Interactions

#####  $\boldsymbol{s\to d}$ Transitions

The bounds on the dipole couplings $\\{\mathbb{C}^{{\mathrm{D}}}_{sd}$,
$\mathbb{C}^{{\mathrm{D}}5}_{sd}\\}$ are set by $K\to\pi+{\mathrm{i}nvis.}$
and hyperon decays, cf. Table 1 and Figure 2. For the two-body decay
$K\to\pi+{\mathrm{i}nvis.}$ we use the bound provided by the NA62
collaboration [34]. For baryon decays there is an upper limit from BES III
[65] on the decay $\Sigma^{+}\to p+{\mathrm{i}nvis.}$ with a massless
invisible. We estimate the potential reach for this search by extending it to
larger invisible masses by assuming that the same experimental limit is valid
for the whole kinematic range. This is indicated by a dashed orange line. For
all other baryon searches, we set upper limits on branching ratios indirectly
as in Ref. [22] by subtracting the measured branching fractions for all
relevant hyperon decay channels from unity. Due to this rather weak limit,
$K\to\pi$ sets a much more stringent constraint than hyperon decays, limiting
the UV scale $\Lambda/\mathbb{C}^{{\mathrm{D}}}_{ij}$ to be at least of the
order $10^{11}\,{\mathrm{G}eV}$. Note however that the search for
$\Sigma^{+}\to p+{\mathrm{i}nvis.}$ strenghtens the upper limit by two orders
of magnitude compared to the conservative limit estimated with the total
lifetime, and thus, out of all baryon decays, it yields the strongest limit of
order $10^{7}\,{\mathrm{G}eV}$ on the scale
$\Lambda/\mathbb{C}^{{\mathrm{D}}}_{ij}$.

Nevertheless baryon decays with missing energy are important for two reasons.
The decays to pseudoscalar, such as $K\to\pi$, are only sensitive to the
$\\{\mathbb{C}^{{\mathrm{D}}}_{ij}$, $\mathbb{C}^{{\mathrm{V}}}_{ij}\\}$
couplings. Thus baryon decays are crucial to constrain the axial coupling
$\Lambda/\mathbb{C}^{{\mathrm{D}}5}_{ij}$ (of the order of a few $\times
10^{7}\,{\mathrm{G}eV}$), as there are no two-body decays to vector particles
in $s\to d$ transitions. Moreover, the decay rates of pseudoscalar processes
are proportional to the LDV mass for the dipole interaction
$\mathscr{L}_{{\mathrm{D}}}$ (cf. Eq. (E.12)), and thus only baryon decays can
constrain $\mathbb{C}^{{\mathrm{D}}}_{ij}$ for small LDV masses. This can be
see in Figure 2 (upper left panel), where the bounds on
$\mathbb{C}^{{\mathrm{D}}}_{sd}$ from hyperon decays dominate for LDV masses
of $m_{V^{\prime}}\approx 0$ yielding a limit of
$\mathcal{O}(10^{7}\,{\mathrm{G}eV})$ on the axial coupling
$\Lambda/\mathbb{C}^{{\mathrm{D}}5}_{ij}$. This provides a strong motivation
for explicit direct searches targeting baryon decays with invisible final
states.

#####  $\boldsymbol{b\to s}$ Transitions

The limits on the dipole couplings
$\\{\mathbb{C}^{{\mathrm{D}}}_{bs},\mathbb{C}^{{\mathrm{D}}5}_{bs}\\}$ are set
by $B$-meson decays $B\to K/K^{*}+{\mathrm{i}nvis.}$ and baryon decays
$\Lambda_{b}\to\Lambda+{\mathrm{i}nvis}$. The limits from the $B$-meson decays
are obtained from our own recast of BaBar data (cf. Appendix C), except for
$B^{+}\rightarrow K^{+}+{\mathrm{i}nvis.}$ for LDV masses
$m_{V^{\prime}}<3\,{\mathrm{G}eV}$ where we use the recast in Ref. [57] of the
recent Belle II measurement of $B^{+}\to K^{+}\overline{\nu}\nu$ [39]. We also
use the recast in Ref. [57] of the BaBar measurement of $B\to
K^{*}\overline{\nu}\nu$ [36] below LDV masses of 3 GeV. The limit on
unobserved $\Lambda_{b}$ decays such as
$\Lambda_{b}\to\Lambda+{\mathrm{i}nvis.}$ is obtained by comparing the SM
prediction for the total lifetime with the experimental one inferred from all
observed channels, ascribing the difference to the allowed value for the two-
body invisible decay [22]. As for $s\to d$ transitions, decays to pseudoscalar
mesons such as $B^{+}\to K^{+}$ can neither constrain the axial coupling
$\mathbb{C}^{{\mathrm{D}}5}_{bs}$, nor $\mathbb{C}^{{\mathrm{D}}}_{bs}$ for
very small LDV masses. Otherwise, however, they do dominate over the
constraint from $\Lambda_{b}\to\Lambda$.

In contrast to $s\to d$ transitions, there is also a decay with vector mesons
in the final-state, $B\to K^{*}$, which constrains both the
$\mathbb{C}^{{\mathrm{D}}}_{bs}$ and the $\mathbb{C}^{{\mathrm{D}}5}_{bs}$
couplings in the entire LDV mass range, if kinematically allowed. Hence, $B\to
K^{*}$ decays are complementary to $B\to K$ decays in constraining
$\Lambda/\mathbb{C}^{{\mathrm{D}}}_{bs}$, setting limits on the UV scale of
the order $10^{8}\,{\mathrm{G}eV}$, and also dominate the bounds on
$\Lambda/\mathbb{C}^{{\mathrm{D}}5}_{bs}$ of similar size, up to a small
region where this channel is kinematically closed and $\Lambda_{b}\to\Lambda$
decays set the strongest limit, of the order $10^{7}\,{\mathrm{G}eV}$. Note
that there is an upper limit of order $10^{8}\,{\mathrm{G}eV}$ on
$\Lambda/\mathbb{C}^{{\mathrm{D}}}_{bs}$ at around $m_{V^{\prime}}\approx
2\,{\mathrm{G}eV}$ coming from $B\to K+V^{\prime}$ decays [57], due to a
$2.8\,\sigma$ excess in the latest Belle II measurement of $B^{+}\to
K^{+}\nu\overline{\nu}$ [39].

#####  $\boldsymbol{b\to d}$ Transitions

The bounds on the dipole couplings
$\\{\mathbb{C}^{{\mathrm{D}}}_{bd},\mathbb{C}^{{\mathrm{D}}5}_{bd}\\}$ are
obtained from $B$-meson decays $B\to\pi/\rho+{\mathrm{i}nvis.}$ and baryon
decays $\Lambda_{b}\to n+{\mathrm{i}nvis.}$ The limit on $B\to\pi$ decays is
obtained from our recast of BaBar data (cf. Appendix C), while a limit on
$B\to\rho$ decays from LEP data [55] has been derived in Ref. [56].
Analogously to $b\to s$ transitions, the pseudoscalar decay $B\to\pi$ does
neither constrain the axial coupling $\mathbb{C}^{{\mathrm{D}}5}_{bd}$ nor
$\mathbb{C}^{{\mathrm{D}}}_{bd}$ for small LDV masses, while the decay to
vector mesons $B\to\rho$ does. Thus the two meson decays are complementary in
setting limits on $\Lambda/\mathbb{C}^{{\mathrm{D}}}_{bd}$, of the order of
$10^{8}\,{\mathrm{G}eV}$, while $B\to\rho$ dominates the bounds on the limits
on $\Lambda/\mathbb{C}^{{\mathrm{D}}5}_{bd}$ of similar size, except for LDV
masses above the kinematic threshold where $\Lambda_{b}\to n$ decays take
over, constraining UV scales up to $10^{7}\,{\mathrm{G}eV}$.

#####  $\boldsymbol{c\to u}$ Transitions

Finally, the constraints on the dipole couplings
$\\{\mathbb{C}^{{\mathrm{D}}}_{cu},\mathbb{C}^{{\mathrm{D}}5}_{cu}\\}$ are set
by $D\to\pi+{\mathrm{i}nvis.}$ and the baryonic process $\Lambda_{c}\to
p+{\mathrm{i}nvis}$. For $D\to\pi$ and LDV masses $m_{V^{\prime}}\lesssim
0.5\,{\mathrm{G}eV}$, we performed a recast of the CLEO data set (analogous to
the $B$-decay recasts in Appendix C). The result is shown as a solid, blue
line in the bottom panel of Fig. 2. CLEO has only collected data up to masses
of $m_{V^{\prime}}\approx 0.5\,{\mathrm{G}eV}$, but we also show the potential
bound that could be obtained above this mass by extrapolating the bound for
massless invisible particles [22] to the whole kinematic range, which we
indicate by a dashed blue line.

For $\Lambda_{c}\to p$ we show two limits in the bottom panel of Fig. 2:
solid, orange lines denote the bound obtained from simply saturating the total
$\Lambda_{c}$ lifetime, i.e., ${\mathrm{B}R}(\Lambda_{c}\to p+V^{\prime})<1$,
while the green line indicates the 95% CL bound obtained from the BES III [40]
result for “massless” invisible particles, ${\mathrm{B}R}(\Lambda_{c}\to
p+V)<8.0\times 10^{-5}$ at 90% CL, which in fact covers invisible masses up to
$316\,{\mathrm{M}eV}$ and are multiplied by a factor 1/2, see the discussion
in the beginning of this section. We estimate the potential reach for a search
extending to larger invisible masses by assuming that the same experimental
limit below $316\,{\mathrm{M}eV}$ is also valid above, and indicate this
extrapolation by a dashed, green line. We observe that the strongest limits on
$\Lambda/\mathbb{C}^{{\mathrm{D}}}_{cu}$ are set by the BES III search for a
“massless” LDV in $\Lambda_{c}\to p$ decays, which are valid for
$m_{V^{\prime}}\lesssim 316\,{\mathrm{M}eV}$ and are of the order of
$10^{7}\,{\mathrm{G}eV}$. Between $316\,{\mathrm{M}eV}\lesssim
m_{V^{\prime}}\lesssim 500\,{\mathrm{M}eV}$ a limit of similar size is
obtained from $D\to\pi$ decays, recasting CLEO data on
$D\to(\tau\to\pi\nu)\nu$. The only available limit on LDV masses above
$0.5\,{\mathrm{G}eV}$ arises from the total $\Lambda_{c}$ lifetime, which sets
limits of order $10^{5}\,{\mathrm{G}eV}$. Naively extrapolating the limits
from CLEO on $D\to\pi$ and BES III on $\Lambda_{c}\to p$ decays to higher LDV
masses instead suggests that present bounds could be strengthened by two
orders of magnitude, if BES III would either analyze the available searches
for $\Lambda_{c}\to p$ decays with extended signal regions, or use available
data on $D\to\pi\nu\overline{\nu}$ to set a limit on the two-body decay.

Currently only $\Lambda_{c}\to p$ decays are capable to set constraints on the
axial coupling $\Lambda/\mathbb{C}^{{\mathrm{D}}5}_{cu}$, of the order of
$10^{5}\,{\mathrm{G}eV}$ and $10^{7}\,{\mathrm{G}eV}$ for LDV masses above and
below $316\,{\mathrm{M}eV}$, respectively. Besides extending the search for
$\Lambda_{c}\to p+V^{\prime}$ to higher LDV masses, this also motivates
dedicated searches for other processes such as $D\to\rho+{\mathrm{i}nvis.}$ or
$D_{s}\to K^{*}+{\mathrm{i}nvis.}$ at current or future experiments.

Figure 2:  Lower limits on quark-flavor violating dipole couplings
$\Lambda/|\mathbb{C}^{{\mathrm{D}}}_{ij}|$ (left column) and
$\Lambda/|\mathbb{C}^{{\mathrm{D}}5}_{ij}|$ (right column) of the LDV for
$s\to d,b\to s,b\to d,c\to u$ transitions @$95\%$ CL${}_{(\text{s})}$. See
text for details.

###  3.2 Dark Vector Interactions

#####  $\boldsymbol{s\to d}$ Transitions

The limits on the vector couplings $\\{\mathbb{C}^{{\mathrm{V}}}_{sd}$,
$\mathbb{C}^{{\mathrm{V}}5}_{sd}\\}$ are shown in Figure 3. As for dipole
couplings, the relevant constraints arise from $K\to\pi$ and hyperon decays,
see Table 1. Analogous to the dipole case, the limit from BES III on the decay
$\Sigma^{+}\to p+{\mathrm{i}nvis.}$ for a massless invisible is tentatively
assumed to be valid for the whole kinematic range. The limit on the scale is
indicated by a dashed orange line. $K\to\pi$ decays dominate the limits on
$\Lambda/\mathbb{C}^{{\mathrm{V}}}_{sd}$, restricting UV scales up to
$10^{12}\,{\mathrm{G}eV}$, but cannot constrain the axial coupling
$\Lambda/\mathbb{C}^{{\mathrm{V}}5}_{sd}$, where hyperon decays set the only
available bounds of the order of $10^{7}\,{\mathrm{G}eV}$. All limits are non-
vanishing when the LDV mass is taken to zero, which is due to the choice of
the prefactor in $\mathscr{L}_{{\mathrm{V}}}$ linear in the LDV mass, see Eq.
(2.2). This corresponds to the gauge-less limit where the longitudinal
polarization of the LDV is essentially a Goldstone boson. With this scaling
the flavor-violating decay is similar to the SM decay $t\to Wb$, which also
remains finite in the gauge-less $g\to 0$ limit, since the top quark
dominantly decays to the charged Goldstone Higgs, which couples only via
Yukawas to the quarks. Different choices for the prefactor, corresponding to
specific UV completions, would result in bounds that would vanish in the limit
of massless LDVs, with a LDV mass dependence that can obtained by rescaling
the limits presented here.

#####  $\boldsymbol{b\to s}$ Transitions

The constraints on the vector couplings $\mathbb{C}^{{\mathrm{V}}}_{bs}$,
$\mathbb{C}^{{\mathrm{V}}5}_{bs}$ are obtained from $B$-meson decays $B\to
K/K^{*}+{\mathrm{i}nvis.}$ and the baryonic decays
$\Lambda_{b}\to\Lambda+{\mathrm{i}nvis.}$ $B^{+}\to K^{+}$ sets the strongest
constraint on $\Lambda/\mathbb{C}^{{\mathrm{V}}}_{bs}$ of the order of
$10^{8}\,{\mathrm{G}eV}$, but cannot constrain the axial coupling
$\Lambda/\mathbb{C}^{{\mathrm{V}}5}_{bs}$. Here the dominant constraints are
set by $B\to K^{*}$ decays, also of the order of $10^{8}\,{\mathrm{G}eV}$,
apart from the region where this channel is kinematically closed and
$\Lambda_{b}\to\Lambda$ takes over and sets limits on the UV scales up to
$10^{6}\,{\mathrm{G}eV}$. Again there is an upper limit of order
$10^{12}\,{\mathrm{G}eV}$ on $\Lambda/\mathbb{C}^{{\mathrm{V}}}_{bs}$ at
around $m_{V^{\prime}}\approx 2\,{\mathrm{G}eV}$ coming from $B\to
K+V^{\prime}$ decays [57], due to a $2.8\,\sigma$ excess from the latest Belle
II measurement of $B^{+}\to K^{+}\nu\overline{\nu}$ [39].

#####  $\boldsymbol{b\to d}$ Transitions

The bounds on the vector couplings $\mathbb{C}^{{\mathrm{V}}}_{bd}$,
$\mathbb{C}^{{\mathrm{V}}5}_{bd}$ arise from $B$-meson decays
$B\to\pi/\rho+{\mathrm{i}nvis.}$ and the baryonic decays $\Lambda_{b}\to
n+{\mathrm{i}nvis.}$. Analogously to $b\to s$ transitions $B\to\pi$ decay sets
the strongest constraint on $\Lambda/\mathbb{C}^{{\mathrm{V}}}_{bd}$ of the
order of $10^{8}\,{\mathrm{G}eV}$, while
$\Lambda/\mathbb{C}^{{\mathrm{V}}5}_{bd}$ is limited to about the same values
by $B\to\rho$ decays, up to LDV masses at the kinematic threshold where
$\Lambda_{b}\to n$ decays dominate the bound of order
$10^{6}\,{\mathrm{G}eV}$.

#####  $\boldsymbol{c\to u}$ Transitions

Finally, the bounds on the vector couplings $\mathbb{C}^{{\mathrm{V}}}_{cu}$,
$\mathbb{C}^{{\mathrm{V}}5}_{cu}$ are set by the decays
$D\to\pi+{\mathrm{i}nvis.}$ and $\Lambda_{c}\to p+{\mathrm{i}nvis.}$ Meson
decays $D\to\pi$ dominate the bound on
$\Lambda/\mathbb{C}^{{\mathrm{V}}}_{cu}$ of order $10^{8}\,{\mathrm{G}eV}$,
while only baryon decays $\Lambda_{c}\to p$ can constrain the axial coupling
$\Lambda/\mathbb{C}^{{\mathrm{V}}5}_{cu}$ at order $10^{5}$ and
$10^{7}\,{\mathrm{G}eV}$, using the total lifetime and the extrapolation of
the BES III measurement, respectively, analogous to the dipole case. Again, it
would be interesting if BES III could extend their search for $\Lambda_{c}\to
p+V^{\prime}$ to higher invisible masses, as this is expected to strengthen
the present bound on the UV scale by two orders of magnitude.

Figure 3:  Lower limits on quark-flavor violating vector couplings
$\Lambda/|\mathbb{C}^{{\mathrm{V}}}_{ij}|$ (left column) and
$\Lambda/|\mathbb{C}^{{\mathrm{V}}5}_{ij}|$ (right column) of the LDV for
$s\to d,b\to s,b\to d,c\to u$ transitions @$95\%$ CL${}_{(\text{s})}$. See
text for details.

##  4 Lepton Phenomenology of Light Dark Vectors

LFV Transition |  Experimental Limit
---|---
$\mu\to e$ |  TWIST [41], Jodidior [73, 18]
$\tau\to e$ |  Belle II [38]
$\tau\to\mu$ |  Belle II [38]
Table 2:  The LFV transitions relevant for the two-body decays
$\ell\to\ell^{\prime}+V^{\prime}$ and the corresponding relevant experimental
measurements. The subindex “$r$” indicates that a recast of experimental data
was needed.

In this section we present the bounds on the flavor-violating couplings in Eq.
(2.1) and (2.2) from LFV decays $\ell\to\ell^{\prime}+V^{\prime}$ for lepton-
flavor transitions $\mu\to e$, $\tau\to e$, and $\tau\to\mu$. There are three
main differences to the quark-sector analysis: i) there is no hadronic input
required, ii) the total decay rates only depend on the combination
$|\mathbb{C}^{{\mathrm{D}}}_{ij}|^{2}+|\mathbb{C}^{{\mathrm{D}}5}_{ij}|^{2}$
and
$|\mathbb{C}^{{\mathrm{V}}}_{ij}|^{2}+|\mathbb{C}^{{\mathrm{V}}5}_{ij}|^{2}$,
and iii) for the case of $\mu\to e$ transitions one can profit from
polarization in order to suppress SM background from Michel decays. This
allows us to distinguish between $\mathbb{C}^{{\mathrm{V}}}_{ij}$ and
$\mathbb{C}^{{\mathrm{V}}5}_{ij}$ using the angular distribution of the
outgoing electron.

Concretely, for $\mu\to e$ we restrict the discussion to three benchmark
scenarios, depending on the angular dependence of the differential two-body
LFV decay rate in the limit of $m_{e}=m_{V^{\prime}}=0$

$\displaystyle\frac{d\Gamma(\mu\to
e+V^{\prime})}{d\cos\theta}\propto(1+A\cos\theta)\,,$ (4.1)

where $\theta$ is the angle between the outgoing electron momentum and the
muon polarization. We distinguish three benchmark cases: isotropic decays
($A=0$), “${\mathrm{V}}-{\mathrm{A}}$” structure $A=-1$, and
“${\mathrm{V}}+{\mathrm{A}}$” structure $A=+1$. Clearly polarization does not
help to distinguish an LFV signal from the SM background for the SM case
$A=-1$. Thus one can only rely on the monochromatic electron as the signal,
which leads to weaker bounds than in the other cases $A=0,+1$ [18].
Interestingly, many proposals have been put forward to look for this decay at
present and future high-luminosity muon facilities [18, 19, 20, 21], which are
sensitive also to invisible LDVs. We take present constraints on LFV
transitions from the references indicated in Table 2, and compare them to the
predictions for (polarized) lepton decay rates calculated in Appendix E.5.

#####  $\boldsymbol{\mu\to e}$ Transitions

The bounds from $\mu\to e+{\mathrm{i}nvis.}$ decays on dipole and vector
couplings are shown in Fig. 4. We derive them employing constraints from
experiments conducted at TRIUMF, both by the TWIST collaboration [41] in 2015
(left panel) and Jodidio et al. [73] in 1986 (right panel). For the latter, we
use the recast of Ref. [18]. The three curves in Fig. 4 show the bounds for
the three benchmark scenarios for chiral structures, corresponding to
$\mathbb{C}^{{\mathrm{D}}}_{e\mu}=0$ or $\mathbb{C}^{{\mathrm{D}}5}_{e\mu}=0$
for $A=0$, and $\mathbb{C}^{{\mathrm{D}}}_{e\mu}=\pm
i\mathbb{C}^{{\mathrm{D}}5}_{e\mu}$ for $A\approx\pm 1$ in the upper panel,
while in the lower panel they correspond to
$\mathbb{C}^{{\mathrm{V}}}_{e\mu}=0$ or $\mathbb{C}^{{\mathrm{V}}5}_{e\mu}=0$
for $A=0$, and
$\mathbb{C}^{{\mathrm{V}}}_{e\mu}=\pm\mathbb{C}^{{\mathrm{V}}5}_{e\mu}$ for
$A\approx\pm 1$. For couplings that are not aligned to the SM, i.e., not
“${\mathrm{V}}-{\mathrm{A}}$”, the dominant constraints on LDVs lighter than
about 5 MeV are set by the Jodidio experiment, which limits UV scales of the
order of $10^{10}\,{\mathrm{G}eV}$. Heavier LDVs are constrained only by
TWIST, setting limits of the order of few $\times 10^{9}\,{\mathrm{G}eV}$.
LDVs with “${\mathrm{V}}-{\mathrm{A}}$” couplings are constrained by TWIST
with bounds of the same order, exceeding the corresponding Jodidio limits also
in the light-mass regime.

#####  $\boldsymbol{\tau\to\mu/e}$ Transitions

The limits from Belle II on $\tau\to\mu/e+{\mathrm{i}nvis.}$ decays constrain
$\tau\to e$ and $\tau\to\mu$ transitions according to Fig. 5, where we shows
the bounds on the dipole $\Lambda/\mathbb{C}^{{\mathrm{D}}}_{\tau\ell}$ (left
panel) and vector couplings $\Lambda/\mathbb{C}^{{\mathrm{V}}}_{\tau\ell}$
(right panel). Constraints on the axial couplings
$\Lambda/\mathbb{C}^{{\mathrm{D}}5}_{\tau\ell}$ and
$\Lambda/\mathbb{C}^{{\mathrm{V}}5}_{\tau\ell}$ are at the same level, as the
difference is suppressed by $m_{\ell}/m_{\tau}$, cf. Appendix E.5. Bounds for
$\tau\to e$ and $\tau\to\mu$ transitions are comparable, limiting UV scales of
the order of few $\times 10^{7}\,{\mathrm{G}eV}$ for dipole couplings, and few
$\times 10^{6}\,{\mathrm{G}eV}$ for vector couplings.

Figure 4: Upper panel: Lower limits on the dipole coupling for $\mu\to e$
transitions $\Lambda/|\mathbb{C}^{{\mathrm{D}}}_{e\mu}|$ from TWIST [41] (left
panel) and Jodidio et al. [73, 18] (right panel). The bounds are shown for
three different choices for $\mathbb{C}^{{\mathrm{D}}5}_{e\mu}$, corresponding
to different angular distributions of the electron momentum, cf. Eq. (4.1):
isotropic decay ($A=0$), alignment to SM decay “$V\\!-\\!A$” ($A=-1$) and
“$V\\!+\\!A$” ($A=+1$). Lower panel: same for the vector coupling
$\Lambda/|\mathbb{C}^{{\mathrm{V}}}_{e\mu}|$. See text for details.

Figure 5: Upper panel: Lower limits on the dipole (left panel) and vector
(right panel) couplings for $\tau\to e$ transitions
$\Lambda/|\mathbb{C}^{{\mathrm{D}}}_{\tau e}|$,
$\Lambda/|\mathbb{C}^{{\mathrm{V}}}_{\tau e}|$ from Belle II [38]. Lower
panel: same for $\tau\to\mu$ transitions
$\Lambda/|\mathbb{C}^{{\mathrm{D}}}_{\tau\mu}|$,
$\Lambda/|\mathbb{C}^{{\mathrm{V}}}_{\tau\mu}|$. Constraints on the axial
couplings $\Lambda/|\mathbb{C}^{{\mathrm{D}}5}_{\tau\ell}|$ and
$\Lambda/|\mathbb{C}^{{\mathrm{V}}5}_{\tau\ell}|$ are essentially of the same
size, as the difference is suppressed by $m_{\ell}/m_{\tau}$, cf. Appendix
E.5.

##  5 Flavor-violating LDVs from the Renormalization Group

In this section we study the phenomenologically interesting scenario in which
LDV interactions with the SM are flavor-universal in the UV theory, so that
flavor-violating couplings are generated only from the SM flavor violation via
the renomalization-group evolution. We start right below the UV scale
$\Lambda$—taken to be much above the electroweak scale—and consider
$SU(2)_{\mathrm{L}}\times U(1)_{Y}$ invariant vector and dipole interactions
of the $V^{\prime}$ to the SM. For vector couplings see the trivial
$SU(2)_{\mathrm{L}}\times U(1)_{Y}$ generalization of Eq. (2.3) and for dipole
couplings see Eq. (2.6). We align possible new sources of flavor-violation of
the $V^{\prime}$ with the flavor violation in the SM by taking the vector
couplings to be flavor-universal, i.e., proportional to the identity matrix in
flavor space, and by taking the dipole couplings to be proportional to the SM
Yukawas. In both cases they are flavor diagonal in the mass basis, such that
flavor-changing interactions with the $V^{\prime}$ are only induced by the
renormalization-group evolution to the EW scale and always proportional to the
CKM matrix. Flavor-violating couplings in the IR thus follow the paradigm of
minimal flavor violation (MFV) [42].

We do not explicitly consider kinetic mixing between the $U(1)^{\prime}$ LDV
and the $U(1)_{Y}$ boson, as it leads only to a shift in the flavor-universal
LDV couplings after diagonalising the photon kinetic terms. By working in this
basis, our results also apply to models with kinetic mixing, upon re-defining
the flavor-universal couplings.

We discuss separately the case of dipole and vector couplings in Section 5.1
and 5.2, respectively.

###  5.1 Dipole interactions

$i-j$ | $2-1$ | $1-2$ | $2-3$ | $3-2$ | $1-3$ | $3-1$
---|---|---|---|---|---|---
$(\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{u})_{i\neq j}/\Lambda$ | $\lambda^{5}y_{b}^{2}y_{c}$ | $\lambda^{5}y_{b}^{2}y_{c}$ | $\lambda^{2}y_{b}^{2}y_{t}$ | $\lambda^{2}y_{b}^{2}y_{t}$ | $\lambda^{3}y_{b}^{2}y_{t}$ | $\lambda^{3}y_{b}^{2}y_{t}$
$(\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{d})_{i\neq j}/\Lambda$ | $\lambda^{5}y_{t}^{2}y_{s}$ | $\lambda^{5}y_{t}^{2}y_{s}$ | $\lambda^{2}y_{t}^{2}y_{b}$ | $\lambda^{2}y_{t}^{2}y_{b}$ | $\lambda^{3}y_{t}^{2}y_{b}$ | $\lambda^{3}y_{t}^{2}y_{b}$
Table 3:  Parametric size of leading flavor-violating contributions at low-
energy in the UV universal scenario for dipole couplings, cf. Eq. (5.3). Here
$\lambda\approx 0.23$ denotes the Wolfenstein parameter and $y_{f}=m_{f}/v$
are SM Yukawas couplings. Up-quark transitions (first line) are proportional
to the high-scale coupling $c^{{\mathrm{D}}}_{d}$, down-quark transitions
(second line) are proportional to the high-scale coupling
$c^{{\mathrm{D}}}_{u}$, and all entries are multiplied by
$v/\Lambda_{6}^{2}\log(\Lambda_{6}/\mu)/(16\pi^{2})$.

In the interaction basis, the dipole interactions of the LDV with SM fermions
are given by (cf. Eq. (2.6))

$\displaystyle\mathscr{L}_{\mathrm{int}}\supset-\left(\overline{Q}\,Y_{u}\widetilde{H}u_{{\mathrm{R}}}+\overline{Q}\,Y_{d}Hd_{{\mathrm{R}}}+\mathrm{h.c.}\right)+\frac{1}{\Lambda^{2}_{6}}V^{\prime}_{\mu\nu}\left(\overline{Q}C^{{\mathrm{D}}}_{u}\sigma^{\mu\nu}\widetilde{H}u_{{\mathrm{R}}}+\overline{Q}C^{{\mathrm{D}}}_{d}\sigma^{\mu\nu}Hd_{{\mathrm{R}}}+\mathrm{h.c.}\right)\,,$
(5.1)

with the SM Yukawa matrices $Y_{f}$, $f=u,d$, and arbitrary $3\times 3$
matrices $C_{f}^{{\mathrm{D}}}$. The one-loop RG equations for the couplings
$C_{f}^{{\mathrm{D}}}$ and the Yukawa matrices $Y_{f}$ are listed in Appendix
B. For the UV universal setup that we consider, the initial conditions at the
UV scale $\Lambda_{6}$ are

$C^{{\mathrm{D}}}_{d}\big{|}_{\mu=\Lambda_{6}}=c^{{\mathrm{D}}}_{d}~{}Y_{d}\big{|}_{\mu=\Lambda_{6}}\,,\qquad
C^{{\mathrm{D}}}_{u}\big{|}_{\mu=\Lambda_{6}}=c^{{\mathrm{D}}}_{u}~{}Y_{u}\big{|}_{\mu=\Lambda_{6}}\,,$
(5.2)

with $c^{{\mathrm{D}}}_{f}\in\mathbb{C}$. By solving the RGE at leading-
logarithmic accuracy and subsequently rotating to the mass basis for the
quarks we find the low-energy dipole couplings in the
${\mathrm{L}}/{\mathrm{R}}$ notation of Eq. (2.3) with $f=u,d$ to be444 Since
the couplings at the UV scale $\Lambda_{6}$ are aligned to the SM Yukawa
matrices, a correction from the Yukawa RGE, given in Appendix B, must be
included.

$\begin{split}\frac{1}{\Lambda}\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{u}(\mu)&=\frac{v}{\Lambda_{6}^{2}}\left(c^{{\mathrm{D}}}_{u}\hat{Y}_{u}-\frac{1}{16\pi^{2}}\left(3c^{{\mathrm{D}}}_{u}\hat{Y}_{u}\hat{Y}_{u}^{\dagger}\hat{Y}_{u}-c^{{\mathrm{D}}}_{d}V_{\text{CKM}}\hat{Y}_{d}\hat{Y}_{d}^{\dagger}V_{\text{CKM}}^{\dagger}\hat{Y}_{u}\right)\log(\Lambda_{6}/\mu)\right)\,,\\\
\frac{1}{\Lambda}\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{d}(\mu)&=\frac{v}{\Lambda_{6}^{2}}\left(c^{{\mathrm{D}}}_{d}\hat{Y}_{d}-\frac{1}{16\pi^{2}}\left(3c^{{\mathrm{D}}}_{d}\hat{Y}_{d}\hat{Y}_{d}^{\dagger}\hat{Y}_{d}-c^{{\mathrm{D}}}_{u}V_{\text{CKM}}^{\dagger}\hat{Y}_{u}\hat{Y}_{u}^{\dagger}V_{\text{CKM}}\hat{Y}_{d}\right)\log(\Lambda_{6}/\mu)\right)\,,\end{split}$
(5.3)

where $V_{\text{CKM}}$ is the CKM matrix, and $\hat{Y}_{f}=m_{f}/v$ are the
diagonal SM Yukawas. The left-handed couplings
$\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{f}$ are related to the ones in Eq.
(5.3) by hermitian conjugation,
$\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{f}=(\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{f})^{\dagger}$.
Note that indeed flavor off-diagonal entries are generated in both the up- and
the down-quark sector at one-loop. They are proportional to the CKM matrix and
the UV coupling of the other sector, i.e.,
$\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{u}\propto c_{d}^{D}$ and
$\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{d}\propto c_{u}^{D}$. Carrying out the
matrix multiplications, one can identify the numerically leading contribution
to a given flavor transition. We show these leading contributions in Table 3
for both sectors.

Using these results, we determine the experimental limits on the high-scale
couplings $c^{{\mathrm{D}}}_{d}$ and $c^{{\mathrm{D}}}_{u}$ in Eq. (5.2) from
the limits on two-body meson decays discussed in Section 3. Note that the
renormalization scale $\mu$ is set to the EW scale since below there is no
Yukawa running.

As expected from the high-level of flavor suppression inherent to the setup,
the resulting bounds are very mild and often weaker than the constraints from
perturbative unitarity. For this reason we only display in Fig. 6 (left panel)
the strongest bounds, which come from $B\to K^{*}$ and require
$\Lambda_{6}\sim\,{\mathrm{T}eV}$ for $c^{{\mathrm{D}}}_{u}=1$ (for
$c^{{\mathrm{D}}}_{d}=1$ the limit on $\Lambda_{6}$ is far below the
electroweak scale and is therefore not shown).

Figure 6: Lower limits on the UV scale in the UV universal scenario for dipole
(left panel) and vector couplings (right panel), only showing the strongest
constraints. See text for details.

###  5.2 Vector interaction

In the interaction basis, the vector interactions of the LDV with the SM
fermions are given by (cf. Eq. (2.3))

$\displaystyle\mathscr{L}_{\mathrm{int}}\supset-\left(\overline{Q}\,Y_{u}\widetilde{H}u_{{\mathrm{R}}}+\overline{Q}\,Y_{d}Hd_{{\mathrm{R}}}+\mathrm{h.c.}\right)+V^{\prime}_{\mu}\left(\overline{Q}C_{Q}^{\mathrm{V}}\gamma^{\mu}Q+\overline{u}_{{\mathrm{R}}}C_{u}^{\mathrm{V}}\gamma^{\mu}u_{{\mathrm{R}}}+\overline{d}_{{\mathrm{R}}}C_{d}^{\mathrm{V}}\gamma^{\mu}d_{{\mathrm{R}}}\right)\,,$
(5.4)

with SM Yukawa matrices $Y_{f}$, $f=u,d$, and arbitrary hermitian $3\times 3$
matrices $C_{X}^{{\mathrm{V}}}$ with $X=Q,u,d$. The one-loop RG equations for
the couplings $C_{X}^{{\mathrm{V}}}$ and the Yukawa matrices $Y_{f}$ are
listed in Appendix B. For the UV universal setup that we consider in this
section, the boundary conditions at the UV scale $\Lambda$ are

$\displaystyle C^{{\mathrm{V}}}_{Q}(\Lambda)$
$\displaystyle=c^{{\mathrm{V}}}_{Q}~{}\mathbbold{1}_{3}\,,$ $\displaystyle
C^{{\mathrm{V}}}_{u}(\Lambda)$
$\displaystyle=c^{{\mathrm{V}}}_{u}~{}\mathbbold{1}_{3}\,,$ $\displaystyle
C^{{\mathrm{V}}}_{d}(\Lambda)$
$\displaystyle=c^{{\mathrm{V}}}_{d}~{}\mathbbold{1}_{3}\,.$ (5.5)

with $c^{{\mathrm{V}}}_{X}$ real numbers.

By solving the RGE at leading-logarithmic accuracy and subsequently rotating
to the mass basis for the quarks, we find the low-energy vector couplings in
the ${\mathrm{L}}/{\mathrm{R}}$ notation of Eq. (2.3) to be

$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{u}(\mu)$
$\displaystyle=c^{{\mathrm{V}}}_{Q}\mathbbold{1}_{3}-\frac{1}{16\pi^{2}}\left(\left(c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{u}\right)\hat{Y}_{u}\hat{Y}_{u}^{\dagger}+\left(c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{d}\right)V_{\text{CKM}}\hat{Y}_{d}\hat{Y}_{d}^{\dagger}V_{\text{CKM}}^{\dagger}\right)\log(\Lambda/\mu)\,,$
$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{d}(\mu)$
$\displaystyle=c^{{\mathrm{V}}}_{Q}\mathbbold{1}_{3}-\frac{1}{16\pi^{2}}\left(\left(c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{u}\right)V_{\text{CKM}}^{\dagger}\hat{Y}_{u}\hat{Y}_{u}^{\dagger}V_{\text{CKM}}+\left(c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{d}\right)\hat{Y}_{d}\hat{Y}_{d}^{\dagger}\right)\log(\Lambda/\mu)\,,$
$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{u}(\mu)$
$\displaystyle=c^{{\mathrm{V}}}_{u}\mathbbold{1}_{3}-\frac{1}{8\pi^{2}}\left(c^{{\mathrm{V}}}_{u}-c^{{\mathrm{V}}}_{Q}\right)\hat{Y}_{u}^{\dagger}\hat{Y}_{u}\log(\Lambda/\mu)\,,$
$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{d}(\mu)$
$\displaystyle=c^{{\mathrm{V}}}_{d}\mathbbold{1}_{3}-\frac{1}{8\pi^{2}}\left(c^{{\mathrm{V}}}_{d}-c^{{\mathrm{V}}}_{Q}\right)\hat{Y}_{d}^{\dagger}\hat{Y}_{d}\log(\Lambda/\mu)\,,$
(5.6)

where $V_{\text{CKM}}$ is the CKM matrix and $\hat{Y}_{f}=m_{f}/v,f=u,d$ are
the diagonal SM Yukawas. Note that the couplings of right-handed interactions
$\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{u},\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{d}$
are always flavor diagonal, while flavor-violating terms in the IR are induced
in the left-handed interactions $\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{u}$,
$\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{d}$ proportionally to
$c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{u}$ and
$c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{d}$. Therefore, if the UV couplings
are also universal among the different sectors, i.e.,
$c^{{\mathrm{V}}}_{Q}=c^{{\mathrm{V}}}_{u}=c^{{\mathrm{V}}}_{d}$, there is no
flavor violation in the IR at one-loop, as in this case the LDV actually
couples to the baryon-number current, which is conserved at tree-level
inducing flavor violation only at two-loop [74].

We now discuss this fact in more detail, before turning to the limits. One can
rewrite the interactions in Eq. (5.4) for the case of flavor-universal UV
boundary conditions in Eq. (5.5) in terms of the tree-level conserved (but
anomalous) $U(1)_{B}$ current
$J_{B}^{\mu}=\sum_{i}\left(\bar{Q}_{i}\gamma^{\mu}Q_{i}+\bar{u}_{{\mathrm{R}}i}\gamma^{\mu}u_{{\mathrm{R}}i}+\bar{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}i}\right)$,
and the two non-conserved currents
$J_{Nd}^{\mu}=\sum_{i}\bar{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}i}$,
and
$J_{Nu}^{\mu}=\sum_{i}\bar{u}_{{\mathrm{R}}i}\gamma^{\mu}u_{{\mathrm{R}}i}$.
As all currents are not conserved beyond tree-level, we take their
coefficients to be proportional to the LDV mass

$\mathscr{L}_{\mathrm{int}}\supset-\left(\overline{Q}\,Y_{u}\widetilde{H}u_{{\mathrm{R}}}+\overline{Q}\,Y_{d}Hd_{{\mathrm{R}}}+\mathrm{h.c.}\right)+\frac{m_{V^{\prime}}}{\Lambda}V^{\prime}_{\mu}\Big{[}C_{B}^{\mathrm{V}}J_{B}^{\mu}+C_{Nd}^{\mathrm{V}}J_{Nd}^{\mu}+C_{Nu}^{\mathrm{V}}J_{Nu}^{\mu}\Big{]}\,,$
(5.7)

Matching to Eqs. (5.4) and (5.5) gives

$\displaystyle\frac{m_{V^{\prime}}}{\Lambda}c_{B}^{\mathrm{V}}$
$\displaystyle=c^{{\mathrm{V}}}_{Q}\,,$
$\displaystyle\frac{m_{V^{\prime}}}{\Lambda}c_{Nd}^{\mathrm{V}}$
$\displaystyle=c^{{\mathrm{V}}}_{d}-c^{{\mathrm{V}}}_{Q}\,,$
$\displaystyle\frac{m_{V^{\prime}}}{\Lambda}c_{Nu}^{\mathrm{V}}$
$\displaystyle=c^{{\mathrm{V}}}_{u}-c^{{\mathrm{V}}}_{Q}\,.$ (5.8)

At the one-loop level there is no flavor violation proportional to
$C_{B}^{\mathrm{V}}$. However, flavor violation does arises due to the non-
conserved currents and is thus proportional to the difference of couplings
$c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{u}$ and
$c^{{\mathrm{V}}}_{Q}-c^{{\mathrm{V}}}_{d}$. Rewriting Eq. (5.2) in terms of
the UV coefficients
$c_{B}^{\mathrm{V}},c_{Nd}^{\mathrm{V}},c_{Nu}^{\mathrm{V}}$ with the proper
LDV mass scaling gives finally

$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{u}(\mu)$
$\displaystyle=\frac{m_{V^{\prime}}}{\Lambda}\left[c^{{\mathrm{V}}}_{B}\mathbbold{1}_{3}+\frac{1}{16\pi^{2}}\left(c_{Nu}^{{\mathrm{V}}}\hat{Y}_{u}\hat{Y}_{u}^{\dagger}+c_{Nd}^{{\mathrm{V}}}V_{\text{CKM}}\hat{Y}_{d}\hat{Y}_{d}^{\dagger}V_{\text{CKM}}^{\dagger}\right)\log(\Lambda/\mu)\right]\,,$
$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{d}(\mu)$
$\displaystyle=\frac{m_{V^{\prime}}}{\Lambda}\left[c^{{\mathrm{V}}}_{B}\mathbbold{1}_{3}+\frac{1}{16\pi^{2}}\left(c_{Nu}^{{\mathrm{V}}}V_{\text{CKM}}^{\dagger}\hat{Y}_{u}\hat{Y}_{u}^{\dagger}V_{\text{CKM}}+c^{{\mathrm{V}}}_{Nd}\hat{Y}_{d}\hat{Y}_{d}^{\dagger}\right)\log(\Lambda/\mu)\right]\,,$
$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{u}(\mu)$
$\displaystyle=\frac{m_{V^{\prime}}}{\Lambda}\left[\left(c^{{\mathrm{V}}}_{B}+c_{Nu}^{{\mathrm{V}}}\right)\mathbbold{1}_{3}-\frac{1}{8\pi^{2}}c_{Nu}^{{\mathrm{V}}}\hat{Y}_{u}^{\dagger}\hat{Y}_{u}\log(\Lambda/\mu)\right]\,,$
$\displaystyle\left(\frac{m_{V^{\prime}}}{\Lambda}\right)\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{d}(\mu)$
$\displaystyle=\frac{m_{V^{\prime}}}{\Lambda}\left[\left(c^{{\mathrm{V}}}_{B}+c_{Nd}^{{\mathrm{V}}}\right)\mathbbold{1}_{3}-\frac{1}{8\pi^{2}}c_{Nd}^{{\mathrm{V}}}\hat{Y}_{d}^{\dagger}\hat{Y}_{d}\log(\Lambda/\mu)\right]\,,$
(5.9)

The numerically leading contributions to a given (hermitian) flavor transition
in left-handed interactions are shown in Table 4 for both sectors. We display
the resulting bounds on $\Lambda$ in the right panel of Fig. 6 for
$c^{\mathrm{V}}_{Nu}=1$ (there is no constraint from $c^{\mathrm{V}}_{Nd}$ at
one-loop), which are of order $\Lambda\geq 10^{3}\,{\mathrm{T}eV}$ for
$K\to\pi$ transitions. These limits are weakened by about an order of
magnitude for LDV masses above $m_{K}-m_{\pi}$, where the dominant constraint
comes from $B\to K$ transitions. In dashed green, we also show the limits
coming from the flavor-violating contribution that is induced at the two-loop
level by the coupling of the LDV to the anomalous baryon current
$J_{B}^{\mu}$. The corresponding limit on the scale $\Lambda$ has been
obtained by rescaling the result for $K\to\pi$ of Fig. (1) from Ref. [74],
giving $\Lambda\geq 3.5\,{\mathrm{T}eV}$ for $c^{{\mathrm{V}}}_{B}=1$ and
$c^{\mathrm{V}}_{Nu}=c^{\mathrm{V}}_{Nd}=0$. This is about three orders of
magnitude weaker than the limit one obtains if the LDV also couples to
currents that are not conserved at tree-level, i.e., taking
$c^{V}_{Nu}=c^{V}_{Nd}=1$.

$i-j$ | $2-1$ | $3-2$ | $3-1$
---|---|---|---
$(\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{u})_{i\neq j}$ | $\lambda^{5}y_{t}^{2}$ | $\lambda^{2}y_{t}^{2}$ | $\lambda^{3}y_{t}^{2}$
$(\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{d})_{i\neq j}$ | $\lambda^{5}y_{b}^{2}$ | $\lambda^{2}y_{b}^{2}$ | $\lambda^{3}y_{b}^{2}$
Table 4:  Parametric size of leading flavor-violating contributions to the
low-energy vector couplings of $V^{\prime}$ in the UV universal scenario, cf.
Eq. (5.2). Here $\lambda\approx 0.23$ denotes the Wolfenstein parameter. Up-
quark transitions (first line) are proportional to the high-scale coupling
$c^{{\mathrm{V}}}_{u}-c^{{\mathrm{V}}}_{Q}=c_{Nu}^{\mathrm{V}}$, down-quark
transitions (second line) are proportional to the high-scale coupling
$c^{{\mathrm{V}}}_{d}-c^{{\mathrm{V}}}_{Q}=c_{Nd}^{\mathrm{V}}$, and all
entries are multiplied by $\log(\Lambda/\mu)/(16\pi^{2})$.

##  6 Summary and Conclusions

In this work we have systematically studied the flavor phenomenology of light
dark vectors (LDVs). We have restricted our analysis to scenarios where the
LDV is directly linked to dark matter, and is either itself invisible or
promptly decays to invisible particles, such that the LDV appears as missing
energy. Working in the context of a general EFT, we have considered both
flavor-violating dipole (see Eq. (2.1)) and vector couplings (see Eq. (2.2))
of the LDV to SM fermions. We have calculated the resulting predictions for
the decay rates of mesons, baryons, and polarized leptons as a function of the
LDV mass, see Section E. These predictions were compared to the experimental
limits on various hadronic processes (Table 1) and LFV transitions (Table 2).
For $B\to\pi/K/K^{*}$ decays experimental limits from B-factories are only
available for three-body decays with two invisible neutrinos, so we have
recasted available data to obtain bounds on the two-body decay with missing
energy as a function of the LDV mass, see Fig. 7. The resulting limits on
general vector and dipole interactions of the LDV are summarized in Figs. 2
and 3 for the quark sector, and in Fig. 4 and 5 for the lepton sector. Vector
couplings are at least dimension-five operators, which results in very
stringent limits on the UV scale, reaching up to $10^{12}\,{\mathrm{G}eV}$ in
$K\to\pi$ decays, $10^{8}\,{\mathrm{G}eV}$ in $B$\- and $D$-meson decays,
$10^{9}\,{\mathrm{G}eV}$ in $\mu\to e$ decays, and $10^{7}\,{\mathrm{G}eV}$ in
$\tau\to\mu/e$ decays. Bounds on dipole couplings are weaker, if viewed as
dimension-six operators above the EW scale, but they still probe UV scales of
order $10^{6}\,{\mathrm{G}eV}$ in $K\to\pi$ and $\mu\to e$ decays.
Importantly, all channels will be improved by present or near-future
experiments, such as NA62, Belle II, BES III, MEG-II or Mu3e. We have also
discussed a scenario where couplings in the UV are flavor-universal, so that
quark-flavor violation is only induced radiatively through the CKM matrix. For
this analysis we derived the relevant renormalization-group equations (RGEs)
for both dipole and minimal couplings in Appendix B, and used our previous
results to convert limits on flavor-changing interactions into limits on
flavor-diagonal couplings, see Figure. 6.

To summarize, the aim of this work is to stress the importance of flavor-
violating transitions for light, dark-matter searches, which is copiously
produced in the lab as missing energy in decays of SM particles. Here we
focused on the LDV as part of a dark sector, and showed that present
constraints from precision flavor experiments already probe UV scales as large
as $10^{12}\,{\mathrm{G}eV}$. This underlines the important role of present
and next-generation flavor factories in hunting down dark matter in the
laboratory.

## Acknowledgments

We would like to thank Adam Falkowski, Jorge Martin Camalich and Zhijun Li for
useful discussions and Jure Zupan for comments on the manuscript. This work
has received support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
860881-HIDDeN and is partially supported by project B3a and C3b of the DFG-
funded Collaborative Research Center TRR257 “Particle Physics Phenomenology
after the Higgs Discovery”.

##  Appendix A UV Motivation of Vector Couplings

In this section we motivate the scaling behavior of the flavor-violating
vector coupling in the Lagrangian of Eq. (2.2), both by EFT considerations and
explicit UV-complete models. In perturbative UV completions, the scaling is at
least linear in the dark $U(1)^{\prime}$ breaking scale, and we will provide
two example scenarios: one that gives linear and one that gives quadratic
scaling. We begin with the EFT discussion of the latter.

###  A.1 EFT Discussion for Quadratic scaling

For the EFT approach it is convenient to consider the coupling to the
Goldstone boson $G$ in the gauge-less limit, rather than the coupling of the
dark vector $V^{\prime}_{\mu}$ itself. They are related by the Goldstone-boson
Equivalence Theorem, which states that at sufficiently high energies, or
equivalently sufficiently small dark vector masses $m_{V}^{\prime}$, the
vector boson coupling is dominated by its longitudinal polarization, which in
the small $m_{V}^{\prime}$ limit becomes the Goldstone boson. Thus one can
work out the couplings of the Goldstone boson and recover the relevant vector-
boson couplings by replacing $\partial_{\mu}G\to-
m_{V^{\prime}}V^{\prime}_{\mu}$ in the interaction Lagrangian.

We, therefore, consider the case where the dark $U(1)^{\prime}$ gauge group is
spontaneously broken by some (SM singlet) scalar field $S$ with charge $+1$
under the $U(1)^{\prime}$. We take the gauge-less limit, so that $G$ is a true
Goldstone boson, contained in $S$ according to

$\displaystyle S=\frac{v^{\prime}}{\sqrt{2}}\exp(iG/v^{\prime})\,,$ (A.1)

where $v^{\prime}/\sqrt{2}$ is the (real) VEV that breaks $U(1)^{\prime}$,
connected to the dark vector mass by $m_{V^{\prime}}=g^{\prime}v^{\prime}$,
and we have ignored the radial mode that obtains its mass around $v^{\prime}$.
This mode, together with all UV fields are taken to be much heavier than the
electroweak scale, so that in the IR there is only the SM and the Goldstone
boson $G\supset S$, which is formally invariant under global $U(1)^{\prime}$
transformations treating $S$ as a spurion with charge $+1$. Note that one can
always realize such a scenario by making $g^{\prime}$ sufficiently small.
Writing down the general EFT for this setup, it is clear that if SM fields are
not charged under $U(1)^{\prime}$, the possible couplings of the Goldstone to
SM fields must involve the same powers of $S^{\dagger}$ and $S$. The first
such bilinear that gives a non-trivial combination containing the Goldstone is
then $S^{\dagger}\overset{\leftrightarrow}{\partial}_{\mu}S\supset
iv^{\prime}\partial_{\mu}G$. This implies that, e.g., right-handed down quarks
can only couple to the Goldstone at the level of dimension-six operators only

$\displaystyle\mathscr{L}^{\text{EFT}}_{\text{quadratic}}\supset\frac{c_{ij}}{\Lambda^{2}}(iS^{\dagger}\overset{\leftrightarrow}{\partial}_{\mu}S)\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)$
$\displaystyle=-\frac{c_{ij}}{\Lambda^{2}}v^{\prime}\partial_{\mu}G\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)\,,$
(A.2)

where $\Lambda$ is the UV scale and in general there is flavor violation in
the (hermitian) EFT coefficients, $c_{i\neq j}\neq 0$. The coupling of the
dark vector in this setup is then recovered by $\partial_{\mu}G\to-
m_{V^{\prime}}V^{\prime}_{\mu}$, so is given by

$\displaystyle\mathscr{L}^{\text{EFT}}_{\text{quadratic}}\supset
c_{ij}\frac{v^{\prime}m_{V}^{\prime}}{\Lambda^{2}}V^{\prime}_{\mu}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)\,.$
(A.3)

This analysis demonstrates that the interactions of dark vectors with SM
fields that are neutral under the $U(1)^{\prime}$ scale at least as
$m_{V}^{\prime}/\Lambda\times v^{\prime}/\Lambda$. In particular they involve
an additional factor of the $U(1)^{\prime}$ breaking scale as compared to Eq.
(2.2). Below we will confirm this expectation in an explicit UV model, see
Section A.3.

###  A.2 EFT Discussion for Linear scaling

In order to have dark-vector couplings with a linear scaling in the
$U(1)^{\prime}$ breaking scale, one necessarily has to charge SM fields under
$U(1)^{\prime}$. In this case the vector boson couples directly to the charged
fields via the dimension-four operator, e.g., for right-handed down quarks

$\displaystyle\mathscr{L}_{\text{linear}}\supset
g^{\prime}V^{\prime}_{\mu}\left(\overline{d}_{{\mathrm{R}}}\gamma^{\mu}X_{d}d_{{\mathrm{R}}}\right)\,.$
(A.4)

where $X_{d}$ is the diagonal $U(1)^{\prime}$ charge matrix. To see how off-
diagonal entries are generated, one has to rotate to the mass basis, which is
governed by the Yukawa couplings. It is clear that there is no flavor
violation if $X_{d}$ is universal, i.e., proportional to the identity matrix.
If instead charges are non-universal, the mass matrix cannot be generic at the
renormalizable level, i.e., it does not yield realistic fermion masses without
breaking $U(1)^{\prime}$. Therefore, insertions of $S$ or $S^{\dagger}$ have
to be considered to obtain realistic fermion masses.

Restricting for simplicity to two generations, and charging only
$d_{{\mathrm{R}}1}$ with charge $+1$, i.e., $X_{d}={\mathrm{D}iag}(1,0)$,
$X_{Q}=X_{H}=0$, the full Yukawa matrix requires higher-dimensional operators
to have full rank

$\displaystyle{\cal L}^{\text{EFT}}_{\text{linear}}\supset-
y_{i}\overline{Q}_{i}Hd_{{\mathrm{R}}2}-z_{i}\frac{S^{\dagger}}{\Lambda}\overline{Q}_{i}Hd_{{\mathrm{R}}1}+{\mathrm{h}.c.}$
(A.5)

Thus the down-quark Yukawa matrix is given by

$\displaystyle Y_{d}=\begin{pmatrix}z_{1}\epsilon&y_{1}\\\
z_{2}\epsilon&y_{2}\end{pmatrix}\quad\text{with}\quad\epsilon=\frac{v^{\prime}}{\sqrt{2}\Lambda}$
(A.6)

We can ignore here the Goldstone in $S$, since we already have the coupling of
the gauge field in Eq. (A.4), which leads to flavor-violating couplings with
$V^{\prime}$ after rotating to the mass basis. Nevertheless we can also
reproduce this coupling with the same arguments as above: in the gauge-less
limit, we rescale $d_{{\mathrm{R}}1}\to d_{{\mathrm{R}}1}e^{iG/v^{\prime}}$,
which removes $G$ from the Yukawa sectors. Ignoring chiral anomalies, this
rescaling only affects the kinetic terms, as it is a local $U(1)^{\prime}$
transformation

$\displaystyle\mathscr{L}_{\text{linear}}^{\text{EFT}}\supset
i\overline{d}_{{\mathrm{R}}1}\not{\partial}d_{{\mathrm{R}}1}\to-\frac{\partial_{\mu}G}{v^{\prime}}\overline{d}_{{\mathrm{R}}1}\gamma^{\mu}d_{{\mathrm{R}}1}\,,$
(A.7)

which reproduces Eq. (A.4) upon $\partial_{\mu}G\to-
m_{V^{\prime}}V_{\mu}=-g^{\prime}v^{\prime}V_{\mu}$.

We are left to diagonalize the Yukawa matrix $Y_{d}$ in Eq. (A.6), or rather
$Y_{d}^{\dagger}Y_{d}$, in order to find the mixing matrix $V_{d}$ of right-
handed down quarks, defined as
$V_{Q}^{\dagger}Y_{d}V_{d}=Y_{d}^{\mathrm{d}iag}$. In the limit when
$\epsilon\ll 1$, one has

$\displaystyle V_{d}\approx\begin{pmatrix}1&z_{2}/y_{2}\epsilon\\\
-z_{2}/y_{2}\epsilon&1\end{pmatrix}\,,$ (A.8)

where we have set $y_{1}=0$ without loss of generality. Rotating the dark-
vector couplings in Eq. (A.4) to the mass basis defined by
$d_{{\mathrm{R}}}\to V_{d}d_{{\mathrm{R}}}$ gives finally

$\displaystyle\mathscr{L}_{\text{linear}}\supset
g^{\prime}V^{\prime}_{\mu}\left(\overline{d}_{{\mathrm{R}}}\gamma^{\mu}V_{d}^{\dagger}X_{d}V_{d}d_{{\mathrm{R}}}\right)=g^{\prime}V^{\prime}_{\mu}(V_{d}^{*})_{1i}(V_{d})_{1j}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)\,,$
(A.9)

so that indeed off-diagonal couplings are generated proportional to
$g^{\prime}(V_{d}^{*})_{11}(V_{d})_{12}\sim g^{\prime}\epsilon\sim
m_{V}^{\prime}/\Lambda$.

To summarize, we have demonstrated that vector interactions of dark vectors
can indeed be proportional to a single power of the $U(1)^{\prime}$ breaking,
and thus scale with the dark-vector mass as in Eq. (A.4), if SM fermions have
non-universal $U(1)^{\prime}$ charges. This situation is quite generic in
models where SM Yukawa hierarchies are explained by non-anomalous abelian
flavor symmetries, for example simple $U(1)_{F}$ Froggatt-Nielsen models [51],
see e.g. Refs. [7] for examples of such models without extra heavy fermions to
cancel anomalies. It is well-known how to build UV completions for such models
[75, 76], and below in Section A.4 we will present an illustrative example.

###  A.3 Explicit UV Model for Quadratic scaling

We first construct an explicit renormalizable model for the scaling of vector
interactions in Eq. (2.2) quadratic in the dark $U(1)^{\prime}$ breaking
scale. We restrict the discussion for simplicity to the down-quark sector with
two generations. The field content is summarized in Table 5, and is clearly
anomaly-free.

| $\boldsymbol{Q_{i}}$ | $\boldsymbol{d_{{\mathrm{R}}i}}$ | $\boldsymbol{H}$ | $\boldsymbol{S}$ | $\boldsymbol{\psi_{{\mathrm{L}}i}}$ | $\boldsymbol{\psi_{{\mathrm{R}}i}}$
---|---|---|---|---|---|---
$SU(2)_{\mathrm{L}}$ | $2$ | $1$ | $2$ | $1$ | $1$ | $1$
$U(1)_{Y}$ | $1/6$ | $-1/3$ | $1/2$ | $0$ | $-1/3$ | $-1/3$
$U(1)^{\prime}$ | $0$ | $0$ | $0$ | $1$ | $-1$ | $-1$
Table 5:  Field content of a renormalizable model featuring quadratic scaling.
We restrict the discussion to the down-quark sector with two generations for
SM quarks and heavy vector-like fermions $\psi_{{\mathrm{L}}i}$,
$\psi_{{\mathrm{R}}i}$, with $i=1,2$ carrying $U(1)^{\prime}$ charges in
addition to the scalar $S$.

The Lagrangian reads

$\displaystyle\mathscr{L}=\mathscr{L}_{\text{kinetic}}+\mathscr{L}_{\text{Yukawa}}+\mathscr{L}_{\text{scalar}}+\mathscr{L}_{\text{int-}V^{\prime}}\,,$
(A.10)

with standard kinetic terms for all fields and

$\displaystyle\mathscr{L}_{\text{Yukawa}}$
$\displaystyle=-Y^{d}_{ij}\overline{Q}_{i}Hd_{{\mathrm{R}}j}-m_{\psi}\overline{\psi}_{{\mathrm{L}}i}\psi_{{\mathrm{R}}i}-\alpha_{ij}\overline{\psi}_{{\mathrm{L}}i}d_{{\mathrm{R}}j}S^{\dagger}+{\mathrm{h}.c.}$
(A.11) $\displaystyle\mathscr{L}_{\text{int-}V^{\prime}}$
$\displaystyle=-g^{\prime}V^{\prime}_{\mu}\left(\overline{\psi}_{{\mathrm{L}}i}\gamma^{\mu}\psi_{{\mathrm{L}}i}+\overline{\psi}_{{\mathrm{R}}i}\gamma^{\mu}\psi_{{\mathrm{R}}i}\right)\,,$
(A.12) $\displaystyle\mathscr{L}_{\text{scalar}}$
$\displaystyle=m_{H}^{2}|H|^{2}+m_{S}^{2}|S|^{2}-\lambda_{H}|H|^{4}-\lambda_{S}|S|^{4}-\lambda_{HS}|H|^{2}|S|^{2}\,.$
(A.13)

For a suitable choice of parameters, the last part in
$\mathscr{L}_{\text{scalar}}$ gives a vacuum expectation value to $S$,
$\langle S\rangle=v^{\prime}/\sqrt{2}$, which sets the mass of the dark vector
boson to

$m_{V^{\prime}}=v^{\prime}g^{\prime}\,,$ (A.14)

and induces a mixing between chiral quarks, $d_{{\mathrm{R}}}$, and vector-
like fermions, $\psi$, from the mixing term in $\mathscr{L}_{\text{Yukawa}}$.
In the limit of $m_{\psi}\gg v^{\prime}\gg v$ we can integrate out the vector-
like fermions using their equations of motion neglecting their kinetic terms

$\displaystyle\psi_{{\mathrm{R}}i}$
$\displaystyle=-\frac{\alpha_{ij}}{m_{\psi}}d_{{\mathrm{R}}j}S^{\dagger}\,,$
$\displaystyle\psi_{{\mathrm{L}}i}$ $\displaystyle=0\,.$ (A.15)

Plugging this back into kinetic terms and $\mathscr{L}_{\text{int}}$ lead to
the EFT

$\displaystyle\mathscr{L}_{\text{int}}$
$\displaystyle\supset-g^{\prime}V^{\prime}_{\mu}\frac{S^{\dagger}S}{m_{\psi}^{2}}C_{ij}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)+\frac{S^{\dagger}S}{m_{\psi}^{2}}C_{ij}\left(i\overline{d}_{{\mathrm{R}}i}\not{\partial}d_{{\mathrm{R}}j}\right)+\frac{Si\partial_{\mu}S^{\dagger}}{m_{\psi}^{2}}C_{ij}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)\,,$
(A.16)

where $C_{ij}=(\alpha^{\dagger}\alpha)_{ij}$. Next we integrate out the radial
mode by substituting $S$ with the Goldstone parametrization in Eq. (A.1) and
use the definition of the dark-vector mass to find

$\displaystyle\mathscr{L}_{\text{int}}$
$\displaystyle\supset-V^{\prime}_{\mu}\frac{m_{V^{\prime}}v^{\prime}}{2m_{\psi}^{2}}C_{ij}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)+\frac{(v^{\prime})^{2}}{2m_{\psi}^{2}}C_{ij}\left(i\overline{d}_{{\mathrm{R}}i}\not{\partial}d_{{\mathrm{R}}j}\right)+\partial_{\mu}G\frac{v^{\prime}}{2m_{\psi}^{2}}C_{ij}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)\,,$
(A.17)

recovering the gauge-invariant555 In our conventions $V^{\prime}_{\mu}\to
V^{\prime}_{\mu}+\partial_{\mu}\beta/g^{\prime}$, $S\to\exp(i\beta)S$, $G\to
G+\beta v^{\prime}$, $\psi\to\exp(-i\beta)\psi$. combination
$V^{\prime}_{\mu}-\partial_{\mu}G/m_{V}^{\prime}$. Without loss of generality
we can assume that $Y^{d}_{ij}$ is diagonal, so that we are already in the
mass basis. Nevertheless, we do need to re-diagonalize the kinetic terms due
to the second term in Eq. (A.17) induced in the EFT. In the limit of
$v^{\prime}\ll m_{\psi}$ this is readily achieved by the rescaling
$d_{{\mathrm{R}}i}\to
d_{{\mathrm{R}}i}-(v^{\prime})^{2}/(4m_{\psi}^{2})C_{ij}d_{{\mathrm{R}}j}$.
This leads to additional small corrections of ${\cal O}(1/m_{\psi}^{4})$ to
the final dark-vector couplings, which can be neglected, such that the leading
couplings from the first term in Eq. (A.17) remain

$\displaystyle\mathscr{L}_{\text{quadratic}}=-\frac{m_{V^{\prime}}v^{\prime}}{2m_{\psi}^{2}}C_{ij}V^{\prime}_{\mu}\left(\overline{d}_{{\mathrm{R}}i}\gamma^{\mu}d_{{\mathrm{R}}j}\right)\,.$
(A.18)

These couplings are indeed quadratic in $v^{\prime}$ and are in general flavor
violating, $C_{i\neq j}\neq 0$. This matches to the EFT term in Eq. (A.3) upon
identifying $C_{ij}/m_{\psi}^{2}=-2c_{ij}/\Lambda^{2}$.

###  A.4 Explicit UV Model for Linear scaling

We now construct an explicit renormalizable model for the minimal scaling of
vector interactions in Eq. (2.2) proportional to a single power of the dark-
vector mass. These types of models are motivated by scenarios addressing the
SM flavor puzzle with non-anomalous abelian horizontal symmetries, see e.g.
Ref. [7]. We restrict the discussion for simplicity to the down-quark sector
with two generations. The field content is summarized in Table 6, and is not
anomaly-free. However, we can always introduce further suitably charged chiral
fermions in the right-handed up- and charged-lepton sector in order to cancel
color and electromagnetic anomalies, respectively. Note that
$\psi_{{\mathrm{R}}}$ and $d_{{\mathrm{R}}2}$ carry the same quantum numbers.

| $\boldsymbol{Q_{i}}$ | $\boldsymbol{d_{{\mathrm{R}}1}}$ | $\boldsymbol{d_{{\mathrm{R}}2}}$ | $\boldsymbol{H}$ | $\boldsymbol{S}$ | $\boldsymbol{\psi_{{\mathrm{L}}}}$ | $\boldsymbol{\psi_{{\mathrm{R}}}}$
---|---|---|---|---|---|---|---
$SU(2)_{\mathrm{L}}$ | $2$ | $1$ | $1$ | $2$ | $1$ | $1$ | $1$
$U(1)_{Y}$ | $1/6$ | $-1/3$ | $-1/3$ | $1/2$ | $0$ | $-1/3$ | $-1/3$
$U(1)^{\prime}$ | $0$ | $1$ | $0$ | $0$ | $1$ | $0$ | $0$
Table 6:  Field content of a renormalizable model featuring linear scaling. We
restrict the discussion to the down-quark sector with two generations for SM
quarks and one family of heavy vector-like fermions $\psi_{L}$, $\psi_{R}$
uncharged under $U(1)^{\prime}$.

The Lagrangian reads

$\displaystyle\mathscr{L}$
$\displaystyle=\mathscr{L}_{\text{kinetic}}+\mathscr{L}_{\text{Yukawa}}+\mathscr{L}_{\text{scalar}}+\mathscr{L}_{\text{int-}V^{\prime}}\,,$
(A.19)

with standard kinetic terms for all fields and

$\displaystyle\mathscr{L}_{\text{Yukawa}}$
$\displaystyle=-y_{i}\overline{Q}_{i}Hd_{{\mathrm{R}}2}-z_{i}\overline{Q}_{i}H\psi_{{\mathrm{R}}}-m_{\psi}\overline{\psi}_{L}\psi_{R}-\alpha\overline{\psi}_{L}d_{{\mathrm{R}}1}S^{\dagger}+{\mathrm{h}.c.}$
(A.20) $\displaystyle\mathscr{L}_{\text{int-}V^{\prime}}$
$\displaystyle=g^{\prime}V^{\prime}_{\mu}\overline{d}_{{\mathrm{R}}1}\gamma^{\mu}d_{{\mathrm{R}}1}\,,$
(A.21) $\displaystyle\mathscr{L}_{\text{scalar}}$
$\displaystyle=m_{H}^{2}|H|^{2}+m_{S}^{2}|S|^{2}-\lambda_{H}|H|^{4}-\lambda_{S}|S|^{4}-\lambda_{HS}|H|^{2}|S|^{2}\,,$
(A.22)

where we have simply defined $\psi_{{\mathrm{R}}}$ to be that field having a
mass term with $\psi_{{\mathrm{L}}}$. This already gives Eq. (A.4) and the
first term in Eq. (A.5) from the EFT discussion, so it only remains to show
that integrating out $\psi_{{\mathrm{L}}},\psi_{{\mathrm{R}}}$ induces the
second term in Eq. (A.5). The equations of motion for the heavy fermions read,
neglecting kinetic terms

$\displaystyle\overline{\psi}_{{\mathrm{L}}}$
$\displaystyle=-\frac{z_{i}}{m_{\psi}}\overline{Q}_{i}H\,,$
$\displaystyle\psi_{{\mathrm{R}}}$
$\displaystyle=-\frac{\alpha}{m_{\psi}}d_{{\mathrm{R}}1}S^{\dagger}\,,$ (A.23)

and, therefore, the resulting EFT Lagrangian term is

$\displaystyle\mathscr{L}_{\text{linear}}\supset\frac{z_{i}\alpha}{m_{\psi}}\overline{Q}_{i}Hd_{{\mathrm{R}}1}S^{\dagger}\,.$
(A.24)

This indeed reproduces Eq. (A.5) with the identification of the UV scale as
$\Lambda=-m_{\psi}/\alpha$. The remaining calculation follows the EFT
discussion, which shows that in these type of UV models the flavor violating
couplings to $V^{\prime}$ scale indeed linearly with $m_{V^{\prime}}/\Lambda$.

##  Appendix B Renormalization Group Equations

In this appendix we collect our results for the renormalization group
equations relevant for the $SU(2)_{L}\times U(1)_{Y}$ interactions of the LDV
with SM quarks as discussed in Section 5. Since in the current work we focused
on the case of the UV universal scenario, in which flavor-violation originates
only from the SM CKM matrix, we present here only the one-loop RGEs
proportional to Yukawa couplings. However, in what follows the matrices
$C^{{\mathrm{D}}}_{u}$, $C^{{\mathrm{D}}}_{d}$ $C^{{\mathrm{V}}}_{Q}$,
$C^{{\mathrm{V}}}_{u}$, and $C^{{\mathrm{D}}}_{d}$ are generic matrices in
flavor space, i.e., we have not assumed any alignment with the SM Yukawas. The
relevant terms in the Lagrangian are the SM Yukawa interaction, the dipole,
and the vector interactions with the LDV. They respectively read:

$\displaystyle\mathscr{L}_{\text{Yukawa}}$
$\displaystyle=-\overline{Q}\,Y_{u}\widetilde{H}u_{{\mathrm{R}}}-\overline{Q}\,Y_{d}Hd_{{\mathrm{R}}}+\mathrm{h.c.}\,,$
(B.1) $\displaystyle\mathscr{L}_{\text{Dipole}}$
$\displaystyle=\frac{1}{\Lambda^{2}}V^{\prime}_{\mu\nu}\left(\overline{Q}C^{{\mathrm{D}}}_{u}\sigma^{\mu\nu}\widetilde{H}u_{{\mathrm{R}}}+\overline{Q}C^{{\mathrm{D}}}_{d}\sigma^{\mu\nu}Hd_{{\mathrm{R}}}+\mathrm{h.c.}\right)\,,$
(B.2) $\displaystyle\mathscr{L}_{\text{Vector}}$
$\displaystyle=V^{\prime}_{\mu}\left(\overline{Q}C_{Q}^{V}\gamma^{\mu}Q+\overline{u}_{{\mathrm{R}}}C_{u}^{V}\gamma^{\mu}u_{{\mathrm{R}}}+\overline{d}_{{\mathrm{R}}}C_{d}^{V}\gamma^{\mu}d_{{\mathrm{R}}}\right)\,.$
(B.3)

The one-loop RGEs for Yukawa running read [77]

$\begin{split}16\pi^{2}\frac{\mathrm{d}Y_{u}}{\mathrm{d}\ln\mu}&=\frac{3}{2}\left(Y_{u}Y_{u}^{\dagger}Y_{u}-Y_{d}Y_{d}^{\dagger}Y_{u}\right)+n_{c}\text{Tr}\left[Y_{u}Y_{u}^{\dagger}+Y_{d}Y_{d}^{\dagger}\right]Y_{u}\,,\\\
16\pi^{2}\frac{\mathrm{d}Y_{d}}{\mathrm{d}\ln\mu}&=\frac{3}{2}\left(Y_{d}Y_{d}^{\dagger}Y_{d}-Y_{u}Y_{u}^{\dagger}Y_{d}\right)+n_{c}\text{Tr}\left[Y_{u}Y_{u}^{\dagger}+Y_{d}Y_{d}^{\dagger}\right]Y_{d}\,,\end{split}$
(B.4)

with $n_{c}=3$ denoting the number of colors. The one-loop running of the
Yukawas is relevant for the dipole analysis because the RG-evolved Yukawas
contribute to the flavor-violating couplings upon rotation to the quark mass-
eigenstates at the EW scale [78].

For the one-loop RGE of the dipole couplings proportional to the SM Yukawas we
find

$\begin{split}16\pi^{2}\frac{\mathrm{d}C^{{\mathrm{D}}}_{u}}{\mathrm{d}\ln\mu}&=\frac{5}{2}Y_{u}Y_{u}^{\dagger}C^{{\mathrm{D}}}_{u}-\frac{3}{2}Y_{d}Y_{d}^{\dagger}C^{{\mathrm{D}}}_{u}-C^{{\mathrm{D}}}_{d}Y_{d}^{\dagger}Y_{u}+2C^{{\mathrm{D}}}_{u}Y_{u}^{\dagger}Y_{u}\\\
&\quad+n_{c}\mathrm{Tr}\left[Y_{u}Y^{\dagger}_{u}+Y_{d}Y^{\dagger}_{d}\right]C^{{\mathrm{D}}}_{u}\,,\\\
16\pi^{2}\frac{\mathrm{d}C^{{\mathrm{D}}}_{d}}{\mathrm{d}\ln\mu}&=\frac{5}{2}Y_{d}Y_{d}^{\dagger}C^{{\mathrm{D}}}_{d}-\frac{3}{2}Y_{u}Y_{u}^{\dagger}C^{{\mathrm{D}}}_{d}-C^{{\mathrm{D}}}_{u}Y_{u}^{\dagger}Y_{d}+2C^{{\mathrm{D}}}_{d}Y_{d}^{\dagger}Y_{d}\\\
&\quad+n_{c}\mathrm{Tr}\left[Y_{u}Y^{\dagger}_{u}+Y_{d}Y^{\dagger}_{d}\right]C^{{\mathrm{D}}}_{d}\,.\end{split}$
(B.5)

For the one-loop RGE of the vector couplings proportional to the SM Yukawas we
find

$\begin{split}16\pi^{2}\frac{\mathrm{d}C^{{\mathrm{V}}}_{Q}}{\mathrm{d}\ln\mu}&=-Y_{u}C^{{\mathrm{V}}}_{u}Y_{u}^{\dagger}-Y_{d}C^{{\mathrm{V}}}_{d}Y_{d}^{\dagger}+\frac{1}{2}\left(Y_{u}Y_{u}^{\dagger}+Y_{d}Y_{d}^{\dagger}\right)C^{{\mathrm{V}}}_{Q}+\frac{1}{2}C^{{\mathrm{V}}}_{Q}\left(Y_{u}Y_{u}^{\dagger}+Y_{d}Y_{d}^{\dagger}\right)\,,\\\
16\pi^{2}\frac{\mathrm{d}C^{{\mathrm{V}}}_{u}}{\mathrm{d}\ln\mu}&=-2Y_{u}^{\dagger}C^{{\mathrm{V}}}_{Q}Y_{u}+Y_{u}^{\dagger}Y_{u}C^{{\mathrm{V}}}_{u}+C^{{\mathrm{V}}}_{u}Y_{u}^{\dagger}Y_{u}\,,\\\
16\pi^{2}\frac{\mathrm{d}C^{{\mathrm{V}}}_{d}}{\mathrm{d}\ln\mu}&=-2Y_{d}^{\dagger}C^{{\mathrm{V}}}_{Q}Y_{d}+Y_{d}^{\dagger}Y_{d}C^{{\mathrm{V}}}_{d}+C^{{\mathrm{V}}}_{d}Y_{d}^{\dagger}Y_{d}\,.\end{split}$
(B.6)

##  Appendix C Recast of Experimental Limits

Figure 7: Upper $95\,\%$ CL${}_{\text{s}}$ limits on the two-body branching
ratios $B\rightarrow K/K^{*}/\pi+V^{\prime}$, as a function of the missing
mass $m_{\mathrm{m}iss}=\sqrt{q^{2}}$, obtained by recasting the experimental
three-body searches at BaBar [35, 36], see text for details.

Experimental collaborations often provide only limits on the $P\rightarrow
P^{\prime}+\mathrm{invisible}$ branching ratios in terms of the three body
decay $P\rightarrow P^{\prime}\nu\overline{\nu}$, as a function of the squared
invariant mass of the di-neutrino system $q^{2}$. In order to get the
experimental limits on the two body decays $P\rightarrow P^{\prime}V$, we use
the event count $n_{B}$ per $q^{2}$-bin information, if provided by the
experimental collaborations. Only the BaBar experiment [35, 36] provides all
information needed to perform a recast for two-body decays $B\rightarrow
K/K^{*}/\pi+V$. For $B\to K^{(*)}$ and dark-vector masses
$m_{V^{\prime}}<3\,{\mathrm{G}eV}$, we use the sophisticated recast of Ref.
[57], otherwise we estimate upper limits on the Wilson coefficients in terms
of the CL${}_{\text{s}}$ method as explained below.

For a given Wilson coefficient $C$, the number of signal events $s$ in a
$q^{2}$-bin $i$ is given as

$\displaystyle s=\mathrm{BR}^{i}_{P\rightarrow P^{\prime}}(C)\times
N_{\mathrm{tot}}\times\epsilon_{i}\,,$ (C.1)

where $N_{\mathrm{tot}}$ is the total number of $P$ mesons and $\epsilon_{i}$
the efficiency associated to bin $i$. Further, $\mathrm{BR}^{i}_{P\rightarrow
P^{\prime}}(C)$ denotes the branching ratio of $P\rightarrow P^{\prime}$
within the $q^{2}$-bin $i$. The $s+b$ likelihood is then given as a Poisson
distribution in the number of signal plus background events. The efficiency
$\epsilon_{i}$ and total number of $P$ mesons $N_{\mathrm{tot}}$ are included
as global observables associated to auxiliary measurements. The uncertainty on
the signal, assumed to be Gaussian, is given by the NP theoretical prediction
and is dominated by the form-factor uncertainty. The systematic uncertainty on
the background is implemented as a Gaussian distribution. With this in mind,
we denote the likelihood as $\mathcal{L}(x|C,\nu)$ with $x$ being the outcome,
i.e., the observed data, $C$ the parameter of interest, i.e., the Wilson
coefficient, and $\nu$ the nuisance parameters. As a test statistics $t_{C}$,
we choose a one-sided profile likelihood. Note that the parameter of interest
is actually $|C|^{2}$ since the branching ratio only depends on $|C|^{2}$ as
we only consider one coupling at a time. The $p$-value $p_{C}$ of the $s+b$
hypothesis for a given value of the Wilson coefficient $C$ is then given by

$\displaystyle
p_{C}=\int_{t_{C}^{\mathrm{obs}}}^{\infty}f(t_{C}|C,\hat{\hat{\nu}}(C))\,\mathrm{d}t_{C}\,,$
(C.2)

where $t_{C}^{\mathrm{obs}}$ denotes the value of the test statistics for the
observed data, $f$ denotes the pdf of the test statistics $t_{C}$, and
$\hat{\hat{\nu}}(C)$ are the values of the nuisance parameter that maximise
the likelihood for a given $C$. The $\alpha\,\%$ CL${}_{\text{s}}$ limit on
the Wilson coefficient is then given by the value of $C$ such that

$\displaystyle\frac{p_{C}}{p_{0}}=1-\frac{\alpha}{100}\,,$ (C.3)

where $p_{0}$ denotes the $p$-value of the background only hypothesis. In
order to evaluate Eq. (C.2), one needs the pdf $f$ of the test statistics
$t_{C}$ for which we use the ROOT toolkit RooStats in order to sample the
distribution by means of a Monte Carlo method.

Taking the $\mathrm{BR}(P\rightarrow P^{\prime}V)$ as a parameter of interest
instead of the Wilson coefficient $C$, we can determine a model independent
limit $\mathrm{BR}_{\mathrm{exp}}(P\rightarrow P^{\prime}V)$ on the two body
branching ratios, see Figure 7.

##  Appendix D Limits in the ${\mathrm{L}}/{\mathrm{R}}$ Basis

In this appendix we present bounds on the couplings in the
${\mathrm{L}}/{\mathrm{R}}$ basis
$\\{\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}$,
$\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{ij}$,
$\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ij}$,
$\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ij}\\}$, which are obtained from the
limits in the ${\mathrm{V}}/{\mathrm{A}}$ basis (discussed in Section 3 and
4)) using Eq. (2.4). As the decay rates are symmetric with respect to
$\mathbb{C}^{{\mathrm{L}}}\leftrightarrow\mathbb{C}^{{\mathrm{R}}}$ the bounds
on both couplings are the same.

###  D.1 Quark Dipole Interactions

Figure 8:  Upper limits on quark-flavor violating dipole couplings
$\Lambda/|\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}}_{ij}|$, for $s\to d,b\to
s,b\to d$ and $c\to u$ transitions. Bounds on
$\Lambda/|\mathbb{C}^{{\mathrm{D}}{\mathrm{R}}}_{ij}|$ are identical.

###  D.2 Quark Vector Interactions

Figure 9:  Upper limits on quark-flavor violating vector couplings
$\Lambda/|\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}}_{ij}|$, for $s\to d,b\to
s,b\to d$ and $c\to u$ transitions. Bounds on
$\Lambda/|\mathbb{C}^{{\mathrm{V}}{\mathrm{R}}}_{ij}|$ are identical.

Figure 10: Upper panel: Lower limits on the dipole (left panel) and vector
(right panel) couplings for $\tau\to e$ transitions
$\Lambda/|\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}({\mathrm{R}})}_{\tau e}|$,
$\Lambda/|\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}({\mathrm{R}})}_{\tau e}|$ from
Belle II [38]. Lower panel: same for $\tau\to\mu$ transitions
$\Lambda/|\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}({\mathrm{R}})}_{\tau\mu}|$,
$\Lambda/|\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}({\mathrm{R}})}_{\tau\mu}|$.

##  Appendix E Two-body decays to Light Dark Vectors

Figure 11: Two-body decays
$\\{P,B,\ell\\}\to\\{P^{\prime},\mathcal{V},B^{\prime},\ell^{\prime}\\}+V^{\prime}$.
The blob represents the non-perturbative QCD effects for the hadronic decays.

In this appendix we present the full expressions for the two-body decays to a
LDV that enter our analysis, namely

  * •

$P\to P^{\prime}+V^{\prime}$: pseudoscalar meson to pseudoscalar meson and
LDV,

  * •

$P\to\mathcal{V}+V^{\prime}$: pseudoscalar meson to vector meson and LDV,

  * •

$B\to B^{\prime}+V^{\prime}$: baryon to baryon and LDV,

  * •

$\ell\to\ell^{\prime}V^{\prime}$: lepton to lepton and LDV.

For the hadronic processes illustrative Feynman diagrams are shown in Figure
1, while throughout this appendix we define the two-body kinematics for all
decays as in Figure 11, namely as

$\text{SM}(p)\to\text{SM}^{\prime}(p^{\prime})+V^{\prime}(q)$ (E.1)

with $q=p-p^{\prime}$ and $q^{2}=(p-p^{\prime})^{2}=m_{V^{\prime}}^{2}$. In
the next subsection we collect the parametrization of all the relevant form
factors for the hadronic processes considered, and in the subsequent
subsections we present the expressions for the rates. The numerical values for
the form-factors are always taken from the most recent work referenced.

###  E.1 Form Factors

#### $\boldsymbol{P\to P^{\prime}+V^{\prime}}$

For these decays the hadronic matrix elements for the vector and axial-vector
currents read [66]

$\begin{split}\langle
P^{\prime}(p^{\prime})|\overline{q}^{\prime}\gamma^{\mu}q|P(p)\rangle&=(p+p^{\prime})^{\mu}f_{+}^{PP^{\prime}}(q^{2})+(p-p^{\prime})^{\mu}f_{-}^{PP^{\prime}}(q^{2})\,,\\\
\langle
P^{\prime}(p^{\prime})|\overline{q}^{\prime}\gamma_{5}\gamma^{\mu}q|P(p)\rangle&=0\,.\end{split}$
(E.2)

The corresponding matrix elements for tensor and pseudo-tensor currents read
[66]

$\begin{split}\langle
P^{\prime}(p^{\prime})|\overline{q}^{\prime}\sigma_{\mu\nu}q|P(p)\rangle&=\frac{2}{m_{P}+m_{P^{\prime}}}\left(p^{\prime}_{\mu}p_{\nu}-p^{\prime}_{\nu}p_{\mu}\right)f_{\mathrm{T}}^{PP^{\prime}}(q^{2})\,,\\\
\langle
P^{\prime}(p^{\prime})|\overline{q}^{\prime}\sigma^{\mu\nu}\gamma_{5}q|P(p)\rangle&=\frac{2i}{m_{P}+m_{P^{\prime}}}\epsilon^{\mu\nu\rho\sigma}p^{\prime}_{\rho}p_{\sigma}f_{\mathrm{T}}^{PP^{\prime}}(q^{2})\,,\end{split}$
(E.3)

where here and throughout we use the $\epsilon_{0123}=-\epsilon^{0123}=+1$
convention for the Levi-Civita tensor.

#### $\boldsymbol{P\to\mathcal{V}+V^{\prime}}$

For the pseudoscalar decays to two vectors with $\mathcal{V}$ denoting the
vector-meson, the hadronic matrix element for the vector and axial-vector
currents are parametrized as [66]

$\displaystyle\langle\mathcal{V}(p^{\prime},\lambda)|\overline{q}^{\prime}\gamma^{\mu}\left(1\mp\gamma_{5}\right)q|P(p)\rangle=P_{1}^{\mu}\mathcal{V}_{1}(q^{2})\pm
P_{2}^{\mu}\mathcal{V}_{2}(q^{2})\pm P_{3}^{\mu}\mathcal{V}_{3}(q^{2})\pm
P_{P}^{\mu}\mathcal{V}_{P}(q^{2})\,,$ (E.4)

where $\lambda$ denotes the polarization of $\mathcal{V}$. The kinematic
functions read

$\displaystyle P_{P}^{\mu}$
$\displaystyle=i(\epsilon^{*}\\!\cdot\\!q)q^{\mu}\,,$ $\displaystyle
P_{1}^{\mu}$
$\displaystyle=2\epsilon^{\mu}_{~{}~{}\alpha\beta\gamma}\epsilon^{*\alpha}p^{\prime\beta}q^{\gamma}\,,$
$\displaystyle P_{2}^{\mu}$
$\displaystyle=i\left[\left(m_{P}^{2}-m_{{\mathcal{V}}}^{2}\right)\epsilon^{*\mu}-(\epsilon^{*}\\!\cdot\\!q)\left(p^{\prime}+p\right)^{\mu}\right]\,,$
$\displaystyle P_{3}^{\mu}$
$\displaystyle=i(\epsilon^{*}\\!\cdot\\!q)\big{[}q^{\mu}-\frac{q^{2}}{m_{P}^{2}-m_{\mathcal{V}}^{2}}(p^{\prime}+p)^{\mu}\big{]}\,,$
(E.5)

where $\epsilon^{*}_{\mu}=\epsilon_{\mu}^{*}(p^{\prime},\lambda)$ denotes the
polarization vector of the outgoing $\mathcal{V}$. The scalar form factors can
be further parametrized as

$\displaystyle\begin{split}\mathcal{V}_{P}(q^{2})&=\frac{-2m_{\mathcal{V}}}{q^{2}}A_{0}(m_{V^{\prime}}^{2})\,,\qquad\mathcal{V}_{1}(q^{2})=\frac{-V(q^{2})}{m_{P}+m_{\mathcal{V}}}\,,\qquad\mathcal{V}_{2}(q^{2})=\frac{-A_{1}(q^{2})}{m_{P}-m_{\mathcal{V}}}\,,\\\
\mathcal{V}_{3}(q^{2})&=\frac{m_{P}+m_{\mathcal{V}}}{q^{2}}A_{1}(q^{2})-\frac{m_{P}-m_{\mathcal{V}}}{q^{2}}A_{2}(q^{2})\equiv\frac{2m_{\mathcal{V}}}{q^{2}}A_{3}(q^{2})\,,\end{split}$
(E.6)

with $A_{3}(0)=A_{0}(0)$, which ensures finite matrix elements at $q^{2}=0$,
i.e., for massless LDV.

The corresponding matrix elements for tensor and pseudo-tensor currents read
[66]

$\begin{split}\langle\mathcal{V}(p^{\prime},\lambda)|\overline{q}^{\prime}\sigma^{\mu\nu}q|P(p)\rangle&=-i\epsilon^{*}_{\alpha}T^{\alpha\mu\nu}(q^{2})\,,\\\
\langle\mathcal{V}(p^{\prime},\lambda)|\overline{q}^{\prime}\sigma_{\mu\nu}\gamma_{5}q|P(p)\rangle&=\frac{1}{2}\epsilon^{*}_{\alpha}\epsilon_{\mu\nu\rho\sigma}T^{\alpha\rho\sigma}(q^{2})\,,\end{split}$
(E.7)

where

$\begin{split}T^{\alpha\mu\nu}(q^{2})&=\epsilon^{\alpha\mu\nu\beta}\left[\left(p_{\beta}+p^{\prime}_{\beta}-q_{\beta}\frac{m_{P}^{2}-m_{\mathcal{V}}^{2}}{q^{2}}\right)T_{1}^{P\mathcal{V}}(q^{2})+q_{\beta}\frac{m_{P}^{2}-m_{\mathcal{V}}^{2}}{q^{2}}T_{2}^{P\mathcal{V}}(q^{2})\right]\\\
&+\frac{2p^{\alpha}}{q^{2}}\epsilon^{\mu\nu\beta\gamma}p_{\beta}p^{\prime}_{\gamma}\left(T_{2}^{P\mathcal{V}}(q^{2})-T_{1}^{P\mathcal{V}}(q^{2})+\frac{q^{2}}{m_{P}^{2}-m_{\mathcal{V}}^{2}}T_{3}^{P\mathcal{V}}(q^{2})\right)\,.\end{split}$
(E.8)

For vanishing momentum transfer $q^{2}=0$, i.e., massless LDV, the scalar
form-factors satisfy

$\displaystyle T_{1}^{P\mathcal{V}}(0)$
$\displaystyle=T_{2}^{P\mathcal{V}}(0)\equiv T\,,$ (E.9)

while the contribution proportional to $T_{3}(0)$ vanishes.

#### $\boldsymbol{B\to B^{\prime}+V^{\prime}}$

For the baryon decays the matrix elements for vector and axial-vector currents
are parametrized by [67, 69, 72]

$\begin{split}\langle
B^{\prime}(p^{\prime})|\overline{q}^{\prime}\gamma_{\mu}q|B(p)\rangle&=\overline{u}_{B^{\prime}}(p^{\prime})\left(f_{1}(q^{2})\gamma_{\mu}-i\frac{f_{2}(q^{2})}{m_{B}}\sigma_{\mu\nu}q^{\nu}+\frac{f_{3}(q^{2})}{m_{B}}q_{\mu}\right){u}_{B}(p)\,,\\\
\langle
B^{\prime}(p^{\prime})|\overline{q}^{\prime}\gamma_{\mu}\gamma_{5}q|B(p)\rangle&=\overline{u}_{B^{\prime}}(p^{\prime})\left(g_{1}(q^{2})\gamma_{\mu}-i\frac{g_{2}(q^{2})}{m_{B}}\sigma_{\mu\nu}q^{\nu}+\frac{g_{3}(q^{2})}{m_{B}}q_{\mu}\right)\gamma_{5}{u}_{B}(p)\,,\end{split}$
(E.10)

with $u_{B}(p)$ and $u_{B^{\prime}}(p^{\prime})$ the spinor functions for $B$
and $B^{\prime}$ respectively. For $\Lambda$ decays the values of the form
factors are taken from [67, 69, 72], while for hyperon decays they are taken
from [62, 63, 64].

The corresponding matrix elements for tensor and pseudo-tensor currents have
the form [32, 79]

$\begin{split}\langle
B^{\prime}(p^{\prime})|\overline{q}^{\prime}\sigma^{\mu\nu}q|B(p)\rangle&=g_{\mathrm{T}}^{BB^{\prime}}\overline{u}_{B^{\prime}}(p^{\prime})\sigma^{\mu\nu}u_{B}(p)\,,\\\
\langle
B^{\prime}(p^{\prime})|\overline{q}^{\prime}\sigma_{\mu\nu}\gamma_{5}q|B(p)\rangle&=\frac{i}{2}g_{\mathrm{T}}^{BB^{\prime}}\epsilon_{\mu\nu\alpha\beta}\overline{u}_{B^{\prime}}(p^{\prime})\sigma^{\alpha\beta}u_{B}(p)\,,\end{split}$
(E.11)

which is an approximation valid for $m_{V^{\prime}}^{2}=0$, which we use for
the hyperon decays. For the baryon $\Lambda_{b}\to\Lambda$, $\Lambda_{b}\to
n$, and $\Lambda_{c}\to p$ decays we use the available full parametrization,
given by [67, 72]

$\begin{split}\langle
B^{\prime}(p^{\prime})|\overline{q}^{\prime}i\sigma^{\mu\nu}q_{\nu}q|B(p)\rangle&=\overline{u}_{B^{\prime}}(p^{\prime})\left(\frac{f_{1}^{{\mathrm{T}V}}(q^{2})}{m_{B}}\left(\gamma^{\mu}q^{2}-q^{\mu}\not{q}\right)-f_{2}^{{\mathrm{T}V}}(q^{2})i\sigma^{\mu\nu}q_{\nu}\right){u}_{B}(p)\,,\\\
\langle
B^{\prime}(p^{\prime})|\overline{q}^{\prime}i\sigma^{\mu\nu}q_{\nu}\gamma_{5}q|B(p)\rangle&=\overline{u}_{B^{\prime}}(p^{\prime})\left(\frac{f_{1}^{{\mathrm{T}A}}(q^{2})}{m_{B}}\left(\gamma^{\mu}q^{2}-q^{\mu}\not{q}\right)-f_{2}^{{\mathrm{T}A}}(q^{2})i\sigma^{\mu\nu}q_{\nu}\right)\gamma_{5}{u}_{B}(p)\,.\end{split}$

Having collected all hadronic input used in the analysis we next present the
full expressions for the two-body rates. We show separately the contributions
from dipole and vector interactions with the LDV, c.f. Eqs. (2.1) and (2.2).
For brevity we drop the argument in all form factors since it is always
$q^{2}=m^{2}_{V^{\prime}}$ in two-body decays. To shorten the expression we
also introduce the notations

$\kappa_{x}\equiv
m^{2}_{x}/M^{2}\qquad\text{and}\qquad\lambda_{xy}\equiv(1-\kappa_{x}-\kappa_{y})^{2}-4\kappa_{x}\kappa_{y}\,,$

with $m_{x}$ indicating the mass of the final-state particle $x$ and $M$ the
mass of the decaying particle.

###  E.2 Partial width for $\boldsymbol{P\to P^{\prime}+V^{\prime}}$

The partial width for the decay ${P\to P^{\prime}+V^{\prime}}$ with an
underlying $q\to q^{\prime}$ flavor-changing transition is given respectively
for dipole and vector interaction by

$\displaystyle\Gamma(P\to P^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}$
$\displaystyle=\frac{\kappa_{V^{\prime}}m_{P}^{3}}{4\pi\Lambda^{2}}\frac{\lambda^{3/2}_{P^{\prime}V^{\prime}}}{(1+\sqrt{\kappa_{P^{\prime}}})^{2}}|f_{\mathrm{T}}^{PP^{\prime}}|^{2}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}\,,$
(E.12) $\displaystyle\Gamma(P\to P^{\prime}V^{\prime})\Big{|}_{\mathrm{V}}$
$\displaystyle=\frac{m_{P}^{3}}{16\pi\Lambda^{2}}\lambda^{3/2}_{P^{\prime}V^{\prime}}|f_{+}^{PP^{\prime}}|^{2}|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}\,,$
(E.13)

Note that due to the parity conservation of strong interactions the rate is
independent of the axial couplings $\mathbb{C}^{{\mathrm{V}}5}_{ij}$ and
$\mathbb{C}^{{\mathrm{D}}5}_{ij}$. Therefore, $P\to P^{\prime}+V^{\prime}$
decays are only sensitive to the
$\mathbb{C}^{{\mathrm{V}}({\mathrm{D}})}_{ij}$ couplings. In the
$\\{{\mathrm{L}},{\mathrm{R}}\\}$ basis, these decays are sensitive to both
$\mathbb{C}^{{\mathrm{D}}{\mathrm{L}}({\mathrm{R}})}_{ij},\mathbb{C}^{{\mathrm{V}}{\mathrm{L}}({\mathrm{R}})}_{ij}$
couplings.

In the limit for massless LDV, the leading in $m_{V^{\prime}}$ contributions
to the decay rates read

$\displaystyle\lim_{m_{V^{\prime}}\to 0}\Gamma(P\to
P^{\prime}+V^{\prime})\Big{|}_{\mathrm{D}}$
$\displaystyle=\frac{m^{2}_{V^{\prime}}m_{P}}{4\pi\Lambda^{2}}\left(1-\sqrt{\kappa_{P^{\prime}}}\right)^{3}|f_{\mathrm{T}}^{PP^{\prime}}|^{2}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}\,,$
(E.14) $\displaystyle\lim_{m_{V^{\prime}}\to 0}\Gamma(P\to
P^{\prime}+V^{\prime})\Big{|}_{\mathrm{V}}$
$\displaystyle=\frac{m_{P}^{3}}{16\pi\Lambda^{2}}\left(1-\kappa_{P^{\prime}}\right)^{3}|f_{+}^{PP^{\prime}}|^{2}|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}\,.$
(E.15)

While the rate originating from dipole interactions vanishes in the massless
limit, the contribution of the vector interaction remains constant due to the
linear scaling $m_{V^{\prime}}/\Lambda$ introduced and discussed in Eq. (2.2).

###  E.3 Partial width for $\boldsymbol{P\to{\mathcal{V}}+V^{\prime}}$

The partial width for the decay ${P\to{\mathcal{V}}+V^{\prime}}$ with an
underlying $q\to q^{\prime}$ flavor-changing transition is given respectively
for dipole and vector interaction by

$\displaystyle\begin{split}\Gamma(P\to\mathcal{V}+V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{P}^{3}}{2\pi\Lambda^{2}}\lambda^{1/2}_{\mathcal{V}V^{\prime}}\left(A_{{\mathrm{D}}}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}+A_{{\mathrm{D}}5}|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\right)\,,\\\
\Gamma(P\to\mathcal{V}+V^{\prime})\Big{|}_{\mathrm{V}}&=\frac{m_{P}^{3}\kappa_{V^{\prime}}^{2}}{8\pi\Lambda^{2}}\lambda^{1/2}_{\mathcal{V}V^{\prime}}\left(A_{{\mathrm{V}}}|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}+A_{{\mathrm{V}}5}|\mathbb{C}^{{\mathrm{V}}5}_{q^{\prime}q}|^{2}\right)\,,\\\
\end{split}$ (E.16)

with the coefficients $A_{X}$ given by

$\displaystyle A_{{\mathrm{D}}}$
$\displaystyle=|T^{P\mathcal{V}}_{1}|^{2}\lambda_{\mathcal{V}V^{\prime}}\,,$
(E.17) $\displaystyle A_{{\mathrm{D}}5}$
$\displaystyle=|T^{P\mathcal{V}}_{2}|^{2}\frac{8\kappa_{\mathcal{V}}\left(1-\kappa_{\mathcal{V}}\right)^{2}+\kappa_{V^{\prime}}\left(1+3\kappa_{\mathcal{V}}\right)^{2}-2\kappa_{V^{\prime}}^{2}(1+3\kappa_{\mathcal{V}})+\kappa_{V^{\prime}}^{3}}{8\kappa_{\mathcal{V}}}$
$\displaystyle+|T^{P\mathcal{V}}_{3}|^{2}\lambda^{2}_{\mathcal{V}V^{\prime}}\frac{\kappa_{V^{\prime}}}{8\kappa_{\mathcal{V}}(1-\kappa_{\mathcal{V}})^{2}}-{\mathrm{R}e}(T^{P\mathcal{V}}_{2}T^{P\mathcal{V*}}_{3})\lambda_{\mathcal{V}V^{\prime}}\frac{\kappa_{V^{\prime}}(1+3\kappa_{\mathcal{V}}-\kappa_{V^{\prime}})}{4\kappa_{\mathcal{V}}(1-\kappa_{\mathcal{V}})}\,,$
(E.18) $\displaystyle A_{{\mathrm{V}}}$
$\displaystyle=|V|^{2}\frac{\lambda_{\mathcal{V}V^{\prime}}}{\left(1+\sqrt{\kappa_{\mathcal{V}}}\right)^{2}}\,,$
(E.19) $\displaystyle A_{{\mathrm{V}}5}$
$\displaystyle=|A_{1}|^{2}\frac{\kappa_{V^{\prime}}^{3}-2\kappa_{V^{\prime}}^{2}(1+3\kappa_{\mathcal{V}})+\kappa_{V^{\prime}}(1+3\kappa_{\mathcal{V}})^{2}+8(1-\kappa_{\mathcal{V}})^{2}\kappa_{\mathcal{V}}}{8\kappa_{\mathcal{V}}\left(1-\sqrt{\kappa_{\mathcal{V}}}\right)^{2}}$
$\displaystyle+|A_{3}|^{2}\frac{\lambda_{\mathcal{V}V^{\prime}}^{2}}{2\kappa_{V^{\prime}}\left(1-\kappa_{\mathcal{V}}\right)^{2}}+{\mathrm{R}e}(A_{1}A_{3}^{*})\frac{\sqrt{1+\kappa_{\mathcal{V}}}}{2\sqrt{\kappa_{\mathcal{V}}}\left(1-\kappa_{\mathcal{V}}\right)^{2}}\lambda_{\mathcal{V}V^{\prime}}\left(1-\kappa_{V^{\prime}}+3\kappa_{\mathcal{V}}\right)\,.$
(E.20)

In the limit of a massless LDV, the decay rates reduce to

$\begin{split}\lim_{m_{V^{\prime}}\to
0}\Gamma(P\to\mathcal{V}V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{P}^{3}}{2\pi\Lambda^{2}}\left(1-\kappa_{\mathcal{V}}\right)^{3}|T|^{2}\left(|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}+|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\right)\,,\\\
\lim_{m_{V^{\prime}}\to
0}\Gamma(P\to\mathcal{V}V^{\prime})\Big{|}_{\mathrm{V}}&=\frac{m_{P}^{3}}{16\pi\Lambda^{2}}\left(1-\kappa_{\mathcal{V}}\right)^{3}\left(|A_{3}|^{2}|\mathbb{C}^{{\mathrm{V}}5}_{q^{\prime}q}|^{2}+\frac{2\kappa_{V^{\prime}}|V|^{2}}{\left(\sqrt{\kappa_{\mathcal{V}}}+1\right)^{2}}|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}\right)\,,\end{split}$
(E.21)

which illustrates that the sensitivity to
$\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}$ weakens for very light LDVs.

###  E.4 Partial width for $\boldsymbol{B\to B^{\prime}+V^{\prime}}$

For baryon decays $B\to B^{\prime}+V^{\prime}$ with an underlying $q\to
q^{\prime}$ transition the contribution to the partial width from the dipole
and vector interaction read

$\displaystyle\begin{split}\Gamma(B\to
B^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{B}^{3}}{4\pi\Lambda^{2}}\lambda_{B^{\prime}V^{\prime}}^{1/2}\Big{[}\left(|f_{1}^{{\mathrm{T}V}}|^{2}\hat{A}_{{\mathrm{D}}1}^{-}+|f_{2}^{{\mathrm{T}V}}|^{2}\hat{A}_{{\mathrm{D}}2}^{-}+\hat{A}_{{\mathrm{D}}12}^{-}{\mathrm{R}e}(f_{1}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}V}*})\right)|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}\\\
&\hskip
50.00008pt+\left(|f_{1}^{{\mathrm{T}A}}|^{2}\hat{A}_{{\mathrm{D}}1}^{+}+|f_{2}^{{\mathrm{T}A}}|^{2}\hat{A}_{{\mathrm{D}}2}^{+}+\hat{A}_{{\mathrm{D}}12}^{+}{\mathrm{R}e}(f_{1}^{{\mathrm{T}A}}f_{2}^{{\mathrm{T}A}*})\right)|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\Big{]}\,,\\\
\Gamma(B\to
B^{\prime}V^{\prime})\Big{|}_{\mathrm{V}}&=\frac{m_{B}^{3}}{16\pi\Lambda^{2}}\lambda^{1/2}_{B^{\prime}V^{\prime}}\Big{[}\left(|f_{1}|^{2}\hat{A}_{{\mathrm{V}}1}^{-}+|f_{2}|^{2}\hat{A}_{{\mathrm{V}}2}^{-}+\hat{A}_{{{\mathrm{V}}12}}^{-}{\mathrm{R}e}(f_{1}f_{2}^{*})\right)|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}\\\
&\hskip
60.00009pt+\left(|g_{1}|^{2}\hat{A}_{{\mathrm{V}}1}^{+}+|g_{2}|^{2}\hat{A}_{{\mathrm{V}}2}^{+}+\hat{A}_{{{\mathrm{V}}12}}^{+}{\mathrm{R}e}(g_{1}g_{2}^{*})\right)|\mathbb{C}^{{\mathrm{V}}5}_{q^{\prime}q}|^{2}\Big{]}\,,\end{split}$
(E.22)

with the kinematic coefficients

$\begin{split}\hat{A}_{{\mathrm{D}}1}^{\pm}&=\kappa_{V^{\prime}}\left(\kappa_{B^{\prime}}^{2}+\kappa_{B^{\prime}}\left(\kappa_{V^{\prime}}-2\right)\pm
6\sqrt{\kappa_{B^{\prime}}}\kappa_{V^{\prime}}-2\kappa_{V^{\prime}}^{2}+\kappa_{V^{\prime}}+1\right)\,,\\\
\hat{A}_{{\mathrm{D}}2}^{\pm}&=2\kappa_{B^{\prime}}^{2}-\kappa_{B^{\prime}}\left(\kappa_{V^{\prime}}+4\right)\pm
6\sqrt{\kappa_{B^{\prime}}}\kappa_{V^{\prime}}-\kappa_{V^{\prime}}^{2}-\kappa_{V^{\prime}}+2\,,\\\
\hat{A}_{{\mathrm{D}}12}^{\pm}&=6\kappa_{V^{\prime}}\left(\sqrt{\kappa_{B^{\prime}}}\mp
1\right)\left(1+\kappa_{B^{\prime}}\pm
2\sqrt{\kappa_{B^{\prime}}}-\kappa_{V^{\prime}}\right)\,,\\\
\hat{A}_{{\mathrm{V}}1}^{\pm}&=\left(1+\kappa_{B^{\prime}}\pm
2\sqrt{\kappa_{B^{\prime}}}-\kappa_{V^{\prime}}\right)\left(1+\kappa_{B^{\prime}}\mp
2\sqrt{\kappa_{B^{\prime}}}+2\kappa_{V^{\prime}}\right)\,,\\\
\hat{A}_{{\mathrm{V}}2}^{\pm}&=\kappa_{V^{\prime}}\left(1+\kappa_{B^{\prime}}\pm
2\sqrt{\kappa_{B^{\prime}}}-\kappa_{V^{\prime}}\right)\left(2+2\kappa_{B^{\prime}}\mp
4\sqrt{\kappa_{B^{\prime}}}+\kappa_{V^{\prime}}\right)\,,\\\
\hat{A}_{{\mathrm{V}}12}^{\pm}&=6\kappa_{V^{\prime}}\left(\kappa_{B^{\prime}}\pm
2\sqrt{\kappa_{B^{\prime}}}-\kappa_{V^{\prime}}+1\right)\left(\sqrt{\kappa_{B^{\prime}}}\mp
1\right)\,.\\\ \end{split}$

In the limit of a massless LDV, the rates reduce to

$\displaystyle\begin{split}\lim_{m_{V^{\prime}}\to 0}\Gamma(B\to
B^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{B}^{3}}{2\pi\Lambda^{2}}\left(1-\kappa_{B^{\prime}}\right)^{3}\Big{[}|f_{2}^{{\mathrm{T}V}}|^{2}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}+|f_{2}^{{\mathrm{T}A}}|^{2}|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\Big{]}\,,\\\
\lim_{m_{V^{\prime}}\to 0}\Gamma(B\to
B^{\prime}V^{\prime})\Big{|}_{\mathrm{V}}&=\frac{m_{B}^{3}}{16\pi\Lambda^{2}}\left(1-\kappa_{B^{\prime}}\right)^{3}\Big{(}|f_{1}|^{2}|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}+|g_{1}|^{2}|\mathbb{C}^{{\mathrm{V}}5}_{q^{\prime}q}|^{2}\Big{)}\,.\end{split}$
(E.23)

For hyperon decays we use the form factor parametrization of Eq. (E.11), valid
for $m_{V^{\prime}}=0$. Nonetheless, we consider a massive LDV for the
kinematics for completeness. The decay rate reads

$\displaystyle\begin{split}\Gamma(B\to
B^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{B}^{3}}{4\pi\Lambda^{2}}\lambda^{1/2}_{B^{\prime}V^{\prime}}(g_{\mathrm{T}}^{BB^{\prime}})^{2}\left(\hat{A}^{-}_{{\mathrm{D}}}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}+\hat{A}^{+}_{{\mathrm{D}}}|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\right)\,,\end{split}$
(E.24)

with the kinematic coefficients

$\begin{split}\hat{A}^{\pm}_{{\mathrm{D}}}&=\left(\kappa_{B^{\prime}}\pm
2\sqrt{\kappa_{B^{\prime}}}-\kappa_{V^{\prime}}+1\right)\left(2\kappa_{B^{\prime}}\mp
4\sqrt{\kappa_{B^{\prime}}}+\kappa_{V^{\prime}}+2\right)\,.\end{split}$ (E.25)

In the limit of a massless LDV, the rate reduces to

$\displaystyle\begin{split}\lim_{m_{V^{\prime}}\to 0}\Gamma(B\to
B^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{B}^{3}}{2\pi\Lambda^{2}}\left(1-\kappa_{B^{\prime}}\right)^{3}|g_{\mathrm{T}}^{BB^{\prime}}|^{2}\Big{(}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}+|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\Big{)}\,.\end{split}$
(E.26)

For a fully polarized initial $B$, the differential width read

$\displaystyle\begin{split}\frac{d\Gamma(B\to
B^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{D}}&=\frac{m_{B}^{3}}{8\pi\Lambda^{2}}\lambda_{B^{\prime}V^{\prime}}^{1/2}\Big{[}\\\
&\quad\phantom{+}\
\left(|f_{1}^{{\mathrm{T}V}}|^{2}\hat{A}_{{\mathrm{D}}1}^{-}+|f_{2}^{{\mathrm{T}V}}|^{2}\hat{A}_{{\mathrm{D}}2}^{-}+\hat{A}_{{\mathrm{D}}12}^{-}{\mathrm{R}e}(f_{1}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}V}*})\right)|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}\\\
&\quad+\left(|f_{1}^{{\mathrm{T}A}}|^{2}\hat{A}_{{\mathrm{D}}1}^{+}+|f_{2}^{{\mathrm{T}A}}|^{2}\hat{A}_{{\mathrm{D}}2}^{+}+\hat{A}_{{\mathrm{D}}12}^{+}{\mathrm{R}e}(f_{1}^{{\mathrm{T}A}}f_{2}^{{\mathrm{T}A}*})\right)|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\\\
&\quad-2\lambda^{1/2}_{B^{\prime}V^{\prime}}\cos\theta\left(\hat{B}_{{\mathrm{D}}11}{\mathrm{I}m}(f_{1}^{{\mathrm{T}V}}f_{1}^{{\mathrm{T}A}*})+\hat{B}_{{\mathrm{D}}12}^{-}{\mathrm{I}m}(f_{1}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}A}*})\right.\\\
&\left.\hskip
70.0001pt+\hat{B}_{{\mathrm{D}}22}{\mathrm{I}m}(f_{2}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}A}*})+\hat{B}_{{\mathrm{D}}12}^{+}{\mathrm{I}m}(f_{2}^{{\mathrm{T}V}}f_{1}^{{\mathrm{T}A}*})\right){\mathrm{R}e}({\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{D}}5*}_{q^{\prime}q})\\\
&\quad-2\lambda^{1/2}_{B^{\prime}V^{\prime}}\cos\theta\left(\hat{B}_{{\mathrm{D}}11}{\mathrm{R}e}(f_{1}^{{\mathrm{T}V}}f_{1}^{{\mathrm{T}A}*})+\hat{B}_{{\mathrm{D}}12}^{-}{\mathrm{R}e}(f_{1}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}A}*})\right.\\\
&\left.\hskip
70.0001pt+\hat{B}_{{\mathrm{D}}22}{\mathrm{R}e}(f_{2}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}A}*})+\hat{B}_{{\mathrm{D}}12}^{+}{\mathrm{R}e}(f_{2}^{{\mathrm{T}V}}f_{1}^{{\mathrm{T}A}*})\right){\mathrm{I}m}({\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{D}}5*}_{q^{\prime}q})\Big{]}\,,\\\
\frac{d\Gamma(B\to
B^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{V}}&=\frac{m_{B}^{3}}{32\pi\Lambda^{2}}\lambda^{1/2}_{B^{\prime}V^{\prime}}\Big{[}\\\
&\quad\phantom{+}\
\left(|f_{1}|^{2}\hat{A}_{{\mathrm{V}}1}^{-}+|f_{2}|^{2}\hat{A}_{{\mathrm{V}}2}^{-}+\hat{A}_{{{\mathrm{V}}12}}^{-}{\mathrm{R}e}(f_{1}f_{2}^{*})\right)|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}\\\
&\quad+\left(|g_{1}|^{2}\hat{A}_{{\mathrm{V}}1}^{+}+|g_{2}|^{2}\hat{A}_{{\mathrm{V}}2}^{+}+\hat{A}_{{{\mathrm{V}}12}}^{+}{\mathrm{R}e}(g_{1}g_{2}^{*})\right)|\mathbb{C}^{{\mathrm{V}}5}_{q^{\prime}q}|^{2}\\\
&\quad-2\lambda^{1/2}_{B^{\prime}V^{\prime}}\cos\theta\left(\hat{B}_{{\mathrm{V}}11}{\mathrm{R}e}(f_{1}g_{1}^{*})+\hat{B}_{{\mathrm{V}}12}^{+}{\mathrm{R}e}(f_{2}g_{1}^{*})\right.\\\
&\left.\hskip
70.0001pt+\hat{B}_{{\mathrm{V}}12}^{-}{\mathrm{R}e}(f_{1}g_{2}^{*})+\hat{B}_{{\mathrm{V}}22}{\mathrm{R}e}(f_{2}g_{2}^{*})\right){\mathrm{R}e}({\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{V}}5*}_{q^{\prime}q})\\\
&\quad+2\lambda^{1/2}_{B^{\prime}V^{\prime}}\cos\theta\left(\hat{B}_{{\mathrm{V}}11}{\mathrm{I}m}(f_{1}g_{1}^{*})+\hat{B}_{{\mathrm{V}}12}^{+}{\mathrm{I}m}(f_{2}g_{1}^{*})\right.\\\
&\left.\hskip
70.0001pt+\hat{B}_{{\mathrm{V}}12}^{-}{\mathrm{I}m}(f_{1}g_{2}^{*})+\hat{B}_{{\mathrm{V}}22}{\mathrm{I}m}(f_{2}g_{2}^{*})\right){\mathrm{I}m}({\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{V}}5*}_{q^{\prime}q})\Big{]}\,,\end{split}$
(E.27)

with the kinematic coefficients

$\displaystyle\hat{B}_{{\mathrm{D}}11}$
$\displaystyle=\kappa_{V^{\prime}}\left(\kappa_{B^{\prime}}+2\kappa_{V^{\prime}}-1\right),$
$\displaystyle\hat{B}_{{\mathrm{D}}22}$
$\displaystyle=2\kappa_{B^{\prime}}+\kappa_{V^{\prime}}-2,$
$\displaystyle\hat{B}_{{\mathrm{D}}12}^{\pm}$
$\displaystyle=-\kappa_{V^{\prime}}\left(3\sqrt{\kappa_{B^{\prime}}}\pm
1\right),$ (E.28) $\displaystyle\hat{B}_{{\mathrm{V}}11}$
$\displaystyle=\kappa_{B^{\prime}}+2\kappa_{V^{\prime}}-1,$
$\displaystyle\hat{B}_{{\mathrm{V}}22}$
$\displaystyle=\kappa_{V^{\prime}}\left(2\kappa_{B^{\prime}}+\kappa_{V^{\prime}}-2\right),$
$\displaystyle\hat{B}_{{\mathrm{V}}12}^{\pm}$
$\displaystyle=\kappa_{V^{\prime}}\left(3\sqrt{\kappa_{B^{\prime}}}\pm
1\right).$ (E.29)

In the limit of a massless LDV, the rate reduces to

$\displaystyle\begin{split}\lim_{m_{V^{\prime}}\to 0}\frac{d\Gamma(B\to
B^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{D}}&=\frac{m_{B}^{3}}{4\pi\Lambda^{2}}\left(1-\kappa_{B^{\prime}}\right)^{3}\Big{[}|f_{2}^{{\mathrm{T}V}}|^{2}|\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}|^{2}+|f_{2}^{{\mathrm{T}A}}|^{2}|\mathbb{C}^{{\mathrm{D}}5}_{q^{\prime}q}|^{2}\\\
&+2\cos\theta\left({\mathrm{I}m}(f_{2}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}A}*}){\mathrm{R}e}({\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{D}}5*}_{q^{\prime}q})+{\mathrm{R}e}(f_{2}^{{\mathrm{T}V}}f_{2}^{{\mathrm{T}A}*}){\mathrm{I}m}({\mathbb{C}^{{\mathrm{D}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{D}}5*}_{q^{\prime}q})\right)\Big{]}\,,\\\
\lim_{m_{V^{\prime}}\to 0}\frac{d\Gamma(B\to
B^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{V}}&=\frac{m_{B}^{3}}{32\pi\Lambda^{2}}\left(1-\kappa_{B^{\prime}}\right)^{3}\Big{(}|f_{1}|^{2}|\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}|^{2}+|g_{1}|^{2}|\mathbb{C}^{{\mathrm{V}}5}_{q^{\prime}q}|^{2}\\\
&+2\cos\theta\left({\mathrm{R}e}(f_{1}g_{1}^{*}){\mathrm{R}e}({\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{V}}5*}_{q^{\prime}q})-{\mathrm{I}m}(f_{1}g_{1}^{*}){\mathrm{I}m}({\mathbb{C}^{{\mathrm{V}}}_{q^{\prime}q}}\mathbb{C}^{{\mathrm{V}}5*}_{q^{\prime}q})\right)\Big{)}\,.\end{split}$
(E.30)

###  E.5 Polarized lepton distributions and rates
$\boldsymbol{\ell\to\ell^{\prime}+V^{\prime}}$

Next we consider the decays $\ell\to\ell^{\prime}+V^{\prime}$ for the case in
which lepton-flavor violating dipole or vector interactions with the LDV are
present. In this case there is experimental sensitivity to the polarization of
the initial lepton by the measurement of the angular distribution of the angle
$\theta$, defined as the angle between the polarization vector of $\ell$ and
the three-momentum of $\ell^{\prime}$. For the different LDV interactions we
find for the differential width of a fully polarized initial $\ell$

$\begin{split}\frac{d\Gamma(\ell\to\ell^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{D}}&=\frac{m_{\ell}^{3}}{{8\pi\Lambda^{2}}}\lambda^{1/2}_{\ell^{\prime}V^{\prime}}\Big{[}(\tilde{A}^{{\mathrm{D}}}_{+}+\tilde{A}^{{\mathrm{D}}}_{-})\left|\mathbb{C}^{{\mathrm{D}}}_{\ell^{\prime}\ell}\right|^{2}+(\tilde{A}^{{\mathrm{D}}}_{+}-\tilde{A}^{{\mathrm{D}}}_{-})\left|\mathbb{C}^{{\mathrm{D}}5}_{\ell^{\prime}\ell}\right|^{2}\\\
&\hskip
60.00009pt+\tilde{A}^{{\mathrm{D}}}_{\theta}\cos\theta\cdot{\mathrm{I}m}(\mathbb{C}^{{\mathrm{D}}}_{\ell^{\prime}\ell}\mathbb{C}^{{\mathrm{D}}5*}_{\ell^{\prime}\ell})\Big{]}\,,\\\
\frac{d\Gamma(\ell\to\ell^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{V}}&=\frac{m_{\ell}^{3}}{32\pi\Lambda^{2}}\lambda^{1/2}_{\ell^{\prime}V^{\prime}}\Big{[}(\tilde{A}^{{\mathrm{V}}}_{+}+\tilde{A}^{{\mathrm{V}}}_{-})\left|\mathbb{C}^{{\mathrm{V}}}_{\ell^{\prime}\ell}\right|^{2}+(\tilde{A}^{{\mathrm{V}}}_{+}-\tilde{A}^{{\mathrm{V}}}_{-})\left|\mathbb{C}^{{\mathrm{V}}5}_{\ell^{\prime}\ell}\right|^{2}\\\
&\hskip
60.00009pt+\tilde{A}^{{\mathrm{V}}}_{\theta}\cos\theta\cdot{\mathrm{R}e}({\mathbb{C}^{{\mathrm{V}}}_{\ell^{\prime}\ell}}\mathbb{C}^{{\mathrm{V}}5*}_{\ell^{\prime}\ell})\Big{]}\,,\end{split}$
(E.31)

with the kinematic coefficients

$\displaystyle\tilde{A}^{{\mathrm{D}}}_{+}$
$\displaystyle=2\left(1-\kappa_{\ell^{\prime}}\right)^{2}-\kappa_{V^{\prime}}\left(1+\kappa_{\ell^{\prime}}\right)-\kappa_{V^{\prime}}^{2}\,,$
$\displaystyle\tilde{A}^{{\mathrm{V}}}_{+}$
$\displaystyle=\left(1-\kappa_{\ell^{\prime}}\right)^{2}+\kappa_{V^{\prime}}\left(1+\kappa_{\ell^{\prime}}\right)-2\kappa_{V^{\prime}}^{2}\,,$
$\displaystyle\tilde{A}^{{\mathrm{D}}}_{-}$
$\displaystyle=-6\sqrt{\kappa_{\ell^{\prime}}}\kappa_{V^{\prime}}\,,$
$\displaystyle\tilde{A}^{{\mathrm{V}}}_{-}$
$\displaystyle=-6\sqrt{\kappa_{\ell^{\prime}}}\kappa_{V^{\prime}}\,,$ (E.32)
$\displaystyle\tilde{A}^{{\mathrm{D}}}_{\theta}$
$\displaystyle=2\lambda^{1/2}_{\ell^{\prime}V^{\prime}}\left(2-2\kappa_{\ell^{\prime}}-\kappa_{V^{\prime}}\right)\,,$
$\displaystyle\tilde{A}^{{\mathrm{V}}}_{\theta}$
$\displaystyle=2\lambda^{1/2}_{\ell^{\prime}V^{\prime}}\left(1-2\kappa_{V^{\prime}}-\kappa_{\ell^{\prime}}\right)\,.$

In the limit of massless LDV, the polarized differential two-body rate reduces
to

$\begin{split}\lim_{m_{V^{\prime}}\to
0}\frac{d\Gamma(\ell\to\ell^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{D}}&=\frac{m_{\ell}^{3}}{{4\pi\Lambda^{2}}}\left(1-\kappa_{\ell^{\prime}}\right)^{3}\left(\left|\mathbb{C}^{{\mathrm{D}}}_{\ell^{\prime}\ell}\right|^{2}+\left|\mathbb{C}^{{\mathrm{D}}5}_{\ell^{\prime}\ell}\right|^{2}+2\cos\theta\cdot{\mathrm{I}m}(\mathbb{C}^{{\mathrm{D}}}_{\ell^{\prime}\ell}\mathbb{C}^{{\mathrm{D}}5*}_{\ell^{\prime}\ell})\right)\\\
\lim_{m_{V^{\prime}}\to
0}\frac{d\Gamma(\ell\to\ell^{\prime}V^{\prime})}{d\cos\theta}\bigg{|}_{\mathrm{V}}&=\frac{m_{\ell}^{3}}{32\pi\Lambda^{2}}\left(1-\kappa_{\ell^{\prime}}\right)^{3}\left(\left|\mathbb{C}^{{\mathrm{V}}}_{\ell^{\prime}\ell}\right|^{2}+\left|\mathbb{C}^{{\mathrm{V}}5}_{\ell^{\prime}\ell}\right|^{2}+2\cos\theta\cdot{\mathrm{R}e}(\mathbb{C}^{{\mathrm{V}}}_{\ell^{\prime}\ell}\mathbb{C}^{{\mathrm{V}}5*}_{\ell^{\prime}\ell})\right)\,.\end{split}$
(E.33)

Finally, after integrating over $\theta$ and averaging over the initial- and
final-state polarizations, the total decay rates read

$\displaystyle\Gamma(\ell\to\ell^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}$
$\displaystyle=\frac{\lambda^{1/2}_{\ell^{\prime}V^{\prime}}m_{\ell}^{3}}{{4\pi\Lambda^{2}}}\left(\left|\mathbb{C}^{{\mathrm{D}}}_{\ell^{\prime}\ell}\right|^{2}(\tilde{A}^{{\mathrm{D}}}_{+}+\tilde{A}^{{\mathrm{D}}}_{-})+\left|\mathbb{C}^{{\mathrm{D}}5}_{\ell^{\prime}\ell}\right|^{2}(\tilde{A}^{{\mathrm{D}}}_{+}-\tilde{A}^{{\mathrm{D}}}_{-})\right)\,,$
(E.34)
$\displaystyle\Gamma(\ell\to\ell^{\prime}V^{\prime})\Big{|}_{\mathrm{V}}$
$\displaystyle=\frac{\lambda^{1/2}_{\ell
V^{\prime}}m_{\ell}^{3}}{16\pi\Lambda^{2}}\left(\left|\mathbb{C}^{{\mathrm{V}}}_{\ell^{\prime}\ell}\right|^{2}(\tilde{A}^{{\mathrm{V}}}_{+}+\tilde{A}^{{\mathrm{V}}}_{-})+\left|\mathbb{C}^{{\mathrm{V}}5}_{\ell^{\prime}\ell}\right|^{2}(\tilde{A}^{{\mathrm{V}}}_{+}-\tilde{A}^{{\mathrm{V}}}_{-})\right)\,,$
(E.35)

which in the limit of massless LDVs reduces to

$\begin{split}\lim_{m_{V^{\prime}}\to
0}\Gamma(\ell\to\ell^{\prime}V^{\prime})\Big{|}_{\mathrm{D}}&=\frac{m_{\ell}^{3}}{{2\pi\Lambda^{2}}}\left(1-\kappa_{\ell^{\prime}}\right)^{3}\left(\left|\mathbb{C}^{{\mathrm{D}}}_{\ell^{\prime}\ell}\right|^{2}+\left|\mathbb{C}^{{\mathrm{D}}5}_{\ell^{\prime}\ell}\right|^{2}\right)\,,\\\
\lim_{m_{V^{\prime}}\to
0}\Gamma(\ell\to\ell^{\prime}V^{\prime})\Big{|}_{\mathrm{V}}&=\frac{m_{\ell}^{3}}{{16\pi\Lambda^{2}}}\left(1-\kappa_{\ell^{\prime}}\right)^{3}\left(\left|\mathbb{C}^{{\mathrm{V}}}_{\ell^{\prime}\ell}\right|^{2}+\left|\mathbb{C}^{{\mathrm{V}}5}_{\ell^{\prime}\ell}\right|^{2}\right)\,.\end{split}$
(E.36)

## References

  * [1] S. Knapen, T. Lin and K.M. Zurek, _Light Dark Matter: Models and Constraints_ , _Phys. Rev. D_ 96 (2017) 115021 [1709.07882].
  * [2] B. Holdom, _Two U(1)’s and Epsilon Charge Shifts_ , _Phys. Lett. B_ 166 (1986) 196.
  * [3] B.A. Dobrescu, _Massless gauge bosons other than the photon_ , _Phys. Rev. Lett._ 94 (2005) 151802 [hep-ph/0411004].
  * [4] T. Hambye, M.H.G. Tytgat, J. Vandecasteele and L. Vanderheyden, _Dark matter from dark photons: a taxonomy of dark matter production_ , _Phys. Rev. D_ 100 (2019) 095018 [1908.09864].
  * [5] E.J. Chun, J.-C. Park and S. Scopel, _Dark matter and a new gauge boson through kinetic mixing_ , _JHEP_ 02 (2011) 100 [1011.3300].
  * [6] M. Fabbrichesi, E. Gabrielli and G. Lanfranchi, _The Dark Photon_ , 2005.01515.
  * [7] B.C. Allanach, J. Davighi and S. Melville, _An Anomaly-free Atlas: charting the space of flavour-dependent gauged $U(1)$ extensions of the Standard Model_, _JHEP_ 02 (2019) 082 [1812.04602].
  * [8] M. Williams, C.P. Burgess, A. Maharana and F. Quevedo, _New Constraints (and Motivations) for Abelian Gauge Bosons in the MeV-TeV Mass Range_ , _JHEP_ 08 (2011) 106 [1103.4556].
  * [9] A. Smolkovič, M. Tammaro and J. Zupan, _Anomaly free Froggatt-Nielsen models of flavor_ , _JHEP_ 10 (2019) 188 [1907.10063].
  * [10] Y. Kahn, G. Krnjaic, S. Mishra-Sharma and T.M.P. Tait, _Light Weakly Coupled Axial Forces: Models, Constraints, and Projections_ , _JHEP_ 05 (2017) 002 [1609.09072].
  * [11] M. Bauer, P. Foldenauer and J. Jaeckel, _Hunting All the Hidden Photons_ , _JHEP_ 07 (2018) 094 [1803.05466].
  * [12] A. Greljo, P. Stangl, A.E. Thomsen and J. Zupan, _On (g $-$ 2)μ from gauged U(1)X_, _JHEP_ 07 (2022) 098 [2203.13731].
  * [13] R. Bause, H. Gisbert, G. Hiller, T. Höhne, D.F. Litim and T. Steudtner, _U-spin-CP anomaly in charm_ , _Phys. Rev. D_ 108 (2023) 035005 [2210.16330].