A bridge to new physics: proposing new – and reviving old – explanations of
$a_{\mu}$

Guilherme Guedes${}^{\,a,b}$ and Pablo Olgoso${}^{\,a}$
a CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de
Granada, Campus de Fuentenueva, E–18071 Granada, Spain
b Laboratório de Instrumentaçao e Física Experimental de Partículas,
Departamento de Física da Universidade do Minho, Campus de Gualtar, 4710-057
Braga, Portugal

###### Abstract

The $4.2\,\sigma$ tension in the combined measurement of the anomalous
magnetic moment of the muon, $a_{\mu}$, and the Standard Model prediction
strongly suggests the existence of beyond the Standard Model physics.
Following the Standard Model Effective Field Theory approach, we study a
particular topology, the _bridge_ diagram, which gives a chirally enhanced
contribution to $a_{\mu}$. We classify all possible 2- and 3-field SM
extensions that can generate this contribution and present the full $a_{\mu}$
result for them. Within our approach, we find that several 2-field fermion-
scalar extensions which had been previously discarded in the literature – when
only the Yukawa-suppressed contribution was considered – can actually be
viable models to explain the observed anomaly. Furthermore, the 3-field
extensions which generate the bridge diagram represent a new class of models
to account for $a_{\mu}$. We explore a particular 3-field extension which,
beyond explaining $a_{\mu}$, can also account for the neutral $B$-meson
anomalies and the Cabibbo angle anomaly. We present the full one-loop matching
for this model and a one-loop phenomenological study.

###### Contents

  1. 1 Introduction
  2. 2 Computation of $a_{\mu}$
  3. 3 General results
    1. 3.1 VLL doublet bridge
    2. 3.2 VLL singlet (triplet) bridge
  4. 4 Two-field extensions
  5. 5 Three-field extensions
  6. 6 General phenomenological considerations
  7. 7 The triple triplet model
  8. 8 Conclusions
  9. A Box diagram results

##  1 Introduction

The recent measurement of the anomalous magnetic moment of the muon,
$a_{\mu}$, by Fermilab Muon g-2 Experiment [1], has sparked immense interest
in the physics community since, together with the previous measurement by
Brookhaven National Laboratory [2], they present a combined $4.2\sigma$
tension with the Standard Model (SM) result [3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38],

$a^{\mathrm{EXP}}_{\mu}-a^{\mathrm{SM}}_{\mu}=(251\pm 59)\times 10^{-11},$
(1.1)

which strongly suggests the existence of new physics (NP) contributions to
this quantity. This value does not take into account the latest lattice
results from the hadronic vacuum polarization SM contributions, presented by
the BMW collaboration [39], which seem to soften the tension in the $a_{\mu}$
measurement; however, since it introduces another tension in
$e^{+}e^{-}\rightarrow$ hadrons cross-section [40, 41, 42, 43], we will
neglect it for now.

From the lens of the Standard Model Effective Field Theory (SMEFT), the new
contribution to $a_{\mu}$ (at tree-level) is given by

$\Delta
a_{\mu}=a_{\mu}^{\mathrm{NP}}-a_{\mu}^{\mathrm{SM}}=\frac{4m_{\mu}v}{\sqrt{2}e}\bigg{(}\mathrm{Re}[\alpha_{eB}^{2,2}]c_{W}-\mathrm{Re}[\alpha_{eW}^{2,2}]s_{W}\bigg{)}\equiv\frac{4m_{\mu}v}{\sqrt{2}e}\alpha^{2,2}_{e\gamma}\,,$
(1.2)

where $m_{\mu}$ is the mass of the muon, $v$ the vacuum expectation value
(VEV) of the Higgs, $e$ the electric charge, $c_{W}$ ($s_{W}$) the co-sine
(sine) of the Weinberg angle and $\alpha^{i,j}_{eB}$ and $\alpha^{i,j}_{eW}$
the coefficients of the dipole operators,

$\displaystyle\mathcal{O}^{i,j}_{eB}$
$\displaystyle=(\overline{\ell}^{i}\sigma^{\mu\nu}e^{j})HB_{\mu\nu}+\mathrm{h.c.},$
(1.3) $\displaystyle\mathcal{O}^{i,j}_{eW}$
$\displaystyle=(\overline{\ell}^{i}\sigma^{\mu\nu}e^{j})\sigma^{I}HW^{I}_{\mu\nu}+\mathrm{h.c.}\,,$
(1.4)

where $i$ and $j$ correspond to flavour indices. Below the electroweak scale,
in the low-energy effective field theory (LEFT), the tree-level contribution
to $a_{\mu}$ is given by the photon dipole operator, whose Wilson coefficient
is $L_{e\gamma}=\frac{v}{\sqrt{2}}\alpha_{e\gamma}$. Results in this paper
will be quoted as contributions to $\alpha_{e\gamma}$.

In spite of contributing at tree-level to $a_{\mu}$, the SMEFT dipole
operators are generated only at loop level by weakly-coupled UV completions
[44, 45, 46] and, as such, a complete one-loop analysis of the contributions
to $a_{\mu}$ needs to include the effects of the one-loop running of tree-
level generated operators and one-loop finite contributions at low-energy. In
Ref. [47] it was shown that, besides the mentioned dipole operators, the only
remaining relevant contribution to $a_{\mu}$ arises as a one-loop effect from
$\mathcal{O}_{\ell equ}^{(3)}$ for an EFT with a cut-off of
$\Lambda=10\;\mathrm{TeV}$. Being already a one-loop contribution, one only
needs to consider the tree-level generation of this operator, which can only
occur by integrating out the scalar leptoquarks $S_{1}$ and $S_{2}$ [45];
since these tree-level contributions are already well known [45, 47], for the
remainder of this paper we will focus solely on the contribution to $a_{\mu}$
arising from the one-loop generation of the dipole operators.

Several completions to the SM (with a different number of new fields) have
been proposed in the literature to generate the dipole operators – see Ref.
[48] (and references therein) for a comprehensive review of the status of such
models. Interest has been given to chirally enhanced solutions, that is,
solutions in which the new contribution to $a_{\mu}$ is not suppressed by the
muon Yukawa coupling, because they can explain the observed $\Delta a_{\mu}$
with masses for the new heavy particles typically large enough to avoid
current experimental constraints. In particular, Refs. [49, 50, 51, 52, 53,
54] studied chirally enhanced contributions for a wide range of 3-field
extensions of the SM.

In this work, we focus on the chirally enhanced contribution to $a_{\mu}$
produced by the topology of Fig. 2, which we will refer to as _bridge_
hereafter. We will study the possible UV completions which can generate this
diagram. Note that the particle connecting the loop and 2 external states (the
bridge) and one of the particles in the loop must be heavy, but the other one
can be heavy or light, and as such, either 2- or 3-field extensions can
generate this type of topology.

While the bridge diagram has been studied in some particular cases of 2-field
extensions with 2 vector-like leptons [55, 56], we perform a complete
classification of all possible 2 and 3 fields extensions of the SM which can
produce this topology. These extensions can sometimes be accompanied by a box
diagram111In the LEFT, when the Higgs takes a VEV, this actually corresponds
to the triangle diagram which is usually considered in the literature. which
one must take into account as well when computing $a_{\mu}$.

Due to our approach in the SMEFT, we consider contributions from several
2-field extensions which have been overlooked in the literature, where only
the lepton Yukawa-suppressed contributions to $a_{\mu}$ were considered
excluding these models as explanations of the anomaly either by direct
searches or due to the negative sign in their contribution. By considering the
bridge contribution from these 2-field models, we are in fact restoring them
as possible explanations for the observed anomaly in $a_{\mu}$.

The 3-field extensions with chirally enhanced contributions considered in
Refs. [50, 51, 52, 53, 54, 49] generated the dipole operators through the box
diagram, shown in Fig. 7. The representations of the heavy fields which fit
into that topology are different from the ones which can generate the bridge
diagram, and, as such, in this work we present a completely new class of
3-field extensions that can account for the observed $\Delta a_{\mu}$.

Given the added flexibility of considering 3 new degrees of freedom, we
consider a particular set of representations for the 3 fields in this class in
which we can also address the neutral flavour anomalies $R_{K}$ and
$R_{K^{*}}$, as well as the Cabibbo angle anomaly. This model is composed of
an $SU(2)$ triplet leptoquark, $S_{3}$, a triplet vector-like lepton,
$\Sigma$, and a triplet vector-like quark, $\Psi_{Q}$. We present here the
full one-loop matching conditions for this model and study its one-loop
phenomenology.

The article is organized as follows. In Section 2 we explore the
technicalities of the computation of the matching conditions for the $a_{\mu}$
contribution. In Section 3 we provide generic results for the bridge topology
for the different combinations of heavy scalar and heavy fermion propagators
that can generate it. Section 4 is devoted to presenting the results of
$\Delta a_{\mu}$ for all 2-field extensions which generate the bridge
topology, featuring novel results for a particular set of completions. In
Section 5 the same is done for 3-field extensions, introducing a new class of
models which can explain $\Delta a_{\mu}$. Phenomenological considerations
which can be applied to all of these models are discussed in Section 6. In
Section 7 we perform the complete one-loop matching for a particular 3-field
extension, in which the neutral flavour anomalies and the Cabibbo angle
anomaly can also be addressed.

##  2 Computation of $a_{\mu}$

In this work we consider fermion and scalar222Scalars can be replaced by heavy
vectors with the same quantum numbers, but we do not present results for those
cases. 2- and 3-field extensions of the SM which generate the bridge topology
of Fig. 2 – with the $W$ or $B$ gauge bosons attached on either of the
internal propagators333Note that in practice one can directly calculate the
diagram with the photon insertion and consider the appropriate electric
charge; however, to keep the language coherent within the SMEFT picture, we
will refer to diagrams with both $W$ and $B$ bosons.. Throughout all
computations we neglect contributions suppressed by lepton Yukawa couplings or
by the Higgs mass (terms of the form $m_{\phi}/M$).

Some of the extensions we consider can also contribute to $a_{\mu}$ through
other diagrams, namely the usual box diagrams shown in Figs. 6 and 7.
Therefore, for completeness, we present in Appendix A the contribution to
$a_{\mu}$ for general representations of heavy fields arising from box
diagrams. With this and the general results from the bridge topology, which
are presented in the next section, one can in principle calculate the full
contribution to $a_{\mu}$ for arbitrary UV extensions of the SM. Furthermore,
note that depending on the particular representation, some 3-field extensions
can also generate diagrams with only 2 or 1 heavy propagators, which have
different kinematic factors and must also be considered in the calculation of
the full $a_{\mu}$ contribution444An example of this are 3-field extensions in
which a heavy Higgs is considered, where one must also consider the diagrams
generated by substituting it by the SM Higgs. Note also that the heavy Higgs
by itself can generate a contribution to $a_{\mu}$.. When we present results
for a particular SM extension (in Sections 4 and 5), we consider all possible
contributions to $a_{\mu}$.

To obtain the one-loop matching conditions of the dipole operators, we compute
the 4 point amplitude between $\overline{\ell}_{L}$, $e_{R}$, $\phi$, and
$B/W$, with all momenta incoming, in the full theory and in the SMEFT. For
simplicity, we take the momentum of the Higgs, $p_{\phi}$, to zero. The
kinematic structure which uniquely defines the dipole operators is
$\not{q}\not{\epsilon}$, where $q$ is the gauge boson momentum and $\epsilon$
its polarization vector. This structure can be traded on-shell by
$\epsilon\cdot p_{e}$, since:

$\displaystyle\overline{v_{\ell}}\,\not{q}\not{\epsilon}\,u_{e}=-\overline{v_{\ell}}\,(\not{p}_{\ell}+\not{p}_{e})\not{\epsilon}\,u_{e}=-2\epsilon\cdot
p_{e}\,\overline{v_{\ell}}\,u_{e}\;,$ (2.1)

where $p_{e(\ell)}$ is the momentum of the right-handed (left-handed)
electron, and $v_{\ell},u_{e}$ are the corresponding external spinors. In the
second equality we considered the on-shell conditions
$\overline{v_{\ell}}\,\not{p}_{\ell}=0$ and $\not{p}_{e}\,u_{e}=0$ (when
applied to the external fields for massless fermions). Therefore,
$\epsilon\cdot p_{e}$ ends up being the only relevant kinematic structure for
the matching calculation.

For this specific case, performing the matching on-shell is particularly
efficient since no other connected diagrams can arise (without inserting
lepton Yukawas). One could think of attaching the gauge bosons to the external
legs of the diagrams instead of in the internal propagators. However, when the
gauge boson is attached to the fermions, the diagram will either result in a
contribution proportional to $\not{p}_{\ell}\not{\epsilon}$ or
$\not{\epsilon}\,\not{p}_{e}$, or, when the photon couples to the Higgs,
proportional to $q\cdot\epsilon=0$ or $p_{\phi}\cdot\epsilon=0$.

The same happens when the gauge boson is attached to the fermionic bridge,
where one can find that all contributions are proportional either to
$\not{p}_{\ell}\not{\epsilon}$ or $\not{\epsilon}\,\not{p}_{e}$, being
therefore zero in light of the arguments presented above. Consequently, we
only compute contributions coming from insertions of gauge bosons in the
particles in the loop. For the same reasons, a mass insertion in the bridge
propagator is needed, fixing the chirality of the coupling between the two
(three) heavy fields.

All results presented in Sections 3 and 4 were cross-checked with
matchmakereft [57] 555When possible we also cross-checked our results against
those in the literature.. This tool calculates the one-loop matching
conditions for UV extensions of the SM diagramatically and off-shell.
Therefore, results in this case are given in terms of operators in a Green’s
basis, which must then be reduced to the Warsaw basis. When using
matchmakereft one must then take

$\displaystyle\alpha_{eB}$
$\displaystyle=\alpha^{G}_{eB}-\frac{g_{1}}{8}\beta^{G}_{eHD2}+\frac{g_{1}}{8}\beta^{G}_{eHD4}-\frac{g_{1}}{2}\beta^{G}_{eHD3}\,,$
(2.2) $\displaystyle\alpha_{eW}$
$\displaystyle=\alpha^{G}_{eW}-\frac{g_{2}}{8}\beta^{G}_{eHD2}+\frac{g_{2}}{8}\beta^{G}_{eHD4}\,,$
(2.3)

where all the coefficients are written following the conventions stated in
matchmakereft, i.e., $\beta$ for redundant operators, and the superscript $G$
corresponds to Wilson coefficients in the Green’s basis. We neglected Yukawa-
suppressed contributions and the evanescent coefficients
$\gamma_{eB},\gamma_{eW}$. Following from Eq. (1.2), one can then write the
contribution to $a_{\mu}$ as a function of the Wilson coefficients in the
Green’s basis as:

$\Delta
a_{\mu}=\frac{4m_{\mu}v}{\sqrt{2}}\bigg{(}\frac{1}{g_{1}}\,\left(\alpha^{G}_{eB}\right)^{2,2}-\frac{1}{g_{2}}\,\left(\alpha^{G}_{eW}\right)^{2,2}-\frac{1}{2}\,\left(\beta^{G}_{eHD3}\right)^{2,2}\bigg{)}.$
(2.4)

##  3 General results

The bridge topologies can be divided according to the particle that runs in
the bridge, where by bridge we are referring to the internal propagator
connecting the loop with 2 external particles. There can never be a SM
particle in the bridge as it would give a Yukawa-suppressed contribution. The
possible heavy particles in the bridges are fixed by the external SM particles
to which it couples to:

  1. 1.

Scalar bridge

The heavy scalar must couple to the left- and right-handed muon, which fixes
the quantum numbers to be the same as the SM Higgs. All the corresponding
contributions to $a_{\mu}$ are zero (they are always proportional to
$\epsilon\cdot q$).

  2. 2.

Fermion bridge coupled to right-handed muon

The heavy fermion must have the same quantum numbers as the SM left-handed
lepton, $\Delta\sim(1,2,-1/2)$. The numbers in parenthesis denote the
representations under $SU(3)_{c}$, $SU(2)_{L}$ and $U(1)_{Y}$, respectively.

  3. 3.

Fermion bridge coupled to left-handed muon

The heavy fermion must either have the quantum numbers of a SM right-handed
lepton, $E\sim(1,1,-1)$, or be an $SU(2)$ triplet, $\Sigma\sim(1,3,-1)$.

We present the results for the contribution to $a_{\mu}$ from the bridge
topologies for generic representations of the heavy fields for these different
vector-like lepton (VLL) bridges in the next subsections.

For concreteness, the results are always presented corresponding to one
specific orientation of the internal propagators shown in the diagrams (in
particular, we always avoid the presence of fermion-number violating
interactions). To translate from these results to those with a flipped
propagator – which may be needed for some choices of the gauge representations
of the heavy fields –, it is sufficient to add a minus sign in the
contribution corresponding to a gauge boson insertion in the flipped
propagator.

###  3.1 VLL doublet bridge

When the bridge particle is a heavy fermion, we are left with 3 possibilities
for the combinations inside the loop: 1 extra heavy fermion, $\Psi$, and the
SM Higgs, Fig. 1(a); 1 heavy fermion and 1 heavy scalar, $\Phi$, Fig. 2; 1
heavy scalar and an SM fermion, Fig. 1(b). The latter case of Fig. 1(b) does
not give a contribution to $a_{\mu}$, and as such, it will be neglected in the
following discussion. The reason is that, since a mass insertion in the bridge
propagator is needed (as explained in Section 2), an extra mass insertion is
required from the fermionic propagator in the loop in order to get the correct
chirality for the external fermions.

The most general Lagrangian, extending the SM with the VLL doublet, $\Delta$,
that can generate a bridge-like contribution to $a_{\mu}$ is the following:

$\displaystyle\mathcal{L}\supset$
$\displaystyle\,gY_{\Psi}\overline{\Psi}\gamma_{\mu}\Psi
B^{\mu}+g_{W}T^{W,\Psi}_{IKI^{\prime}}\overline{\Psi_{I}}\gamma_{\mu}\Psi_{I^{\prime}}W^{\mu}_{K}$
(3.1) $\displaystyle-
igY_{\Phi}B^{\mu}(\partial_{\mu}\Phi^{\dagger}\Phi-\Phi^{\dagger}\partial_{\mu}\Phi)-ig_{W}T^{W,\Phi}_{JKJ^{\prime}}W^{\mu}_{K}(\partial_{\mu}\Phi^{\dagger}_{J}\Phi_{J^{\prime}}-\Phi^{\dagger}_{J}\partial_{\mu}\Phi_{J^{\prime}})$
$\displaystyle+y_{M}\overline{\Delta}\,e_{R}\,\phi+T_{IKJ}\left(y_{b}^{R}\,\overline{\Psi}_{I}\,P_{R}\,\Delta_{K}\Phi_{J}+y_{b}^{L}\,\overline{\Psi}_{I}\,P_{L}\,\Delta_{K}\Phi_{J}\right)+y_{F}\,T^{\prime}_{KIJ}\,\overline{\ell_{L,K}}\Psi_{I}\Phi^{\dagger}_{J}+\mathrm{h.c.}\,,$

where $\Phi$ can stand generically for a heavy scalar or the SM Higgs ($\phi$
always stands for the SM Higgs), $\Psi$ is a heavy fermion and $\ell_{L}$ is
the left-handed SM lepton. We do not write family indices in the couplings
with the SM leptons as we assume only the needed couplings to muons. The
indices $I^{(^{\prime})}$, $J^{(^{\prime})}$, $K^{(^{\prime})}$ correspond to
the SU(2) components of the fields, $Y_{\Psi(\Phi)}$ denotes the hypercharge
of $\Psi(\Phi)$, and $T^{W}$, $T$ and $T^{\prime}$ are the Clebsch-Gordan
coefficients of the fields in the corresponding interaction term. For
instance, for a colourless SU(2) triplet $\Psi$ and SU(2) doublet $\Phi$, one
could have $T^{W,\Psi}_{IKJ}=i\epsilon_{IKJ}$,
$T^{W,\Phi}_{IKJ}=\sigma^{K}_{IJ}/2$, $T_{IKJ}=(\epsilon\sigma^{I})_{KJ}$,
$T^{\prime}_{KIJ}=(\sigma^{I}\epsilon)_{KJ}$, with $\epsilon$ denoting the
Levi-Civita tensor and $\sigma^{i}$ Pauli matrices.

With this notation, and defining $T^{\gamma,\psi}_{ij}\equiv
Y_{\psi}\delta_{ij}+T^{W,\psi}_{i3j}$, where $\psi$ represents any particle,
we can write the generic result for $\alpha_{e\gamma}$ from the bridge
topology as:

$\alpha_{e\gamma}^{2,2}=\frac{iNe}{4}y_{M}y_{F}y_{b}^{R}\sum_{IJ}T_{I2J}\left[\gamma_{\Psi}T^{\gamma,\Psi}_{I^{\prime}I}T^{\prime}_{2JI^{\prime}}+\gamma_{\Phi}T^{\gamma,\Phi}_{JJ^{\prime}}T^{\prime}_{2IJ^{\prime}}\right]\,,$
(3.2)

with $\gamma_{\Psi,\Phi}$ being kinematic factors which will be defined below,
corresponding to the insertion of the gauge bosons on the fermion and scalar,
respectively; their explicit expression depends on the number of heavy
propagators. $N$ designs the dimension of the $SU(3)$ representation of $\Psi$
(and $\Phi$, by extension, when denoting a heavy scalar). $T^{\gamma,\psi}$
would be diagonal and proportional to the electric charge if the charge
eigenstate basis is chosen for the $\psi$ multiplet, i.e., $T^{W,\psi}$ is
diagonal.

(a)

(b)

Figure 1: _Left:_ Bridge topology for the fermionic bridge with an extra heavy
fermion and the SM Higgs. _Right:_ Bridge topology for the fermionic bridge
with an extra heavy scalar and a SM fermion. Double lines represent heavy
particles whereas single lines are SM particles. The gauge boson ($B$ or $W$)
is represented outside the diagram since it can be attached to any of the
internal propagators.

The kinematic factors $\gamma_{\Psi,\Phi}$ defined throughout Section 3 can
always be expressed as these two functions of the masses:

$\displaystyle f(M_{A},M_{B},M_{C})$
$\displaystyle\equiv-\frac{iM_{B}}{(4\pi)^{2}M_{A}}\,\frac{M_{B}^{4}-4M_{B}^{2}M_{C}^{2}+3M_{C}^{4}+2M_{C}^{4}\operatorname{Log}{[M_{B}^{2}/M_{C}^{2}]}}{(M_{B}^{2}-M_{C}^{2})^{3}}\,,$
(3.3) $\displaystyle h(M_{A},M_{B},M_{C})$
$\displaystyle\equiv-\frac{iM_{B}}{(4\pi)^{2}M_{A}}\,\frac{M_{B}^{4}-M_{C}^{4}-2M_{B}^{2}M_{C}^{2}\operatorname{Log}{[M_{B}^{2}/M_{C}^{2}]}}{(M_{B}^{2}-M_{C}^{2})^{3}}\,.$
(3.4)

For the case in which inside the loop we have another heavy fermion, $\Psi$
and the SM Higgs, Fig. 1(a), the kinematic factors in Eq. (3.2) are given by:

$\gamma_{\Psi}=\gamma_{\Phi}=\lim_{M_{\Phi}\rightarrow
0}f(M_{\Delta},M_{\Psi},M_{\Phi})=\frac{-i}{(4\pi)^{2}M_{\Delta}M_{\Psi}}.$
(3.5)

Figure 2: Bridge topology for the fermionic bridge with an extra heavy fermion
and a heavy scalar. Double lines represent heavy particles, whereas single
lines are SM particles. The gauge boson ($B$ or $W$) is represented outside
the diagram since it can be attached to any of the internal propagators.

If the bridge diagram with three heavy propagators is generated, Fig. 2, the
contribution to $\alpha_{e\gamma}$ is calculated with:

$\displaystyle\gamma_{\Psi}=$
$\displaystyle\,f(M_{\Delta},M_{\Psi},M_{\Phi})\,,$ (3.6)
$\displaystyle\gamma_{\Phi}=$
$\displaystyle\,h(M_{\Delta},M_{\Psi},M_{\Phi})\,.$ (3.7)

###  3.2 VLL singlet (triplet) bridge

The relevant topologies for the case of a VLL singlet or triplet bridge are
the same ones as explored in the previous section for the doublet VLL bridge,
changing only the fermionic current since the bridge is now connected to the
left-handed muon.

The relevant Lagrangian for an extension of the SM with a triplet, $\Sigma$,
that generates the bridge diagram is given by:

$\displaystyle\mathcal{L}$ $\displaystyle\supset
y_{M}\overline{\ell}_{L}\sigma^{I}\phi
P_{R}\Sigma_{I}+y_{F}\overline{\Psi}_{I}\Phi_{I}e_{R}+T_{KIJ}\left(y_{b}^{R}\,\overline{\Sigma}_{K}P_{R}\Psi_{I}\Phi_{J}^{\dagger}+y_{b}^{L}\,\overline{\Sigma}_{K}P_{L}\Psi_{I}\Phi_{J}^{\dagger}\right)+\mathrm{h.c.},$
(3.8)

where $\sigma^{I}$ denotes the Pauli matrices and we use the same gauge
conventions and general notation introduced in Eqs. (3.1) and (3.2).

We can write the general result for the bridge contribution to
$\alpha_{e\gamma}$ as:

$\alpha_{e\gamma}^{2,2}=\frac{-iNe}{4}y_{M}y_{F}y_{b}^{R}\sum_{IJ}T_{3IJ}\left[\gamma_{\Psi}T^{\gamma,\Psi}_{IJ}+\gamma_{\Phi}T^{\gamma,\Phi}_{IJ}\right]\,.$
(3.9)

For the singlet bridge case, $E$, the relevant Lagrangian is the following:

$\displaystyle\mathcal{L}$ $\displaystyle\supset y_{M}\overline{\ell}_{L}\phi
P_{R}E+y_{F}\overline{\Psi}_{I}\Phi_{I}e_{R}+y_{b}^{R}\overline{E}P_{R}\Psi_{I}\Phi_{I}^{\dagger}+y_{b}^{L}\overline{E}P_{L}\Psi_{I}\Phi_{I}^{\dagger}+\mathrm{h.c.}\,,$
(3.10)

where once again we use the same conventions as in Eqs. (3.1),(3.2). The
contributions to $\alpha_{e\gamma}$ are given by:

$\alpha_{e\gamma}^{2,2}=\frac{iNe}{4}y_{M}y_{F}y_{b}^{R}\left(\operatorname{Tr}{[T^{\gamma,\Psi}]}\gamma_{\Psi}+\operatorname{Tr}{[T^{\gamma,\Phi}]}\gamma_{\Phi}\right)\,.$
(3.11)

For the diagram with one heavy fermion and the SM Higgs in the loop, the
kinematic factors on both Eqs. (3.9) and (3.11) and are given by:

$\gamma_{\Psi}=\gamma_{\Phi}=\lim_{M_{\Phi}\rightarrow
0}f(M_{E(\Sigma)},M_{\Psi},M_{\Phi})\,,$ (3.12)

whereas for the case in which a heavy fermion and a heavy scalar are in the
loop, the kinematic factors read:

$\displaystyle\gamma_{\psi}=$
$\displaystyle\,f(M_{E(\Sigma)},M_{\Psi},M_{\Phi})\,,$ (3.13)
$\displaystyle\gamma_{\Phi}=$
$\displaystyle\,h(M_{E(\Sigma)},M_{\Psi},M_{\Phi})\,.$ (3.14)

##  4 Two-field extensions

There is a finite number of two-field extensions of the SM which can generate
the topologies discussed previously, and as such, we will present the final
$\alpha_{e\gamma}$ – defined in Eq. (1.2) – contribution for all of them.

In Table 1 we present all the possible completions which in principle generate
a contribution to $\alpha_{e\gamma}$ from a bridge topology with only one
heavy propagator besides the bridge, Fig. 1(a). Note that once the particle in
the bridge is fixed, the quantum numbers of the other heavy particle are also
fixed.

Table 1: 2-field UV completions which generate the bridge topology, with only two heavy propagators. However we consider all possible topologies for the presented $a_{\mu}$ result both those coming from single field topologies and from the box diagrams. Bridge | Other Fermion |  $a_{\mu}$ result |
---|---|---|---
$E\sim(1,1,-1)$ | $\Delta\sim(1,2,-1/2)$ | Eq. (4.1) |
$\Delta_{3}\sim(1,2,-3/2)$ | Eq. (4.2) |
$\Delta\sim(1,2,-1/2)$ | $E\sim(1,1,-1)$ | Eq. (4.1) |
$\Sigma\sim(1,3,-1)$ | Eq. (4.3) |
$N\sim(1,1,0)$ | Eq. (4.4) |
$\Sigma_{0}\sim(1,3,0)$ | Eq. (4.5) |
$\Sigma\sim(1,3,-1)$ | $\Delta\sim(1,2,-1/2)$ | Eq. (4.3) |
$\Delta_{3}\sim(1,2,-3/2)$ | Eq. (4.6) |

The obtained contributions to $\alpha_{e\gamma}$ are given below, with all
results to be understood as divided by the loop factor $(16\pi^{2})$:

  1. 1.

$E\,\sim(1,1,-1)~{}\text{and}~{}\Delta\,\sim(1,2,-1/2)$

There are two different bridge diagrams that contribute in this case: one with
the singlet on the bridge and the doublet in the loop, and vice-versa.

$\alpha_{e\gamma}^{2,2}=-\frac{e\,y_{M}y_{F}y_{b}^{R}}{4M_{E}M_{\Delta}}\,.$
(4.1)

The couplings above can be interpreted within the Lagrangian in Eq. (3.1) or
Eq. (3.10).

  2. 2.

$E\sim(1,1,-1)$ and $\Delta_{3}\sim(1,2,-3/2)$

$\alpha_{e\gamma}^{2,2}=-\frac{5e\,y_{M}y_{F}y_{b}^{R}}{4M_{E}M_{\Delta_{3}}}\,;$
(4.2)

  3. 3.

$\Delta\,\sim(1,2,-1/2)$ and $\Sigma\,\sim(1,3,-1)$

There are two bridge diagrams relevant for this case: one with the doublet on
the bridge and the triplet in the loop, and vice-versa.

$\alpha_{e\gamma}^{2,2}=-\frac{9e\,y_{M}y_{F}y_{b}^{R}}{4M_{\Delta}M_{\Sigma}}\,.$
(4.3)

Both (3.1) and (3.8) can be used to interpret this result.

  4. 4.

$\Delta\sim(1,2,-1/2)$ and $N\sim(1,1,0)$

$\alpha_{e\gamma}^{2,2}=0\,.$ (4.4)

This zero has been extensively discussed in the literature in Refs. [55, 56].

  5. 5.

$\Delta\sim(1,2,-1/2)$ and $\Sigma_{0}\sim(1,3,0)$

$\alpha_{e\gamma}^{2,2}=-\frac{e\,y_{M}y_{F}y_{b}^{R}}{2M_{\Delta}M_{\Sigma_{0}}}\,;$
(4.5)

  6. 6.

$\Sigma\sim(1,3,-1)$ and $\Delta_{3}\sim(1,2,-3/2)$

$\alpha_{e\gamma}^{2,2}=-\frac{5e\,y_{M}y_{F}y_{b}^{R}}{4M_{\Sigma}M_{\Delta_{3}}}\,.$
(4.6)

These results were cross-checked against those in [58, 56, 59], and found
agreement except for the numerical factors in Eq. (4.8) from Ref. [58] for
VLLs with doubly charged components.

A 2-field extension can also generate a bridge topology with 3 heavy
propagators, Fig. 2, if the fermion in the bridge is the same as the fermion
that propagates in the loop. The extensions for which this can occur are
listed in Table 2. The corresponding contributions to $\alpha_{e\gamma}$ are:

  1. 1.

$E\sim(1,1,-1)$ and $\mathcal{S}_{0}\sim(1,1,0)$

$\alpha_{e\gamma}^{2,2}=-ey_{b}^{R}y_{M}y_{F}\frac{M_{E}^{4}-4M_{E}^{2}M_{\mathcal{S}_{0}}^{2}+3M_{\mathcal{S}_{0}}^{4}+2M_{\mathcal{S}_{0}}^{4}\operatorname{Log}{[M_{E}^{2}/M_{\mathcal{S}_{0}}^{2}]}}{4(M_{E}^{2}-M_{\mathcal{S}_{0}}^{2})^{3}}\,;$
(4.7)

  2. 2.

$E\sim(1,1,-1)$ and $\mathcal{S}_{2}\sim(1,1,-2)$

$\alpha_{e\gamma}^{2,2}=ey_{b}^{R}y_{M}y_{F}\frac{3M_{E}^{4}-4M_{E}^{2}M_{\mathcal{S}_{2}}^{2}+M_{\mathcal{S}_{2}}^{4}+(2M_{\mathcal{S}_{2}}^{4}-4M_{E}^{2}M_{\mathcal{S}_{2}}^{2})\operatorname{Log}{[M_{E}^{2}/M_{\mathcal{S}_{2}}^{2}]}}{2(M_{E}^{2}-M_{\mathcal{S}_{2}}^{2})^{3}}\,;$
(4.8)

  3. 3.

$\Delta\sim(1,2,-1/2)$ and $\mathcal{S}_{0}\sim(1,1,0)$

$\alpha_{e\gamma}^{2,2}=-ey_{b}^{R}y_{M}y_{F}\frac{M_{\Delta}^{4}-4M_{\Delta}^{2}M_{\mathcal{S}_{0}}^{2}+3M_{\mathcal{S}_{0}}^{4}+2M_{\mathcal{S}_{0}}^{4}\operatorname{Log}{[M_{\Delta}^{2}/M_{\mathcal{S}_{0}}^{2}]}}{4(M_{\Delta}^{2}-M_{\mathcal{S}_{0}}^{2})^{3}}\,;$
(4.9)

  4. 4.

$\Delta\sim(1,2,-1/2)$ and $\mathcal{S}_{1}\sim(1,1,-1)$

$\alpha_{e\gamma}^{2,2}=0\,;$ (4.10)

  5. 5.

$\Delta\sim(1,2,-1/2)$ and $\Xi_{0}\sim(1,3,0)$

$\alpha_{e\gamma}^{2,2}=ey_{b}^{R}y_{M}y_{F}\frac{M_{\Delta}^{4}+4M_{\Delta}^{2}M_{\Xi_{0}}^{2}-5M_{\Xi_{0}}^{4}-(4M_{\Xi_{0}}^{2}M_{\Delta}^{2}+2M_{\Xi_{0}}^{4})\operatorname{Log}{[M_{\Delta}^{2}/M_{\Xi_{0}}^{2}]}}{4(M_{\Delta}^{2}-M_{\Xi_{0}}^{2})^{3}}\,;$
(4.11)

  6. 6.

$\Delta\sim(1,2,-1/2)$ and $\Xi_{1}\sim(1,3,-1)$

$\alpha_{e\gamma}^{2,2}=-ey_{b}^{R}y_{M}y_{F}\frac{7M_{\Delta}^{4}-8M_{\Delta}^{2}M_{\Xi_{1}}^{2}+M_{\Xi_{1}}^{4}+(-10M_{\Xi_{1}}^{2}M_{\Delta}^{2}+4M_{\Xi_{1}}^{4})\operatorname{Log}{[M_{\Delta}^{2}/M_{\Xi_{1}}^{2}]}}{2(M_{\Delta}^{2}-M_{\Xi_{1}}^{2})^{3}}\,;$
(4.12)

  7. 7.

$\Sigma\sim(1,3,-1)$ and $\Xi_{0}\sim(1,3,0)$

$\alpha_{e\gamma}^{2,2}=-ey_{b}^{R}y_{M}y_{F}\frac{M_{\Sigma}^{2}-M_{\Xi_{0}}^{2}+M_{\Xi_{0}}^{2}\operatorname{Log}{[M_{\Xi_{0}}^{2}/M_{\Sigma}^{2}]}}{(M_{\Sigma}^{2}-M_{\Xi_{0}}^{2})^{2}}\,;$
(4.13)

  8. 8.

$\Sigma\sim(1,3,-1)$ and $\Xi_{2}\sim(1,3,-2)$

$\alpha_{e\gamma}^{2,2}=0\,.$ (4.14)

A few comments are in order concerning these results. First, we are
considering here only one family of heavy particles for simplicity. Therefore,
in this type of completions, where the bridge coupling $y_{b}$ involves two
fields that are equal, it will be zero when the gauge structure is
antisymmetric. This is the case for completions (4.10) and (4.14). Note that
this is not true in general when dealing with multiple families for heavy
fields. On the other hand, for models (4.7), (4.9), (4.11) and (4.13), the
couplings $y^{R}_{b}$ and $y^{L}_{b}$ are related by hermitian conjugation;
therefore, we redefine $y_{b}^{R}\equiv y_{b}^{R}+y_{b}^{L\,*}$ as the
effective coupling with the right-handed chirality and write our results using
such convention.

These models have been considered previously in the literature [60, 49, 59],
apart from the ones that involve fermion number violating (FNV) vertices
which, to the best of our knowledge, are first explored here. However, only
the Yukawa-proportional contribution was considered, which made it so that
most of the models in Tab. 2 were excluded as explanations of $\Delta
a_{\mu}$, since the new particles had to be lighter than what was allowed by
experiments. Since we are performing our calculations in the unbroken phase of
the SM, it becomes easier to see the chirally enhanced contribution coming
from the bridge diagram, which allows for heavier particles to explain
$a_{\mu}$, and as such, opens up this class of models as possible explanations
of the anomaly.

A particularly interesting example is the contribution of the model with
$\Delta\sim(1,2,-1/2)$ \+ $\Xi_{0}\sim(1,3,0)$ extension where the
contribution is taken to be always negative in the literature (and as such
excluded as an explanation of the observed anomaly). However, from Eq. (4.11),
we see that, given the dependence on the couplings, we have the freedom to
make the contribution positive in order to account for the observed $\Delta
a_{\mu}$.

Table 2: 2 field fermion-scalar UV extensions which generate the bridge topology with 3 heavy propagators. Completions in gray color involve fermion number violating interactions.  Fermion | Scalar | Result
---|---|---
$E\sim(1,1,-1)$ | $\mathcal{S}_{0}\sim(1,1,0)$ | Eq. (4.7)
$\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\mathcal{S}_{2}\sim(1,1,-2)$ | Eq. (4.8)
$\Delta\sim(1,2,-1/2)$ | $\mathcal{S}_{0}\sim(1,1,0)$ | Eq. (4.9)
$\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\mathcal{S}_{1}\sim(1,1,-1)$ | Eq. (4.10)
$\Xi_{0}\sim(1,3,0)$ | Eq. (4.11)
$\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\Xi_{1}\sim(1,3,-1)$ | Eq. (4.12)
$\Sigma\sim(1,3,-1)$ | $\Xi_{0}\sim(1,3,0)$ | Eq. (4.13)
$\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\Xi_{2}\sim(1,3,-2)$ | Eq. (4.14)

##  5 Three-field extensions

The possible three-field extensions which can generate the bridge diagram of
Fig. 2 are infinite due to the existence of a heavy loop. As such, in this
section we present the conditions set on the gauge representations of the new
fields, for the bridge contribution to $a_{\mu}$ to be generated. These
conditions are once again fixed by the particle which runs in the bridge.

All conditions are defined considering no FNV vertices. In the presence of
those interactions, one can simply check whether the conjugate version of the
fields respects the following conditions.

The extra heavy scalar, $\Phi$, and heavy fermion, $\Psi$, must respect:

  1. 1.

VLL singlet

$\displaystyle Y_{\Psi}-Y_{\Phi}$ $\displaystyle=-1\,,$ (5.1) $\displaystyle
SU(2)_{\Phi}\otimes SU(2)_{\Psi}$ $\displaystyle=1\,;$ (5.2)

  2. 2.

VLL doublet

$\displaystyle Y_{\Psi}-Y_{\Phi}$ $\displaystyle=-1/2\,,$ (5.3) $\displaystyle
SU(2)_{\Phi}\otimes SU(2)_{\Psi}$ $\displaystyle=2\,;$ (5.4)

  3. 3.

VLL triplet

$\displaystyle Y_{\Psi}-Y_{\Phi}$ $\displaystyle=-1\,,$ (5.5) $\displaystyle
SU(2)_{\Phi}\otimes SU(2)_{\Psi}$ $\displaystyle=3\,,$ (5.6) $\displaystyle
SU(2)_{\Phi}\otimes SU(2)_{\Psi}$ $\displaystyle=1\,,$ (5.7)

where Eq. (5.6) refers to the coupling with the bridge triplet and Eq. (5.7)
to the coupling with the SM right-handed muon.

As for the color charge, we always need to form a singlet with the two fields
in the loop, $\Phi$ and $\Psi$. Larger color representations will result in an
enhancement factor to the diagram, as explored in [53] for completions with
box diagrams.

Limiting ourselves to, at most, triplet representations of $SU(2)$, the UV
completions which can generate the bridge topology are listed in Table 3.

Table 3: Three-field UV extensions which generate the bridge topology, considering only singlet, doublet and triplet $SU(2)$ representations. We show only the $SU(2)$ representations of the 2 extra fields, $\Phi$ and $\Psi$, since the color representation must be the conjugate of each other and their hypercharge must respect the conditions specified above. Switching the assigned $SU(2)$ representations between $\Phi$ and $\Psi$ is also a possible extension. Bridge | ($SU(2)_{\Psi}\,$,$\,SU(2)_{\Phi}$) | Result
---|---|---
$E\sim(1,1,-1)$ | (1,1) | Eq. (5.8)
(2,2) | Eq. (5.9)
(3,3) | Eq. (5.10)
$\Delta\sim(1,2,-1/2)$ | (2,1) | Eqs. (5.11), (5.12)
(2,3) | Eqs. (5.13), (5.14)
$\Sigma\sim(1,3,-1)$ | (2,2) | Eq. (5.15)
(3,3) | Eq. (5.16)

It is evident that none of these completions is among the ones explored in
Refs. [49, 50, 51, 52, 53, 54] and as such constitute novel extensions of the
SM which can in principle contribute to $a_{\mu}$. Let us present the final
expression for these models, using the notation introduced in Eqs. (3.1),
(3.8) and (3.10). Again, a factor $1/16\pi^{2}$ is omitted in all cases and we
neglect contributions proportional to the lepton Yukawa. We present the
results in terms of a generic hypercharge for $\Psi$, $Y_{\Psi}$, and use the
notation $(\Psi,\Phi)\sim(SU(2)_{\Psi},SU(2)_{\Phi})$ to indicate the $SU(2)$
representations of the fields, as listed in Table 3.

  1. 1.

$E\sim(1,1,-1)+(\Psi,\Phi)\sim(1,1)$

$\displaystyle\begin{split}\alpha_{e\gamma}^{2,2}=\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{4M_{E}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(1-2Y_{\Psi})+M_{\Psi}^{2}(1+2Y_{\Psi}))\bigg{.}\\\
\bigg{.}-2(-M_{\Phi}^{4}Y_{\Psi}+M_{\Psi}^{2}M_{\Phi}^{2}(1+Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.8)

  2. 2.

$E\sim(1,1,-1)+(\Psi,\Phi)\sim(2,2)$

$\begin{split}\alpha_{e\gamma}^{2,2}=\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{2M_{E}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(1-2Y_{\Psi})+M_{\Psi}^{2}(1+2Y_{\Psi}))\\\
-2(-M_{\Phi}^{4}Y_{\Psi}+M_{\Psi}^{2}M_{\Phi}^{2}(1+Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.9)

  3. 3.

$E\sim(1,1,-1)+(\Psi,\Phi)\sim(3,3)$

$\begin{split}\alpha_{e\gamma}^{2,2}=\frac{3eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{4M_{E}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(1-2Y_{\Psi})+M_{\Psi}^{2}(1+2Y_{\Psi}))\\\
-2(-M_{\Phi}^{4}Y_{\Psi}+M_{\Psi}^{2}M_{\Phi}^{2}(1+Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.10)

  4. 4.

$\Delta\sim(1,2,-1/2)+(\Psi,\Phi)\sim(2,1)$

$\begin{split}\alpha_{e\gamma}^{2,2}=\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{4M_{\Delta}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}2(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(1-Y_{\Psi})+M_{\Psi}^{2}Y_{\Psi})\\\
-(M_{\Phi}^{4}(1-2Y_{\Psi})+M_{\Psi}^{2}M_{\Phi}^{2}(1+2Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.11)

  5. 5.

$\Delta\sim(1,2,-1/2)+(\Psi,\Phi)\sim(1,2)$

$\begin{split}\alpha_{e\gamma}^{2,2}=\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{4M_{\Delta}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(1-2Y_{\Psi})+M_{\Psi}^{2}(1+2Y_{\Psi}))\\\
-2(-M_{\Phi}^{4}Y_{\Psi}+M_{\Psi}^{2}M_{\Phi}^{2}(1+Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.12)

  6. 6.

$\Delta\sim(1,2,-1/2)+(\Psi,\Phi)\sim(2,3)$

$\begin{split}\alpha_{e\gamma}^{2,2}=\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{4M_{\Delta}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}2(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(1-3Y_{\Psi})+M_{\Psi}^{2}(2+3Y_{\Psi}))\\\
+(M_{\Phi}^{4}(1+6Y_{\Psi})-M_{\Psi}^{2}M_{\Phi}^{2}(7+6Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.13)

  7. 7.

$\Delta\sim(1,2,-1/2)+(\Psi,\Phi)\sim(3,2)$

$\begin{split}\alpha_{e\gamma}^{2,2}=-\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{4M_{\Delta}(M_{\Psi}^{2}-M_{\Phi}^{2})^{3}}\bigg{\\{}(M_{\Psi}^{2}-M_{\Phi}^{2})(M_{\Phi}^{2}(7-6Y_{\Psi})+M_{\Psi}^{2}(-1+6Y_{\Psi}))\\\
-2(M_{\Phi}^{4}(2-3Y_{\Psi})+M_{\Psi}^{2}M_{\Phi}^{2}(1+3Y_{\Psi}))\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.14)

  8. 8.

$\Sigma\sim(1,3,-1)+(\Psi,\Phi)\sim(2,2)$

$\begin{split}\alpha_{e\gamma}^{2,2}=-\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{2M_{\Sigma}(M_{\Psi}^{2}-M_{\Phi}^{2})^{2}}\bigg{\\{}M_{\Psi}^{2}-M_{\Phi}^{2}-M_{\Phi}^{2}\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,;\end{split}$
(5.15)

  9. 9.

$\Sigma\sim(1,3,-1)+(\Psi,\Phi)\sim(3,3)$

$\begin{split}\alpha_{e\gamma}^{2,2}=-\frac{eNM_{\Psi}y_{M}y_{F}y_{b}^{R}}{M_{\Sigma}(M_{\Psi}^{2}-M_{\Phi}^{2})^{2}}\bigg{\\{}M_{\Psi}^{2}-M_{\Phi}^{2}-M_{\Phi}^{2}\operatorname{Log}{[M_{\Psi}^{2}/M_{\Phi}^{2}]}\bigg{\\}}\,.\end{split}$
(5.16)

##  6 General phenomenological considerations

Among the new particles introduced, the one expected to be the most
constrained by experiment is the VLL in the bridge, as it is the only one
which must have a coupling to 2 SM particles.

The mixing of VLLs with the SM muon is bounded by electroweak precision
observables (EWPO) [61, 62]:

$\displaystyle\frac{v}{M_{E}}y_{M}$ $\displaystyle\lesssim 0.03\,(0.04)\,,$
(6.1) $\displaystyle\frac{v}{M_{\Delta}}y_{M}$ $\displaystyle\lesssim
0.065\,(0.075)\,,$ (6.2) $\displaystyle\frac{v}{M_{\Sigma}}y_{M}$
$\displaystyle\lesssim 0.1\,(0.11)\,,$ (6.3)

for the singlet, doublet and triplet of $SU(2)$ respectively at 1 (2) $\sigma$
confidence levels.

Direct searches at colliders for these VLLs set lower limits on their masses.
Ref. [63] set a limit of $M_{E}\gtrsim 175\,\mathrm{GeV}$ for a singlet VLL
decaying through muon or electron channels. However, Ref. [64] estimates that
the HL phase of the LHC could exclude masses lighter than $800\;\mathrm{GeV}$.
The doublet has more recently been probed by CMS [65], where masses below
$\sim 800\,\mathrm{GeV}$ are excluded; these are conservative results for our
study as only tau decays were considered, whereas in order to explain
$a_{\mu}$ the VLL must couple to muons. For the triplet case, Ref. [66]
estimates the discovery reach of the LHC at $3\,\mathrm{ab}^{-1}$ at
$5\,\sigma$ to be of approximately $1.4\;\mathrm{TeV}$.

Other direct bounds could be taken from the particles that run in the loop,
through EW or QCD (when possible) pair production. However, this – and other
possible probes – are clearly dependent on the choice of model and it is
beyond the scope of this paper to go over the particular phenomenological
implications of every model.

Another common issue with chirally enhanced solutions to $a_{\mu}$ has to do
with the large contribution to the muon Yukawa through the same diagrams which
explain $a_{\mu}$ but without the gauge boson insertion. This is not a problem
for the bridge diagram with a VLL triplet since, in order to have a non-zero
contribution, an insertion of a $W$-boson is needed. For the case of the
singlet and doublet VLL bridges, indeed one can expect a sizable contribution
to the muon Yukawa; however, this could only be bounded by fine-tuning
arguments to account for a cancellation between this contribution and a
possible tree-level coefficient. One can further explore possible UV scenarios
in which the Yukawa couplings and the dipole operators have the same origin
[67, 68, 69].

##  7 The triple triplet model

So far, we have focused only on the contributions of SM extensions towards
$a_{\mu}$. However, given the degrees of freedom we introduce by considering
multi-field extensions, we have enough flexibility to try to accommodate other
anomalous observations. The purpose of this section is to show a particular
example of how the bridge topology can connect explanations of different
anomalies.

Of particular interest within the latest measurements is the suggestion of
lepton flavour universality violation in $B$-meson decays. The ratio

$R_{K}^{(*)}=\frac{\mathrm{BR}(B\rightarrow
K^{(*)}\overline{\mu}\mu)}{\mathrm{BR}(B\rightarrow K^{(*)}\overline{e}e)}$
(7.1)

is close to 1 in the SM, but the observations of these ratios show a combined
deviation from the SM prediction of more than $4\,\sigma$ [70, 71, 72, 73,
74].

Another tension within the SM is the possible violation of unitarity in the
first row of the CKM matrix, known as the Cabibbo Angle Anomaly (CAA). Direct
measurements of $V_{us}$ from leptonic kaon decays seem to be in tension with
those which assume CKM unitarity coming from super-allowed $\beta$ decays [75,
76]. The significance of this tension is quoted as $3$ or $5\,\sigma$
depending on the parametrization of the $\beta$ decays. We parametrize this
tension as the deviation from unity of the $R(V_{us})$ observable, which is
defined as the ratio between the extraction of $V_{us}$ from purely muonic
kaon decays over the one extracted from $\beta$ decays assuming unitarity [77,
78]. In [77, 78] it is shown how this deviation is directly related to a
correction in the muonic vertex with the $W$ boson, which we denote by
$\epsilon_{\mu\mu}$:

$\mathcal{L}\supset\frac{g}{\sqrt{2}}W_{\mu}^{-}\overline{\ell}_{i}\gamma^{\mu}P_{L}\nu_{j}(\delta_{ij}+\epsilon_{ij}),$
(7.2)

where $i,j$ run over lepton families.

With the goal of also explaining these two anomalies, we consider one specific
realization of the last class of models in Table 3, in which the SM is
extended with the vector-like lepton triplet, $\Sigma$, a triplet scalar
leptoquark with hypercharge $-1/3$, $S_{3}$, and a triplet vector-like quark,
$\Psi_{Q}$, with hypercharge $-4/3$.

The triplet leptoquark is well-known to be a good solution for the neutral
flavour anomalies at tree-level [79, 80, 81, 82], whereas the vector-like
lepton triplet can explain the CAA, in spite of creating some tension with
EWPO [62, 78]666The latest CDF II measurement of the $W$-boson mass [83] has
significantly increased the tension between CAA and EWPO (see [84] for a
recent analysis). However, addressing this anomaly is beyond the scope of this
paper.. In order to explain $a_{\mu}$ with the bridge topology including these
two particles, we fix the quantum numbers of $\Psi_{Q}$.

The Lagrangian of this model is the following:

$\displaystyle\begin{split}\mathcal{L}&\supset
y_{T}^{i}\overline{\ell}_{Li}\phi\sigma^{I}\Sigma^{I}_{R}+y_{Q}^{i}\overline{\Psi}^{I}_{QL}S_{3}^{I}\,e_{Ri}+iy_{b}^{L}\epsilon^{IJK}\overline{\Sigma}^{I}_{R}\Psi_{Q,L}^{J}S_{3}^{K\dagger}+iy_{b}^{R}\epsilon^{IJK}\overline{\Sigma}^{I}_{L}\Psi_{Q,R}^{J}S_{3}^{K\dagger}\\\
&+\lambda_{S}^{ij}\overline{q}^{c}_{Li}i\sigma^{2}\sigma^{I}\ell_{Lj}S_{3}^{I\dagger}+\lambda_{U}^{i}\overline{u_{R}}_{i}\,\Sigma^{c,I}S_{3}^{I}+\mathrm{h.c.},\end{split}$
(7.3)

with $q^{c}\equiv\mathcal{C}\overline{q}^{T}$ and $\mathcal{C}$ the charge
conjugation matrix. The SM $SU(2)$ doublets $q_{L}$ and $\ell_{L}$ are in the
down-quark and charged lepton diagonal basis, respectively.

In the simplest version of this model, we will consider the minimal set of
couplings which allow for an explanation of the observed discrepancies in
$B$-meson decays, $V_{\mathrm{CKM}}$ unitarity and the anomalous magnetic
moment of the muon. Therefore we suppose just one family of heavy particles
and assume that new physics only couples to second generation leptons. This
also helps to avoid some constraints like the Lepton Flavour Violating (LFV)
decay $\mu\rightarrow e\gamma$. With respect to quarks, we will only allow for
second and third generation couplings in $\lambda_{S}^{ij}$, namely
$\lambda_{S}^{s\mu}$ and $\lambda_{S}^{b\mu}$.

The aforementioned anomalies are explained in this model by the generation of
$\mathcal{O}_{\ell q}^{(1),(3)}$ at tree-level by $S_{3}$ exchange,
$\mathcal{O}_{H\ell}^{(3)}$ also at tree-level by $\Sigma$ exchange and a
bridge-like one-loop contribution to $\Delta a_{\mu}$. The expressions for the
relevant Wilson coefficients are:

$[\alpha_{\ell
q}^{(1)}]_{i,j,k,l}=\frac{3\lambda_{S}^{*ki}\lambda_{S}^{lj}}{4M_{S_{3}}^{2}}+\mathcal{O}(\frac{1}{16\pi^{2}})\,,$
(7.4) $[\alpha_{\ell
q}^{(3)}]_{i,j,k,l}=\frac{\lambda_{S}^{*ki}\lambda_{S}^{lj}}{4M_{S_{3}}^{2}}+\mathcal{O}(\frac{1}{16\pi^{2}})\,,$
(7.5)
$[\alpha_{H\ell}^{(3)}]_{i,j}=\frac{y_{T}^{i}y_{T}^{*j}}{4M_{\Sigma}^{2}}+\mathcal{O}(\frac{1}{16\pi^{2}})\,,$
(7.6) $[\alpha_{eB}]_{i,j}\simeq 0\,,$ (7.7)
$[\alpha_{eW}]_{i,j}\simeq\frac{3g_{W}y_{b}^{R}\,y_{T}^{i}y_{Q}^{j}}{16\pi^{2}}\frac{M_{\Psi_{Q}}}{M_{\Sigma}}\left(\frac{M_{\Psi_{Q}}^{2}-M_{S_{3}}^{2}+M_{S_{3}}^{2}\,\operatorname{Log}{\left[\frac{M_{S_{3}}^{2}}{M_{\Psi_{Q}}^{2}}\right]}}{(M_{\Psi_{Q}}^{2}-M_{S_{3}}^{2})^{2}}\right)\,,$
(7.8)

where the $\simeq$ means that we are neglecting Yukawa-suppressed
contributions and the notation of the Wilson coefficients follows the
convention of matchmakereft [57].

Explaining $R^{(*)}_{K}$ and CAA essentially fixes the ratios
$x_{S}\equiv\lambda_{S}^{*s\mu}\lambda_{S}^{b\mu}/M_{S_{3}}^{2}$ and
$x_{T}\equiv y_{T}^{\mu}/M_{\Sigma}$, up to one-loop corrections that break
this scale invariance in couplings over masses. However, the loop factor
suppression assures that observables are approximately flat on the values of
the masses (in a certain range). The $x_{T}$ ratio also enters in the
expression for $\Delta a_{\mu}$, but we have enough freedom with the couplings
$y_{b}^{R}$ and $x_{F}\equiv y_{Q}^{\mu}/M_{\Psi_{Q}}$ to fix both observables
to the desired value. Note also that, since they couple two and three heavy
fields, both $y_{b}$ and $x_{F}$ always generate coefficients at one-loop
order, so in principle one expects a wider parameter space in comparison to
other couplings.

In order to study the one-loop low-energy phenomenology of the model, we
computed the complete one-loop matching with matchmakereft [57]777The output
from matchmakereft is provided in an auxiliary file. and used smelli [85, 86,
87, 88] to construct a $\chi^{2}$ function from the observables at low energy
in terms of the Wilson coefficients matched at a high scale. We then performed
a fit, minimizing the $\chi^{2}$ function using the iminuit [89] python
package; for this fit we considered the observables given in smelli in the
classes of leptonic observables (which include magnetic dipole moments for
leptons), lepton flavour universality for neutral currents (for anomalies in
$B$ decays such as $R^{(*)}_{K}$), EWPO (which contain observables sensitive
to deviations in the electroweak vertices) and quark flavour related
observables (which include meson decays and mixing)888Further details on all
the observables included in these classes can be found in Appendix D of [85]
where they are listed.. Besides these observables taken directly from smelli,
we also added $\epsilon_{\mu\mu}$.

In addition to the flavour assumptions commented above, we further imposed the
couplings to be lower than 1, and fixed the masses to $M_{\Sigma}=3.4$ TeV,
$M_{S_{3}}=2$ TeV and $M_{\Psi_{Q}}=4.6$ TeV. As explained, the observables
were essentially flat in masses between 1-5 TeV, so we chose this hierarchy as
an example that avoided current experimental detection limits but that could
be reached by upcoming searches. Other hierarchies and values for the masses
between 1-5 TeV are also feasible and yield similar results.

The best fit point in this setup is:

$\begin{aligned} &x_{F}=0.2\;\operatorname{TeV}{}^{-1},\\\
&x_{T}=0.17\;\operatorname{TeV}{}^{-1},\\\
&y_{b}^{L}=0.10,\end{aligned}\qquad\qquad\begin{aligned}
&x_{S}=0.00078\;\operatorname{TeV}{}^{-2},\\\ &\lambda_{S}^{b\mu}=0.07,\\\
&y_{b}^{R}=0.13,\end{aligned}$ (7.9)

which corresponds to a global pull from the SM of 6.2 $\sigma$. For the
calculation of this pull, we considered the observables that were fitted,
i.e., the ones available in smelli in the classes EWPO, leptonic observables,
lepton flavour universality for neutral currents and quark flavour
observables; we do not include $\epsilon_{\mu\mu}$ since we did not consider
its correlations with the observables in the stated classes. In Table 4, we
collect some of the individual pulls, both from experiment and SM, for the
most relevant observables.

Table 4: Individual values for the SM prediction, model prediction,
experimental measure and pulls of the most relevant observables as given by
smelli. Definitions of these observables (and updated values) can be read from
the flavio [88] documentation, which smelli uses to calculate contributions
from the Wilson coefficients to low-energy observables.

Observable |  SM Prediction |  Model Prediction | Experiment |  Pull model ($\sigma$) | Pull SM ($\sigma$)
---|---|---|---|---|---
$a_{\mu}$ | 0.0011659181(4) | 0.0011659201(4) | 0.0011659206(4) | 0.82 | 4.22
$\langle R_{\mu e}\rangle(B^{\pm}\rightarrow K^{\pm}\ell^{+}\ell^{-})^{[1.0,6.0]}$ | 1 | 0.79 | 0.85(5) | 1.41 | 3.21
$\langle R_{\mu e}\rangle(B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-})^{[0.045,1.1]}$ | 0.93 | 0.87 | 0.65(12) | 1.98 | 2.39
$\langle R_{\mu e}\rangle(B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-})^{[1.1,6.0]}$ | 0.99 | 0.79 | 0.68(12) | 1.04 | 2.55
$\epsilon_{\mu\mu}$ | 0 | 0.40e-3 | 0.58(15)e-3 | 1.20 | 3.87
$\Delta M_{s}$ | 1.25(8)e-11 | 1.25(8)e-11 | 1.1688(14)e-11 | 1.08 | 1.07
$\Delta M_{d}$ | 3.9(5)e-13 | 3.9(5)e-13 | 3.33(15)e-13 | 1.25 | 1.25
$M_{W}$ | 80.36 | 80.35 | 80.379(12) | 2.28 | 1.72
$A_{e}$ | 0.147 | 0.146 | 0.151(2) | 2.77 | 2.22

In Figs. 3 and 4 we show the 1- and 2-$\sigma$ regions from the best-fit point
for the model parameters that generate tree-level Wilson coefficients using
the global likelihood provided by smelli. For each point in the grid, we
minimize the value of $\chi^{2}$ by varying the other parameters of the model.
The profiles in the rest of the variables that contribute only at one-loop are
very similar to what one would expect from only taking the tree level
solutions (i.e. they feature somewhat flat directions), showing that the one-
loop effects give us enough freedom to explain $\Delta a_{\mu}$ without
spoiling the anomalies independently explained at tree level. These flat
directions are illustrated in the plots of Fig. 5, where the coupling on the
$x$-axis only contributes at one-loop; note that a vanishing $y_{b}^{L}$
coupling is allowed since it does not contribute to the relevant anomalies.

Figure 3: The 1 (2)-$\sigma$ regions in green (yellow) around the model’s best
fit point. For each $x_{S}$ and $x_{T}$ point in the plot, the other couplings
were marginalized in order to minimize the $\chi^{2}$. The observables
included in the fit were the ones available in smelli in the classes EWPO,
leptonic observables, lepton flavour universality for neutral currents and
quark flavour observables, and $\epsilon_{\mu\mu}$. Figure 4: The 1
(2)-$\sigma$ regions in green (yellow) around the model’s best fit point. For
each point in the plot, the other couplings were marginalized in order to
minimize the $\chi^{2}$. The observables included in the fit were the ones
available in smelli in the classes EWPO, leptonic observables, lepton flavour
universality for neutral currents and quark flavour observables, and
$\epsilon_{\mu\mu}$. Values of $\lambda_{S}^{b\mu}$ very close to zero were
not plotted because that would imply $\lambda_{S}^{s\mu}$ larger than 1 for a
fixed $x_{S}$ and $M_{S_{3}}$.

Figure 5: The 1 (2)-$\sigma$ regions in green (yellow) around the model’s best
fit point. For every plot, the vertical axis represents a coupling that enters
observables a tree-level, whereas the horizontal axis represents one that only
contributes at one-loop. For each point in the plot, the other couplings were
marginalized in order to minimize the $\chi^{2}$. The observables included in
the fit were the ones available in smelli in the classes EWPO, leptonic
observables, lepton flavour universality for neutral currents and quark
flavour observables, and $\epsilon_{\mu\mu}$.

##  8 Conclusions

We have studied in detail the chirally enhanced contribution to $a_{\mu}$
arising from the bridge topology shown in Fig. 2. Results for this
contribution have been presented for arbitrary extensions of the SM, in terms
of the gauge representations of the heavy fields.

The particular representations of the combination of heavy fields which can
generate the bridge diagram have been classified in terms of 2- or 3-field
extensions. The full results for the contribution to $a_{\mu}$ of all possible
2-field extensions have been presented; for 3-field extensions, due to the
infinite number of possibilities, we presented the results for extensions with
singlet, doublet and triplet representations $SU(2)$. For higher
representations, the general results presented in Section 3 (and in Appendix A
for the box diagrams) can be used to compute the contribution to $a_{\mu}$.

Within these results, we arrive at a class of 2-field extensions with a
fermion and a scalar which had previously been discarded in the literature but
that, with this chirally enhanced contribution from the bridge diagram, can in
principle be viable solutions to the $a_{\mu}$ anomaly. The studied 3-field
extensions which generate the bridge topology also represent a completely new
class of models to explain the $a_{\mu}$ anomaly.

An interesting avenue to pursue is to study the specific phenomenology of
these models to explore the parameter space in which they can explain
$a_{\mu}$ and also other anomalous observations. For example, within the
2-field fermion-scalar extensions, some include a scalar singlet which could
play the role of Dark Matter.

An exercise in this direction was also performed in this work, in which we
explored the full one-loop phenomenology of a 3-field extension which not only
generates the bridge topology but can also explain the neutral $B$-anomalies,
$R_{K}$ and $R_{K^{*}}$, and the CAA. This model, which we denote by the
triple triplet, is constituted by a triplet leptoquark – responsible for
explaining the $B$-anomalies at tree-level –, a triplet vector-like lepton –
which accounts for the CAA at tree-level – and a vector-like quark – needed to
generate the bridge topology.

Using matchmakereft [57] to perform the full one-loop matching of this model –
which we provide in an auxiliary file – and smelli [85], we have shown the
allowed parameter region which not only explains the aforementioned anomalies
but also respects bounds from an array of other observations.

## Acknowledgments

We are grateful to José Santiago, Javier Fuentes-Martín, Manuel Morales,
Achilleas Lazopoulos, Mikael Chala, Susanne Westhoff and Maria Ramos for
useful discussions. G.G is supported by LIP (FCT, COMPETE2020-Portugal2020,
FEDER, POCI-01-0145-FEDER-007334) as well as by INCD under the project
CPCA-A1-401197-2021, through Proyecto de Excelencia (P18-FR-4314) grant and by
FCT under the project CERN/FIS-PAR/0032/2021 and under the grant
SFRH/BD/144244/2019. P.O is supported by the Ministry of Science and
Innovation and SRA (10.13039/501100011033) under grant PID2019-106087GB-C22,
by the Junta de Andalucía grant FQM 101 and by an FPU grant from the Spanish
government.

##  Appendix A Box diagram results

(a)

(b)

Figure 6: _Left:_ Box diagram contribution to $\alpha_{e\gamma}$ with 2 heavy
fermion propagators. _Right:_ Box diagram contribution to $\alpha_{e\gamma}$
with 1 heavy fermion propagator and 1 heavy scalar propagator. Double lines
represent heavy particles whereas single lines are SM particles. The gauge
boson ($B$ or $W$) is represented outside the diagram since it can be attached
to any of the internal propagators.  Figure 7: Box diagram contribution to
$\alpha_{e\gamma}$ with all heavy internal propagators. Double lines represent
heavy particles, whereas single lines are SM particles. The gauge boson ($B$
or $W$) is represented outside the diagram since it can be attached to any of
the internal propagators.

For completeness, we present in this appendix the generic results for the
contribution to $a_{\mu}$ by the box diagrams. Note that, while in our study
these results are important for only some SM gauge representations of the BSM
fields, these contributions are the ones commonly considered in the
literature.

For the box diagram with two heavy fermion propagators, Fig. 6(a), the
schematic Lagrangian reads

$\displaystyle\mathcal{L}\supset$
$\displaystyle\,y_{R}T_{IJ}\overline{\Psi_{1}}_{I}\phi_{J}e_{R}+y_{L}T_{IJK}\overline{\ell_{L}}_{,I}\phi^{\dagger}_{J}{\Psi_{2}}_{K}+y_{H}^{R}T^{H}_{IJK}\overline{\Psi_{2}}_{I}\phi_{J}P_{R}{\Psi_{1}}_{K}$
$\displaystyle+T^{H}_{IJK}y_{H}^{L}\overline{\Psi_{2}}_{I}\phi_{J}P_{L}{\Psi_{1}}_{K}+\mathrm{h.c.}\,,$
(A.1)

where we use the same conventions for the gauge interactions of $\Psi$ and
$\Phi$ as in Eq. (3.1).

The contribution to $\alpha_{e\gamma}$ is given by:

$\displaystyle\alpha_{e\gamma}^{2,2}=$
$\displaystyle\left(\frac{i}{4}\right)ey_{R}y_{L}\sum_{\chi=R,L}y_{H}^{\chi}\left[T_{IJ}T_{2JK}T^{\gamma}_{I^{\prime}I}T^{H}_{K2I^{\prime}}\gamma_{\Psi_{1}}^{\chi}\right.$
$\displaystyle\left.+T_{IJ}T_{2JK}T^{H}_{K^{\prime}2I}T^{\gamma}_{KK^{\prime}}\gamma_{\Psi_{2}}^{\chi}+T_{IJ}T_{2J^{\prime}K}T^{H}_{K2I}T^{\gamma}_{J^{\prime}J}\gamma_{\phi}^{\chi}\right],$
(A.2)

where $\chi$ sums over the right- and left-handed chiralities and the
kinematic factors read:

$\displaystyle\gamma_{\Psi_{1}}^{L}=0\,,$
$\displaystyle\gamma_{\Psi_{1}}^{R}=-\frac{i}{16\pi^{2}}\frac{M_{\Psi_{2}}\left(M_{\Psi_{1}}^{2}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)-M_{\Psi_{1}}^{2}+M_{\Psi_{2}}^{2}\right)}{M_{\Psi_{1}}\left(M_{\Psi_{1}}^{2}-M_{\Psi_{2}}^{2}\right)^{2}}\,,$
$\displaystyle\gamma_{\Psi_{2}}^{L}=0\,,$
$\displaystyle\gamma_{\Psi_{2}}^{R}=-\frac{i}{16\pi^{2}}\frac{M_{\Psi_{1}}\left(-M_{\Psi_{2}}^{2}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)+M_{\Psi_{1}}^{2}-M_{\Psi_{2}}^{2}\right)}{M_{\Psi_{2}}\left(M_{\Psi_{1}}^{2}-M_{\Psi_{2}}^{2}\right)^{2}}\,,$
$\displaystyle\gamma_{\Phi}^{L}=0\,,$
$\displaystyle\gamma_{\Phi}^{R}=\frac{-i}{16\pi^{2}M_{\Psi_{1}}M_{\Psi_{2}}}\,.$
(A.3)

For the box diagram with a light fermion in which the heavy fermion couples
with the right-handed muon, the Lagrangian reads:

$\displaystyle\mathcal{L}\supset$ $\displaystyle
y_{R}T^{e}_{IJ}\overline{\Psi_{1}}_{I}\Phi_{J}e_{R}+y_{L}T^{1}_{IJ}\overline{\psi}\phi_{J}P_{L}{\Psi_{1}}_{I}+y_{\Phi}T^{\Phi}_{IJ}\overline{\ell_{LI}}\Phi_{J}^{\dagger}P_{R}\psi+\mathrm{h.c.}\,,$
(A.4)

where $\psi$ is any light SM fermion which fits with the heavy field
representations. The resulting contribution to $\alpha_{e\gamma}$ is:

$\displaystyle\alpha_{e\gamma}^{22}=$
$\displaystyle\left(\frac{i}{4}\right)eNy_{R}y_{L}y_{\Phi}\left[T^{e}_{IJ}T^{\gamma}_{I^{\prime}I}T^{1}_{I^{\prime}2}T^{\Phi}_{2J}\gamma_{\Psi}\right.$
$\displaystyle\left.+T^{e}_{IJ}T^{1}_{I2}Y_{\psi}T^{\Phi}_{2J}\gamma_{\psi}+T^{e}_{IJ}T^{1}_{I2}T^{\gamma}_{JJ^{\prime}}T^{\Phi}_{2J^{\prime}}\gamma_{\Phi}\right],$
(A.5)

with the following kinematic factors:

$\displaystyle\gamma_{\Psi}=-\frac{M_{\Phi}^{2}\left(\left(M_{\Psi}^{2}+M_{\Phi}^{2}\right)\operatorname{Log}\left(\frac{M_{\Psi}^{2}}{M_{\Phi}^{2}}\right)-2M_{\Psi}^{2}+2M_{\Phi}^{2}\right)}{\left(M_{\Phi}^{2}-M_{\Psi}^{2}\right)^{3}}\,,$
$\displaystyle\gamma_{\psi}=\frac{-M_{\Phi}^{2}\operatorname{Log}\left(\frac{M_{\Psi}^{2}}{M_{\Phi}^{2}}\right)+M_{\Psi}^{2}-M_{\Phi}^{2}}{\left(M_{\Psi}^{2}-M_{\Phi}^{2}\right)^{2}}\,,$
$\displaystyle\gamma_{\Phi}=\frac{M_{\Psi}^{4}-2M_{\Psi}^{2}M_{\Phi}^{2}\operatorname{Log}\left(\frac{M_{\Psi}^{2}}{M_{\Phi}^{2}}\right)-M_{\Phi}^{4}}{\left(M_{\Psi}^{2}-M_{\Phi}^{2}\right)^{3}}\,.$
(A.6)

In the case the heavy fermion couples with the left-handed muon, the relevant
Lagrangian can be written as:

$\displaystyle\mathcal{L}\supset$
$\displaystyle\,y_{R}T^{\ell}_{IJK}\overline{\ell_{L}}_{I}\Phi_{J}^{\dagger}\Psi_{1K}+y_{L}T^{2}_{IJK}\overline{\Psi}_{I}\phi_{J}P_{L}\psi_{K}+y_{\Phi}T^{\Phi}_{IJ}\overline{\psi}_{I}\Phi_{J}e_{R}+\mathrm{h.c.}\,,$
(A.7)

resulting in the following contribution to $\alpha_{e\gamma}$:

$\displaystyle\alpha_{e\gamma}^{22}=$
$\displaystyle\left(\frac{i}{4}\right)eNy_{R}y_{L}y_{\Phi}\left[T^{\ell}_{2JI}T^{\gamma}_{II^{\prime}}T^{2}_{I^{\prime}2K}T^{\Phi}_{KJ}\gamma_{\Psi}\right.$
$\displaystyle\left.+T^{\ell}_{2JI}T^{2}_{I2K}T^{\gamma}_{KK^{\prime}}T^{\Phi}_{K^{\prime}J}\gamma_{\psi}+T^{\ell}_{2JI}T^{2}_{I2K}T^{\Phi}_{KJ^{\prime}}T^{\gamma}_{J^{\prime}J}\gamma_{\Phi}\right],$
(A.8)

where

$\displaystyle\gamma_{\Psi}=\frac{M_{\Phi}^{2}\left(\left(M_{\Psi_{1}}^{2}+M_{\Phi}^{2}\right)\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Phi}^{2}}\right)-2M_{\Psi_{1}}^{2}+2M_{\Phi}^{2}\right)}{\left(M_{\Psi_{1}}^{2}-M_{\Phi}^{2}\right)^{3}}\,,$
$\displaystyle\gamma_{\psi}=-\frac{\left(M_{\Psi_{1}}^{2}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Phi}^{2}}\right)-M_{\Psi_{1}}^{2}+M_{\Phi}^{2}\right)}{\left(M_{\Psi_{1}}^{2}-M_{\Phi}^{2}\right)^{2}}\,,$
$\displaystyle\gamma_{\Phi}=\frac{M_{\Psi}^{4}-2M_{\Psi}^{2}M_{\Phi}^{2}\operatorname{Log}\left(\frac{M_{\Psi}^{2}}{M_{\Phi}^{2}}\right)-M_{\Phi}^{4}}{\left(M_{\Psi}^{2}-M_{\Phi}^{2}\right)^{3}}\,.$
(A.9)

When there are only heavy propagators in the box diagram, Fig. 7, the relevant
Lagrangian reads:

$\displaystyle\mathcal{L}$ $\displaystyle\supset
y_{R}T^{1}_{IJ}\overline{\Psi_{1}}_{I}\Phi_{J}e_{R}+y_{L}T^{2}_{IJ}\overline{\ell_{L}}\Psi_{2I}\Phi^{\dagger}_{J}+y^{R}_{H}T_{IJK}\overline{\Psi_{2}}_{I}\phi_{J}P_{R}{\Psi_{1}}_{K}$
$\displaystyle+y^{L}_{H}T_{IJK}\overline{\Psi_{2}}_{I}\phi_{J}P_{L}{\Psi_{1}}_{K}+\mathrm{h.c.},$
(A.10)

and the resulting $\alpha_{e\gamma}$ is given by:

$\displaystyle\alpha_{e\gamma}^{22}=$
$\displaystyle\left(\frac{i}{4}\right)y_{R}y_{L}\sum_{\chi=R,L}y_{H}^{\chi}\left[T^{2}_{IJ}T^{H}_{I2K}T^{\gamma}_{KK^{\prime}}T^{1}_{K^{\prime}J}\gamma_{\Psi_{1}}^{\chi}\right.$
$\displaystyle\left.+T^{2}_{IJ}T^{\gamma}_{II^{\prime}}T_{I^{\prime}2K}T^{1}_{KJ}\gamma_{\Psi_{2}}^{\chi}+T^{2}_{IJ}T_{I2K}T_{KJ^{\prime}2}T^{\gamma}_{JJ^{\prime}}\gamma_{\Phi}^{\chi}\right],$
(A.11)

where

$\displaystyle\gamma_{\Psi_{1}}^{L}=$
$\displaystyle\frac{i}{16\pi^{2}}M_{\Phi}^{2}\bigg{[}(M_{\Psi_{2}}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Psi_{2}})\left(M_{\Phi}^{2}(M_{\Psi_{2}}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Psi_{2}})\left(M_{\Psi_{1}}^{2}\left(M_{\Phi}^{2}-2M_{\Psi_{2}}^{2}\right)\right.\right.\bigg{.}$
$\displaystyle\bigg{.}\left.\left.+M_{\Phi}^{4}\right)\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Phi}^{2}}\right)-(M_{\Phi}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Phi})(M_{\Psi_{2}}-M_{\Phi})(M_{\Psi_{2}}+M_{\Phi})\left(M_{\Psi_{1}}^{2}\left(M_{\Psi_{2}}^{2}-2M_{\Phi}^{2}\right)\bigg{.}\right.\right.$
$\displaystyle\bigg{.}\left.\left.+M_{\Psi_{2}}^{2}M_{\Phi}^{2}\right)\right)+M_{\Psi_{2}}^{4}\left(M_{\Psi_{1}}^{2}-M_{\Phi}^{2}\right)^{3}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)\bigg{]}\times$
$\displaystyle\frac{1}{(M_{\Psi_{1}}-M_{\Psi_{2}})^{2}(M_{\Psi_{1}}+M_{\Psi_{2}})^{2}\left(M_{\Phi}^{2}-M_{\Psi_{1}}^{2}\right)^{3}(M_{\Psi_{2}}-M_{\Phi})^{2}(M_{\Psi_{2}}+M_{\Phi})^{2}}\,,$
(A.12)

$\displaystyle\gamma^{R}_{\Psi_{1}}=$
$\displaystyle\frac{i}{16\pi^{2}}M_{\Psi_{1}}\left(M_{\Psi_{2}}(M_{\Psi_{2}}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Psi_{2}})\left((M_{\Psi_{1}}-M_{\Phi})(M_{\Psi_{1}}+M_{\Phi})\right.\right.$
$\displaystyle\left.\left.(M_{\Psi_{2}}-M_{\Phi})(M_{\Psi_{2}}+M_{\Phi})\left(M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{2}-3M_{\Psi_{2}}^{2}M_{\Phi}^{2}+2M_{\Phi}^{4}\right)+M_{\Phi}^{4}(M_{\Psi_{2}}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Psi_{2}})\right.\right.$
$\displaystyle\left.\left.\left(M_{\Psi_{1}}^{2}+2M_{\Psi_{2}}^{2}-3M_{\Phi}^{2}\right)\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Phi}^{2}}\right)\right)+M_{\Psi_{2}}^{3}\left(M_{\Psi_{1}}^{2}-M_{\Phi}^{2}\right)^{3}\left(M_{\Psi_{2}}^{2}-2M_{\Phi}^{2}\right)\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)\right)\times$
$\displaystyle\frac{1}{(M_{\Psi_{1}}-M_{\Psi_{2}})^{2}(M_{\Psi_{1}}+M_{\Psi_{2}})^{2}(M_{\Psi_{1}}-M_{\Phi})^{3}(M_{\Psi_{1}}+M_{\Phi})^{3}(M_{\Psi_{2}}-M_{\Phi})^{2}(M_{\Psi_{2}}+M_{\Phi})^{2}}\,,$
(A.13)

$\displaystyle\gamma^{L}_{\Psi_{2}}=-\frac{i}{16\pi^{2}}\left[M_{\Psi_{1}}^{6}\left(-\left(M_{\Psi_{2}}^{2}-M_{\Phi}^{2}\right)^{3}\right)\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Phi}^{2}}\right)+M_{\Psi_{1}}^{4}(M_{\Psi_{1}}-M_{\Phi})(M_{\Psi_{1}}+M_{\Phi})\right.$
$\displaystyle\left.(M_{\Psi_{2}}-M_{\Phi})^{3}(M_{\Psi_{2}}+M_{\Phi})^{3}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)+(M_{\Psi_{1}}-M_{\Phi})(M_{\Psi_{1}}+M_{\Phi})\left(M_{\Phi}^{2}(M_{\Psi_{1}}-M_{\Psi_{2}})(M_{\Psi_{1}}+M_{\Psi_{2}})\right.\right.$
$\displaystyle\left.\left.(M_{\Psi_{2}}-M_{\Phi})(M_{\Psi_{2}}+M_{\Phi})\left(M_{\Psi_{1}}^{2}\left(M_{\Psi_{2}}^{2}+M_{\Phi}^{2}\right)-2M_{\Psi_{2}}^{2}M_{\Phi}^{2}\right)+\left(M_{\Psi_{1}}^{4}M_{\Psi_{2}}^{6}-M_{\Phi}^{6}\left(M_{\Psi_{1}}^{2}-M_{\Psi_{2}}^{2}\right)^{2}\right.\right.\right.$
$\displaystyle\left.\left.\left.+M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{4}M_{\Phi}^{2}\left(M_{\Psi_{2}}^{2}-3M_{\Psi_{1}}^{2}\right)+M_{\Psi_{2}}^{2}M_{\Phi}^{4}\left(M_{\Psi_{1}}^{4}+M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{2}-M_{\Psi_{2}}^{4}\right)\right)\operatorname{Log}\left(\frac{M_{\Psi_{2}}^{2}}{M_{\Phi}^{2}}\right)\right)\right]\times$
$\displaystyle\frac{1}{(M_{\Psi_{1}}-M_{\Psi_{2}})^{2}(M_{\Psi_{1}}+M_{\Psi_{2}})^{2}(M_{\Psi_{1}}-M_{\Phi})^{2}(M_{\Psi_{1}}+M_{\Phi})^{2}(M_{\Psi_{2}}-M_{\Phi})^{3}(M_{\Psi_{2}}+M_{\Phi})^{3}}\,,$
(A.14)

$\displaystyle\gamma^{R}_{\Psi_{2}}=\frac{i}{16\pi^{2}}M_{\Psi_{1}}M_{\Psi_{2}}\left(2M_{\Psi_{1}}^{2}(M_{\Psi_{1}}-M_{\Phi})(M_{\Psi_{1}}+M_{\Phi})\left(M_{\Phi}^{2}-M_{\Psi_{2}}^{2}\right)^{3}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)\right.$
$\displaystyle\left.+M_{\Psi_{1}}^{4}\left(M_{\Psi_{2}}^{2}-M_{\Phi}^{2}\right)^{3}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Phi}^{2}}\right)+(M_{\Phi}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Phi})\left((M_{\Psi_{2}}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Psi_{2}})\right.\right.$
$\displaystyle\left.\left.(M_{\Psi_{2}}-M_{\Phi})(M_{\Psi_{2}}+M_{\Phi})\left(M_{\Psi_{1}}^{2}\left(M_{\Psi_{2}}^{2}-3M_{\Phi}^{2}\right)+2M_{\Phi}^{4}\right)+\left(M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{6}+M_{\Psi_{2}}^{4}M_{\Phi}^{2}\left(M_{\Psi_{2}}^{2}-3M_{\Psi_{1}}^{2}\right)\right.\right.\right.$
$\displaystyle\left.\left.\left.+M_{\Phi}^{4}\left(-2M_{\Psi_{1}}^{4}+6M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{2}-3M_{\Psi_{2}}^{4}\right)\right)\operatorname{Log}\left(\frac{M_{\Psi_{2}}^{2}}{M_{\Phi}^{2}}\right)\right)\right)\times$
$\displaystyle\frac{1}{(M_{\Psi_{1}}-M_{\Psi_{2}})^{2}(M_{\Psi_{1}}+M_{\Psi_{2}})^{2}(M_{\Psi_{1}}-M_{\Phi})^{2}(M_{\Psi_{1}}+M_{\Phi})^{2}(M_{\Psi_{2}}-M_{\Phi})^{3}(M_{\Psi_{2}}+M_{\Phi})^{3}}\,,$
(A.15)

$\displaystyle\gamma^{L}_{\Phi}=\frac{iM_{\Phi}^{2}}{16\pi^{2}}\left[\frac{1}{\left(M_{\Psi_{2}}^{2}-M_{\Phi}^{2}\right)^{3}}\left(2\left(M_{\Psi_{1}}^{4}M_{\Psi_{2}}^{4}+M_{\Phi}^{6}\left(M_{\Psi_{1}}^{2}+M_{\Psi_{2}}^{2}\right)-3M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{2}M_{\Phi}^{4}\right)\operatorname{Log}\left(\frac{M_{\Psi_{2}}^{2}}{M_{\Phi}^{2}}\right)\right.\right.$
$\displaystyle\left.\left.-(M_{\Phi}-M_{\Psi_{1}})(M_{\Psi_{1}}+M_{\Phi})(M_{\Psi_{2}}-M_{\Phi})(M_{\Psi_{2}}+M_{\Phi})\left(M_{\Phi}^{2}\left(M_{\Psi_{1}}^{2}+M_{\Psi_{2}}^{2}\right)-3M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{2}+M_{\Phi}^{4}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{2M_{\Psi_{1}}^{4}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)}{M_{\Psi_{1}}^{2}-M_{\Psi_{2}}^{2}}\right)\right]\times\frac{1}{\left(M_{\Psi_{1}}-M_{\Phi}\right)^{3}\left(M_{\Psi_{1}}+M_{\Phi}\right)^{3}}\,,$
(A.16)

$\displaystyle\gamma^{R}_{\Phi}=\frac{i}{16\pi^{2}}M_{\Psi_{1}}M_{\Psi_{2}}\left(2M_{\Psi_{1}}^{2}M_{\Phi}^{2}\left(M_{\Psi_{2}}^{2}-M_{\Phi}^{2}\right)^{3}\operatorname{Log}\left(\frac{M_{\Psi_{1}}^{2}}{M_{\Psi_{2}}^{2}}\right)+(M_{\Psi_{1}}-M_{\Psi_{2}})(M_{\Psi_{1}}+M_{\Psi_{2}})\right.$
$\displaystyle\left.\left((M_{\Psi_{1}}-M_{\Phi})(M_{\Psi_{1}}+M_{\Phi})(M_{\Psi_{2}}-M_{\Phi})(M_{\Psi_{2}}+M_{\Phi})\left(M_{\Phi}^{2}\left(M_{\Psi_{1}}^{2}+M_{\Psi_{2}}^{2}\right)+M_{\Psi_{1}}^{2}M_{\Psi_{2}}^{2}-3M_{\Phi}^{4}\right)\right.\right.$
$\displaystyle\left.\left.-2M_{\Phi}^{2}\left(M_{\Psi_{1}}^{4}M_{\Psi_{2}}^{2}+M_{\Psi_{1}}^{2}\left(M_{\Psi_{2}}^{4}-3M_{\Psi_{2}}^{2}M_{\Phi}^{2}\right)+M_{\Phi}^{6}\right)\operatorname{Log}\left(\frac{M_{\Psi_{2}}^{2}}{M_{\Phi}^{2}}\right)\right)\right)\times$
$\displaystyle\frac{1}{(-M_{\Psi_{1}}^{2}+M_{\Psi_{2}}^{2})(M_{\Psi_{1}}^{2}-M_{\Phi}^{2})^{3}(M_{\Psi_{2}}^{2}-M_{\Phi}^{2})^{3}}\,.$
(A.17)

We cross-checked this last result of the box diagram with only heavy internal
propagators with Eq. (4.4) of Ref. [53] in the limit of degenerate masses and
found perfect agreement.

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