IFIC/22-12, FTUV-22-0331

Anomaly-free ALP from non-Abelian flavor symmetry

C<EMAIL_ADDRESS>M.L. López-
<EMAIL_ADDRESS>A<EMAIL_ADDRESS>O.
<EMAIL_ADDRESS>J.M.
<EMAIL_ADDRESS>
a School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
b Universidad Politécnica de Madrid, ETSIST, C/Nikola Tesla s/n, 28031 Madrid,
Spain.
c Laboratory of High Energy and Computational Physics, NICPB, Rävala 10, 10143
Tallinn, Estonia
d Departament de Física Tèorica, Universitat de València & IFIC, Universitat
de València & CSIC,
Dr. Moliner 50, E-46100 Burjassot (València), Spain.
e CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics
Chinese Academy of Sciences, Beijing 100190, P. R. China.
f School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, P. R. China

###### Abstract

Motivated by the Xenon1T excess in electron-recoil measurements, we
investigate the prospects of probing axion-like particles (ALP) in lepton
flavor violation experiments. In particular, we identify such ALP as a pseudo-
Goldstone from the spontaneous breaking of the flavor symmetries that explain
the mixing structure of the Standard Model leptons. We present the case of the
flavor symmetries being a non-Abelian $U(2)$ and the ALP originating from its
$U(1)$ subgroup, which is anomaly-free with the Standard Model group. We build
two explicit realistic examples that reproduce leptonic masses and mixings and
show that the ALP which is consistent with Xenon1T anomaly could be probed by
the proposed LFV experiments.

##  1 Introduction

The structure of fermion masses and mixings remains one of the more mysterious
and difficult questions still unanswered in high energy physics. The Standard
Model (SM) accommodates the observed flavor structures through the Yukawa
couplings of the fermions with the Higgs, but it can not explain their
intricate structures or even the existence of three families of fermions.

The use of flavor symmetries, which generate the Yukawa couplings after
spontaneous symmetry breaking, seems to be the most promising path to provide
answers to these questions. In flavor models, we extend the SM with a flavor
symmetry, under which the SM fermions are charged. The SM Yukawa couplings are
forbidden by the symmetry, but higher dimensional operators with additional
scalars, the flavons, and fermionic mediators are possible. After spontaneous
symmetry breaking, the flavon vevs, normalized by the mediator masses,
generate the dimensionless Yukawa couplings in a Froggatt-Nielsen mechanism
[1]. In the literature, there is a large variety of models with different
flavor symmetries [1, 2, 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], either continuous or
discrete, Abelian or non-Abelian, that try to explain the structure of Yukawas
and neutrino masses. All these flavor symmetries can be gauged or global
symmetries but both possibilities have different physical consequences. In
particular, as it is well-known, when the flavor symmetry is global, its
spontaneous breaking generates light (pseudo-)Goldstone bosons that can have
important effects in the low-energy phenomenology.

Recently, the Xenon1T experiment reported an excess on electronic recoil
events [31], indicating the possible existence of an axion-like particle (ALP)
coupling to the electrons [32, 33, 34]. Then, it would be interesting to
identify such ALP as the "flavored" pseudo-Goldstone boson accounting for the
mixing structure of lepton sectors. However, to avoid the constraints from
astrophysics observations, the global symmetry generating the ALP should be
anomaly free with electromagnetic fields, resulting in additional requirements
on the structure of this global symmetry. Actually such leptophilic ALP has
attracted much attentions in these days [35, 36, 37, 34, 38, 39, 40, 41, 42,
43].

In a recent paper [33], we studied an axion-like particle (ALP) with family-
dependent charges as a possible source for the recent Xenon1T excess. In these
models, we have a $U(1)_{f}$ global symmetry, part of a bigger flavor
symmetry, spontaneously broken by the vev of a complex scalar field, $\phi$,
whose angular component is identified with an ALP. The couplings of this
$\phi$ field are flavor dependent (in the lepton sector) and thus the ALP will
have lepton flavor-changing couplings. These couplings are basically
determined by the mixing matrices diagonalizing the charged lepton Yukawas
from the flavor basis with diagonal $U(1)_{f}$ charges. If this $U(1)_{f}$
symmetry is the only responsible for the observed hierarchy among the charged-
lepton generations, the charged-lepton mixing matrices tend to be CKM-like.
Then, ALP flavor changing effects are small and out of reach for the expected
sensitivity of proposed experiments like Mu3e or MEGII-fwd.

However, from the observed PMNS matrix [44, 45], we know that leptonic mixings
are large and, in fact, nearly maximal in some sectors. Thus, we can naturally
expect large mixings both in the neutrino and charged-lepton mass matrices.
Therefore, it is equally legitimate to explore the possibility that these
large mixings come mainly from the charged-lepton or from the neutrino sector.
In this paper, we follow the large charged-lepton mixing possibility and
explore the associated phenomenology in the presence of a "flavored" pseudo-
Goldstone boson. As pointed out in [33], in this case, we would have larger
LFV effects that could be observable in the near-future proposed experiments.
We will explore two different realistic flavor models with large charged-
lepton mixings, able to reproduce lepton masses and mixings. Both examples are
based in a non-Abelian $U(2)_{f}\,=\,SU(2)_{f}\,\times\,U(1)_{f}$ flavor
symmetry, where the $U(1)_{f}$ is global and anomaly-free, with two different
assignments for the $U(2)_{f}$ representations.

The paper is organized as follows. In the next section, we obtain the ALP
couplings to SM leptons and show that they depend only on the charges and
charged-lepton mixing matrices. Then we list the constraints and sensitivity
from LFV experiments and Xenon1T observations. Section 3 analyzes the required
ingredients to have sizeable LFV ALP couplings in flavor models. In section 4
we build two different realizations of $U(2)_{f}$, assigning to a doublet the
first and second families in 4.1 and the second and third families in 4.2.
Finally, in section 5 we present our conclusions.

##  2 ALP couplings and phenomenology

In this section, we present the ALP couplings with the SM fermions and the
phenomenological constraints that must be taken into account for our analysis.

In general, if we have several flavon scalars charged under the $U(1)_{f}$
group, the pseudoscalar parts of those fields, $a_{i}$, mix to produce the
physical ALP, $a$, as

$a~{}=~{}\sum_{i}\,\frac{Q_{i}\,v_{i}\,a_{i}}{\sqrt{\sum\,Q^{2}_{j}\,v_{j}^{2}}},$
(1)

where $Q_{i}$ refers to the charge of the flavons under $U(1)_{f}$ and $v_{i}$
to the vevs.

Due to the Nambu-Goldstone nature of the ALP field, the interactions between
the SM fermions and the ALP are derivative. In this case, they are flavor
dependent if $U(1)_{f}$ couplings are family-dependent, and, in the mass
basis, are given by

$-{\cal
L}_{ae}~{}=~{}i\frac{\partial_{\mu}a}{2f_{a}}\,\bar{e}_{i}\,\gamma^{\mu}\left(V^{e}_{ij}\,+\,\gamma^{5}A^{e}_{ij}\right)\,e_{j},$
(2)

with $f_{a}\simeq{\mathcal{O}}(v_{i})$ the ALP decay constant and the axial
and vector couplings defined as

$\displaystyle V^{e}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(U^{e\,\dagger}_{R}x_{R}U^{e}_{R}\;+\;U^{e\,\dagger}_{L}x_{L}U^{e}_{L}\right),$
(3) $\displaystyle A^{e}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(U^{e\,\dagger}_{R}x_{R}U^{e}_{R}\;-\;U^{e\,\dagger}_{L}x_{L}U^{e}_{L}\right).$
(4)

Here the diagonal matrices $x_{L}$ and $x_{R}$ are determined by the
$U(1)_{f}$ charges of the corresponding left- (LH) and right-handed (RH)
charged leptons:

$x_{L}\>=\>{\rm Diag}\left(Q_{1},\,Q_{2},\,Q_{3}\right),\hskip
28.45274ptx_{R}\>=\>{\rm Diag}\left(q_{1},\,q_{2},\,q_{3}\right),$ (5)

and $U^{e}_{L}$, $U^{e}_{R}$ are the two rotations that diagonalize the lepton
mass matrix

$U^{e\,\dagger}_{L}\;M_{e}\;U^{e}_{R}~{}=~{}{\rm
diag}\left(m_{e},\,m_{\mu},\,m_{\tau}\right).$ (6)

By using unitarity of $U^{e}_{L}$ and $U^{e}_{R}$, eqs. (3) and (4) can be
written as:

$\displaystyle V^{e}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(q_{3}+Q_{3})\,\delta_{ij}\;+\;(q_{1}-q_{3})\,{U^{e\,*}_{R}}_{1i}{U^{e}_{R}}_{1j}\;+\;(q_{2}-q_{3})\,{U^{e\,*}_{R}}_{2i}{U^{e}_{R}}_{2j}\right.$
(7)
$\displaystyle\left.\;+\;(Q_{1}-Q_{3})\,{U^{e\,*}_{L}}_{1i}{U^{e}_{L}}_{1j}\;+\;(Q_{2}-Q_{3})\,{U^{e\,*}_{L}}_{2i}{U^{e}_{L}}_{2j}\right],$
$\displaystyle A^{e}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(q_{3}-Q_{3})\,\delta_{ij}\;+\;(q_{1}-q_{3})\,{U^{e\,*}_{R}}_{1i}{U^{e}_{R}}_{1j}\;+\;(q_{2}-q_{3})\,{U^{e\,*}_{R}}_{2i}{U^{e}_{R}}_{2j}\right.$
(8)
$\displaystyle\left.\;-\;(Q_{1}-Q_{3})\,{U^{e\,*}_{L}}_{1i}{U^{e}_{L}}_{1j}\;-\;(Q_{2}-Q_{3})\,{U^{e\,*}_{L}}_{2i}{U^{e}_{L}}_{2j}\right].$

ALP interactions, at scales much below the symmetry breaking scale, are given
by these couplings, $V^{e}$ and $A^{e}$. From eqs. (7) and (8), it can be seen
that, for non-equal charges ${\mathcal{O}}(1)$, the size of flavor-changing
effects is basically determined by the LH and RH mixing, $U^{e}_{L}$ and
$U^{e}_{R}$.

As discuss at refs.[46, 35], flavored ALPs can be searched through LFV
processes such as $\ell_{j}\to\ell_{i}\,a$, whose branching ratio is given by

${\rm
BR}\left({\ell_{i}\to\ell_{j}a}\right)\,=\,\frac{m_{\ell_{i}}^{3}}{16\pi\Gamma(\ell_{j})}\frac{\left|C_{ij}^{e}\right|^{2}}{4\,f_{a}^{2}}\left(1-\frac{m_{a}^{2}}{m^{2}_{\ell_{i}}}\right)^{2},$
(9)

with
$\left|C^{e}_{ij}\right|^{2}=\left|V^{e}_{ij}\right|^{2}+\left|A^{e}_{ij}\right|^{2}$.
Limits from the LFV decays $\ell_{j}\to\ell_{i}\gamma$ and $\ell_{j}\to
3\ell_{i}$ can be imposed over the ALP process in such a way that, once the
ALP couplings to leptons are known, they can be translated into bounds on
$f_{a}$. Present and expected sensitivities on the LFV transitions involving
the ALP are collected at Table 1. The strongest restrictions come from the
$\mu\to e\,a$ decay and imply,

$\displaystyle f_{a}^{\text{Jodidio}}$ $\displaystyle\geq$
$\displaystyle\left(2.7\times
10^{9}\,\text{GeV}\right)\left|C_{21}^{e}\right|,$ (10) $\displaystyle
f_{a}^{\text{Mu3e}}$ $\displaystyle\geq$ $\displaystyle\left(1.6\times
10^{10}\,\text{GeV}\right)\left|C_{21}^{e}\right|.$ (11)

Lepton decay |  BR limit | Experiment
---|---|---
Present best limits
${\rm BR}\left({\mu\to e\,a}\right)$ | $<2.6\cdot 10^{-6}$ |  Jodidio et al. [47]
${\rm BR}\left({\mu\to e\,a}\right)$ | $<2.1\cdot 10^{-5}$ |  TWIST [48]
${\rm BR}\left({\mu\to e\,a\,\gamma}\right)$ | $<1.1\cdot 10^{-9}$ |  Crystal Box [49]
${\rm BR}\left({\tau\to e\,a}\right)$ | $<2.7\cdot 10^{-3}$ |  ARGUS [50]
${\rm BR}\left({\tau\to\mu\,a}\right)$ | $<4.5\cdot 10^{-3}$ |  ARGUS [50]
Projections of running experiments
${\rm BR}\left({\mu\to e\,a}\right)$ | $<1.3\cdot 10^{-7}$ | MEGII-fwd [51, 35]
${\rm BR}\left({\tau\to e\,a}\right)$ | $<8.4\cdot 10^{-6}$ | Belle-II
${\rm BR}\left({\tau\to\mu\,a}\right)$ | $<1.6\cdot 10^{-5}$ | Belle-II
Projections of planned experiments
${\rm BR}\left({\mu\to e\,a}\right)$ | $<7.3\cdot 10^{-8}$ |  Mu3e [52]

Table 1:  Limits over the axion decay constant from lepton decays. Belle-II
limits are derived from the simulated result at Belle [53] by rescaling the
luminosity [35].

In addition, there is an interesting proposal at PSI, the MEGII-fwd [35]. This
proposal is based on a detector in the forward direction, where the SM
background is suppressed, to collect energetic forward positrons. If we assume
perfect $\mu^{+}$ polarization and with the detector in the forward direction,
this decay is only sensitive to $|V^{e}_{21}+A^{e}_{21}|^{2}$ and not to
$(V-A)$ couplings [35]. Therefore, the bound on $f_{a}$ would be,

$\displaystyle f_{a}^{\text{MEGII-fwd}}$ $\displaystyle\geq$
$\displaystyle\left(1.2\times
10^{10}\,\text{GeV}\right)\left|V_{21}^{e}+A_{21}^{e}\right|,$ (12)

where the most stringent constraint is obtained in the case of
$V_{21}^{e}=A_{21}^{e}$. Moreover, if we require the ALP to explain the
Xenon1T excess [31], it can only constitute the $\sim 7\%$ of the total dark
matter abundance and its coupling to electrons must be666 In ref.[32], eq.(2)
and discussion below, it is explained that the Xenon1T excess implies
$f_{a}/q_{e}\simeq\sqrt{r}\,10^{10}\,{\rm GeV}$, with $r$ the fraction of DM
constituted by the ALP. Then, $g_{ae}\simeq 5\times 10^{-14}/\sqrt{r}$. In our
notation, $g_{ae}$ is the pseudoscalar coupling of the axion with the
electron, $P_{11}^{e}/f_{a}$, with $P_{ij}^{e}=(m_{i}+m_{j})A_{ij}^{e}$. Then,
for $r=0.07$, $A_{11}^{e}\simeq 2\times
10^{-13}/(2\,m_{e})\,f_{a}=10^{-13}\,f_{a}/m_{e}$. As our models fix
$A_{11}^{e}$, a prediction for $f_{a}$ can be derived as
$f_{a}=A_{11}^{e}\,m_{e}/10^{-13}$.  [32]:

$A^{e}_{11}\simeq 10^{-13}\frac{f_{a}}{m_{e}}\,,\hskip 14.22636pt{\rm
for}~{}m_{a}\in[2,3]\,{\rm keV}.$ (13)

Furthermore, for this range of masses, the ALP has to be anomaly-free,
$\sum_{i}Q_{i}=\sum_{i}q_{i}=0$, so that its interaction with photons is very
suppressed and can evade present limits on X-ray emissions.

##  3 Building anomaly-free global flavour models

As seen in the previous section, ALP flavor-changing couplings in the lepton
sector are strongly dependent on the unitary matrices diagonalizing the
charged-lepton Yukawa matrix. Additionally, the mass of the ALP and its decay
constant are fixed if we require it to explain the Xenon1T excess. The absence
of the electromagnetic anomaly to evade X-ray constraints puts an additional
restriction on the fermionic charges. Following the strategy of our previous
paper [33], we require anomaly cancellation with the SM particle content,
without the addition of new fermions, and this leads us to consider flavor
dependent charges.

In our scenario, the flavor symmetries describing fermion masses must include
a global $U(1)$ group with non-universal charges and these charges must cancel
the electromagnetic anomaly. Therefore, we have to assign different $U(1)$
charges to the three charged-leptons above the scale of flavor symmetry
breaking. This implies that the three charged-leptons can not belong to a
single triplet of flavor, which would have a single $U(1)$ charge, but there
must be three different representations of dimension 1 or, at most, a doublet
and a singlet in the left and right-handed sectors. Hence, we will consider a
$U(1)_{f}$ and a $U(2)_{f}=SU(2)_{f}\times U(1)_{f}$ symmetries as examples of
these two situations.

###  3.1 A $U(1)_{f}$ flavor symmetry

$U(1)$ flavor symmetries were used in the first attempts to explain the
structure of fermion masses and mixings and they remain a viable option [1, 2,
3, 6, 7, 9, 11, 12, 18]. However, we must impose the additional requirement of
anomaly cancellation with the leptonic charges, $-Q_{1}-Q_{2}=Q_{3}$ and
$-q_{1}-q_{2}=q_{3}$.

If the Yukawa Lagrangian is

$\displaystyle\mathcal{L}_{Y}$ $\displaystyle\supset$ $\displaystyle
Y^{e}_{ij}\,\bar{\ell}_{iL}\,\widetilde{H}_{2}\,e_{jR}+\;Y^{\nu}_{ij}\,\bar{\ell}_{iL}\,H_{2}\,\nu_{jR}\;+\;M^{\nu}_{ij}\,\nu_{iR}\,\nu_{jR}^{c},$
(14)

we can always use the Higgs charge to make the (3,3) element ${\cal O}(1)$ or
the required power, i.e $-Q_{3}+q_{3}+q_{H}\geq 0$, and we take the flavon
charge $q=-1$. The required hierarchy imposes then,
$(-Q_{3})\leq(-Q_{2})\leq(-Q_{1})$ and $q_{3}\leq q_{2}\leq q_{1}$ with
$(-Q_{1}),q_{1}\geq 0$ and $(-Q_{3}),q_{3}\leq 0$ (remember that we impose
that the sum of charges must be zero, independently for left and right
fields). Now, we can consider, for simplicity, the 2–3 sector, and we have,

$Y^{e}/y_{\tau}~{}=~{}\begin{pmatrix}\varepsilon^{Q_{3}-Q_{2}+q_{2}-q_{3}}&\varepsilon^{Q_{3}-Q_{2}}\\\
\varepsilon^{q_{2}-q_{3}}&1\end{pmatrix}\quad=\quad\begin{pmatrix}\varepsilon^{-Q_{1}-2Q_{2}+q_{1}+2q_{2}}&\varepsilon^{-Q_{1}-2Q_{2}}\\\
\varepsilon^{q_{1}+2q_{2}}&1\end{pmatrix},$ (15)

where, we have suppressed ${\cal O}(1)$ coefficients in all the entries. Now,
we must require that the second eigenvalue, the muon mass, is ${\cal
O}(\lambda_{c}^{2})$. If we take $\varepsilon=\lambda_{c}$, we must have
$-Q_{1}-2Q_{2}+q_{1}+2q_{2}=2$, and the only possibilities would be: i)
$-Q_{1}-2Q_{2}=1$ and $q_{1}+2q_{2}=1$, ii) $-Q_{1}-2Q_{2}=0$ and
$q_{1}+2q_{2}=2$ and iii) $-Q_{1}-2Q_{2}=2$ and $q_{1}+2q_{2}=0$. From here,
we can see that only the solution ii) would give large leptonic mixing in the
left-handed charged lepton sector. However, in this case, we have
$Q_{1}=-2Q_{2}$ and $q_{1}=2-2q_{2}$. Given that
$(-Q_{1}),q_{1}\geq(-Q_{2}),q_{2}$, we have necessarily $(-Q_{2}),q_{2}\leq
0$. Moreover, if we require $m_{e}/m_{\mu}\simeq\lambda_{c}^{2}$
($m_{e}/m_{\mu}\simeq\lambda_{c}^{3}$), the only possibilities would be
$Q_{2}=0$ and $q_{2}=0$ ($Q_{2}=0$ and $q_{2}=-1/3$).

Notice this structure is not changed if we add the first generation, as we
have chosen the charges hierarchical and the first row or the first column can
be at most of the same order as the elements in this submatrix. Therefore they
can not change the order of magnitude of these elements. Furthermore, the
charges of the first generation are already fixed by this submatrix and the
anomaly cancellation condition. The full $3\times 3$ matrix with $Q_{2}=0$ and
$q_{2}=-1/3$ is

$Y^{e}/y_{\tau}~{}=~{}\begin{pmatrix}\varepsilon^{5}&\varepsilon^{2}&1\\\
\varepsilon^{5}&\varepsilon^{2}&1\\\
\varepsilon^{5}&\varepsilon^{2}&1\end{pmatrix}.$ (16)

As we can see, this matrix gives rise to ${\mathcal{O}}(1)$ LH mixings, but,
in principle, small, CKM-like RH mixing. Neverthless, the large LH mixings are
due to $Q_{1}=Q_{2}=Q_{3}$ and therefore, from eqs. (7)–(8), left-handed
mixings disappear in offdiagonal $V^{e}$ and $A^{e}$ ALP couplings. Thus, ALP
couplings will depend only on the small RH mixings and therefore will be out
of reach for the near future experiments. Unfortunately, the situation is the
same in case iii) with ${\mathcal{O}}(1)$ RH mixings, but equal RH charges and
even case i), where we have medium, but still small, LH and RH mixings, can
not reach the required size for the near future experiments.

###  3.2 A $U(2)_{f}$ flavor symmetry

Let us consider now the flavour group
$U(2)_{f}\,=\,SU(2)_{f}\,\times\,U(1)_{f}$ [8, 10, 54, 55, 56, 57, 58, 59, 60,
61, 62]. Here we group the three generations in a doublet and a singlet of
$SU(2)_{f}$ with different $U(1)_{f}$ charges, giving then rise to FC
couplings. We consider two possibilities:

i) 12-doublet case — LH and RH fermions of the third generation transform as
singlets of $SU(2)_{f}$ whilst those of the first and second generation belong
to doublets,

$\bar{\ell}_{L}\,\equiv\,\begin{pmatrix}\bar{e}_{L}\\\
\bar{\mu}_{L}\end{pmatrix},\qquad\bar{\nu}_{L}\,\equiv\,\begin{pmatrix}\bar{\nu}_{eL}\\\
\bar{\nu}_{\mu L}\end{pmatrix},\qquad\ell_{R}\,\equiv\,\begin{pmatrix}e_{R}\\\
\mu_{R}\end{pmatrix}.$ (17)

ii) 23-doublet case — LH and RH fermions of the first generation are singlets
while those of the second and third generation transform as doublets,

$\bar{\ell}_{L}\,\equiv\,\begin{pmatrix}\bar{\mu}_{L}\\\
\bar{\tau}_{L}\end{pmatrix},\qquad\bar{\nu}_{L}\,\equiv\,\begin{pmatrix}\bar{\nu}_{\mu
L}\\\ \bar{\nu}_{\tau
L}\end{pmatrix},\qquad\ell_{R}\,\equiv\,\begin{pmatrix}\mu_{R}\\\
\tau_{R}\end{pmatrix}.$ (18)

Considering $Q$ and $q$ the $U(1)_{f}$ charges of the LH and RH fields, the
following charge matrices determine the ALP couplings to charged leptons in
the 12– and 23–doublet case, respectively:

$x_{L}^{(12)}~{}=~{}{\rm Diag}\left(Q,\,Q,\,-2Q\right),\hskip
28.45274ptx_{R}^{(12)}~{}=~{}{\rm Diag}\left(q,\,q,\,-2q\right).$ (19)
$x_{L}^{(23)}~{}=~{}{\rm Diag}\left(-2Q,\,Q,\,Q\right),\hskip
28.45274ptx_{R}^{(23)}~{}=~{}{\rm Diag}\left(-2q,\,q,\,q\right).$ (20)

Notice that eqs. (7) and (8) can be simplified if the universal part is
separated from the $x_{L}$ and $x_{R}$ charge matrices:

$\displaystyle V^{e}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(q+Q)\,\delta_{ij}\;-\;3\,q\,U^{e\,*}_{R,\,{\bf
s}i}\;U^{e}_{R,\,{\bf s}j}\;-\;3\,Q\,U^{e\,*}_{L,\,{\bf s}i}\;U^{e}_{L,\,{\bf
s}j}\right],$ (21) $\displaystyle A^{e}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(q-Q)\delta_{ij}\;-\;3\,q\,U^{e\,*}_{R,\,{\bf
s}i}\;U^{e}_{R,\,{\bf s}j}\;+\;3\,Q\,U^{e\,*}_{L,\,{\bf s}i}\;U^{e}_{L,\,{\bf
s}j}\right],$ (22)

with ${\bf s}=3$ for the 12–doublet case and ${\bf s}=1$ for the 23–doublet
case. From eq. (22), the Xenon1T requirement in eq. (13) can be recast as:

$\displaystyle f_{a}^{\rm X1T}$ $\displaystyle=$ $\displaystyle(2.5\times
10^{9}\,\text{GeV})\,\left|\,q\>-\>Q\>+\>3\,Q\,\left|U^{e}_{L,\,{\bf
s}1}\right|^{2}\;-\;3\,q\,\left|U^{e}_{R,\,{\bf s}1}\right|^{2}\,\right|.$
(23)

A general view can be obtained from eqs. (21)-(23) by setting $Q=-q=1$,
assuming $U^{e}_{L}=U^{3\sigma}_{\rm PMNS}$ and $U^{e}_{R}$ totally general.
Figure 1 displays, in these conditions, the prediction of the ALP decay
constant compatible with the Xenon1T result in the 12-doublet (${\bf s}=3$,
blue region) and 23-doublet case (${\bf s}=1$, violet region). The dashed and
continuous black ellipses correspond to the regions where the RH mixing is
CKM- and PMNS-like, respectively. The different shape of the blue and violet
regions can be understood from eqs. (21)-(23). For $Q=-q=1$,

$\displaystyle A^{e}_{11}$ $\displaystyle=$
$\displaystyle\left(-1+3/2\left(\left|U^{e}_{L,\,{31}}\right|^{2}\;+\;\left|U^{e}_{R,\,{31}}\right|^{2}\right)\right)\,,$
(24) $\displaystyle C^{e}_{12}$ $\displaystyle=$
$\displaystyle\frac{3\sqrt{2}}{2}\left(\left|U^{e\,*}_{R,\,31}\;U^{e}_{R,\,32}\right|^{2}+\left|U^{e\,*}_{L,\,31}\;U^{e}_{L,\,32}\right|^{2}\right)^{1/2}\,,$
(25)

in the 12-doublet case or

$\displaystyle A^{e}_{11}$ $\displaystyle=$
$\displaystyle\left(-1+3/2\left(\left|U^{e}_{L,\,{11}}\right|^{2}\;+\;\left|U^{e}_{R,\,{11}}\right|^{2}\right)\right)\,,$
(26) $\displaystyle C^{e}_{21}$ $\displaystyle=$
$\displaystyle\frac{3\sqrt{2}}{2}\left(\left|U^{e\,*}_{R,\,11}\;U^{e}_{R,\,12}\right|^{2}+\left|U^{e\,*}_{L,\,11}\;U^{e}_{L,\,12}\right|^{2}\right)^{1/2}\,,$
(27)

in the 23-doublet case.

Given that $U^{e}_{L}$ is presumed to be PMNS-like, $|U^{e}_{L,\,{31}}|\sim
0.5$, $|U^{e}_{L,\,{32}}|\sim 0.6$, $|U^{e}_{L,\,{11}}|\sim 0.8$ and
$|U^{e}_{L,\,{12}}|\sim 0.55$. If $U^{e}_{R}$ ranges from small to large
values, it is understood that in the 23-doublet case $A^{e}_{11}$ can reach a
factor of two larger than in the 12-doublet case. Similarly, as $f_{a}=5\times
10^{9}~{}{\rm GeV}|A^{e}_{11}|$, the slope of the region is given by $5\times
10^{9}~{}{\rm GeV}/|C^{e}_{12}|$ and in the 12-doublet case, for small
$U^{e}_{R}$, the slope is proportional to $1/|C^{e}_{12}|\simeq 1/0.3\simeq
3.3$ while, in the 23-doublet case, it is $1/|C^{e}_{12}|\simeq 1/0.45\simeq
2.3$.

The horizontal lines display the bounds in eqs. (10)-(11), where the
continuous line is the current bound while the dashed lines indicate the
limits expected from future experiments. It is observed that for both
scenarios most of the allowed region by present bounds is testable at future
experiments. Only for the 12-doublet case, a small area above the expected
Mu3e limit remains unconstrained.

Figure 1:  Values of the ALP decay constant compatible with the Xenon 1T
result assuming a $U(2)_{f}$ flavor symmetry in the 12-doublet-case (blue
region) and 23-doublet-case (violet region) and $Q=-q=1$. The horizontal lines
display the bounds in eqs. (10)-(11), where the continuous line is the current
bound while the dashed lines indicate the limits expected from future
experiments. The MEGII-fwd line has been drawn, as a reference, in the limit
$V_{21}^{e}=A_{21}^{e}$. The full constraint must be obtained in each case
from eq. (11). The LH-mixing is fixed to reproduce the PMNS at the 3$\sigma$
level while the RH-mixing is completely general. The circles identify the
regions where the RH-rotation is CKM-like (dashed circles) and PMNS-like
(continuous circles). The stars show the predictions corresponding to the
benchmark points in Table 3 and 6.

##  4 Explicit $U(2)_{f}=SU(2)_{f}\times U(1)_{f}$ flavour models

Let us consider now explicit models with $U(2)_{f}$ flavor group. These models
must reproduce lepton masses and mixings and simultaneously explain the
Xenon1T excess. Additionally, we will investigate lepton flavor violation
effects in this models and look for points that could be observed at the next
generation of experiments.

As stated before, two possibilities for the leptonic sector are considered:
one, where the first and second generation constitute a doublet and the third
transforms as singlet, and another, where the role of the first and third
generations are exchanged.

In this case, both Higgs doublets, which corresponds to a type-X 2HDM 777In
type-X 2HDM, $H_{1}$ and $H_{2}$ couple only to quarks and leptons
respectively due to an additional $Z_{2}$ symmetry. Our models are formulated
in the large $\tan\beta=v_{1}/v_{2}$ regime where the leptophobic doublet can
be identified with the SM Higgs in good approximation. , transform as singlets
under $SU(2)_{f}$ but have different $U(1)_{f}$ charges.

In either case three flavons are added: two transforming as doublets under
$SU(2)_{f}$, $\Phi_{1}$ and $\Phi_{2}$, and one as a singlet, $\chi$. The
flavor symmetry is spontaneously broken when the flavons get non-zero vevs in
the directions:

$\langle\Phi_{1}\rangle~{}\propto~{}\Lambda\,\begin{pmatrix}0\\\
\varepsilon_{1}\end{pmatrix},\qquad\langle\Phi_{2}\rangle~{}\propto~{}\Lambda\,\begin{pmatrix}\varepsilon_{2}\\\
\varepsilon_{2}\end{pmatrix},\qquad\langle\chi\rangle~{}\propto~{}\Lambda\,\varepsilon_{\chi}.$
(28)

As we show below, the three vevs are of the same order,
$\varepsilon_{1}\sim\varepsilon_{2}\sim\varepsilon_{\chi}$ 888The alignment of
these vevs, in particular of $\langle\Phi_{2}\rangle$, is motivated by the
presence of large leptonic mixing. This non-trivial structure can be obtained
with an adequate scalar potential, (see for instance [17, 63, 64, 65, 66]).
Here we note that all the examples below introduce three RH neutrinos that
generate the neutrino masses through a type-I see-saw mechanism.

###  4.1 The 12-doublet case

Fields | $H_{1}$ | $H_{2}$ | $\bar{\ell}_{L},\bar{\nu}_{L}$ | $\ell_{R}$ | $\nu_{eR}$ | $\nu_{\mu R}$ | $\bar{\tau}_{L},\bar{\nu}_{\tau L}$ | $\tau_{R}$ | $\nu_{\tau R}$ | $\Phi_{1}$ | $\Phi_{2}$ | $\chi$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\rm SU(2)$ | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1
$\rm U(1)$ | 0 | 2/3 | -1/3 | -1/3 | -1/6 | 1/3 | 2/3 | 2/3 | -2/3 | 1 | -2/3 | -1/3

Table 2:  Particle content and charge assginment under $U(2)_{f}$ for the
12-doublet case.

The spectrum and quantum numbers for this version of the $U(2)_{f}$ model are
detailed in table 2. From the symmetries, the most general set of non-
renormalizable operators for charged leptons, at leading order in the flavon
vevs, is given by the following couplings with $\Phi_{2}$:

$\displaystyle-{\cal L}_{e}^{(12)}$ $\displaystyle=$
$\displaystyle\frac{c_{1}^{e}}{\Lambda^{2}}\left(\bar{\ell}_{L}\,\widetilde{\Phi}_{2}\right)\left(\ell_{R}\,\widetilde{\Phi}_{2}\right)\widetilde{H}_{2}\;+\;\frac{c_{2}^{e}}{\Lambda^{2}}\left(\widetilde{\Phi}_{2}\,\bar{\ell}_{L}\right)\chi\,\tau_{R}\,\widetilde{H}_{2}$
(29) $\displaystyle+$
$\displaystyle\frac{c_{3}^{e}}{\Lambda^{2}}\bar{\tau}_{L}\left(\widetilde{\Phi}_{2}\,\ell_{R}\right)\chi\,\widetilde{H}_{2}\;+\;\frac{c_{4}^{e}}{\Lambda^{2}}\;\bar{\tau}_{L}\,\tau_{R}\,\chi^{2}\;\widetilde{H}_{2}\;+\;{\rm
h.c.}\,,$

where $\widetilde{\phi}=i\sigma_{2}\phi^{*}$. The parenthesis in the operators
make explicit the $SU(2)_{f}$ contractions of the fields. The first order
corrections to the terms in eq.(29) come from the substitution
$\widetilde{\Phi}_{2}\rightarrow\Phi_{1}\chi$ as well as
$\chi\rightarrow\widetilde{\Phi}_{1}\widetilde{\Phi}_{2}$:

$\displaystyle-\delta{\cal L}_{e}^{(12)}$ $\displaystyle=$
$\displaystyle\frac{c^{e}_{5}}{\Lambda^{3}}\left(\bar{\ell}_{L}\,\Phi_{1}\right)\left(\widetilde{\Phi}_{2}\,\ell_{R}\right)\chi\,\widetilde{H}_{2}\;+\;\frac{c^{e}_{6}}{\Lambda^{3}}\left(\bar{\ell}_{L}\,\widetilde{\Phi}_{2}\right)\left(\Phi_{1}\,\ell_{R}\right)\chi\,\widetilde{H}_{2}$
$\displaystyle+$
$\displaystyle\frac{c^{e}_{7}}{\Lambda^{3}}\left(\bar{\ell}_{L}\,\Phi_{1}\right)\chi^{2}\,\tau_{R}\,\widetilde{H}_{2}\;+\;\frac{c^{e}_{8}}{\Lambda^{3}}\;\bar{\tau}_{L}\left(\ell_{R}\,\Phi_{1}\right)\chi^{2}\widetilde{H}_{2}\;+\;{\rm
h.c.}\,.$

From eqs.(29)-(4.1), the resulting mass matrices are

$\displaystyle M_{e}^{(12)}$ $\displaystyle=$
$\displaystyle\frac{v_{H_{2}}}{\sqrt{2}}\;\frac{\varepsilon^{2}_{2}}{2}\begin{pmatrix}c^{e}_{1}&c^{e}_{1}&\sqrt{2}\,c^{e}_{2}\,\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\\\
c^{e}_{1}&c^{e}_{1}&\sqrt{2}\,c^{e}_{2}\,\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\\\
\sqrt{2}\,c^{e}_{3}\,\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}&\sqrt{2}\,c^{e}_{3}\,\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}&c^{e}_{4}\,\frac{\varepsilon_{\chi}^{2}}{\varepsilon_{2}^{2}}\\\
\end{pmatrix},$ (30) $\displaystyle\delta M_{e}^{(12)}$ $\displaystyle=$
$\displaystyle\frac{v_{H_{2}}}{\sqrt{2}}\;\frac{\varepsilon_{1}\,\varepsilon_{2}\,\varepsilon_{\chi}}{2}\begin{pmatrix}c^{e}_{5}+c^{e}_{6}&~{}c^{e}_{5}~{}&~{}\sqrt{2}\,c^{e}_{7}\,\frac{\varepsilon_{\chi}}{\varepsilon_{2}}~{}\\\\[3.0pt]
c^{e}_{6}&0&0\\\\[3.0pt]
~{}\sqrt{2}\,c^{e}_{8}\,\frac{\varepsilon_{\chi}}{\varepsilon_{2}}~{}&0&\sqrt{2}\,c^{e}_{9}\\\
\end{pmatrix}.$ (31)

In the neutral sector, the Yukawa and Majorana terms that contribute at
leading order are999Equations (32) and (33) receive NLO corrections from the
replacement of flavons discussed above eq. (4.1). However, these NLO operators
contribute to exactly the same entries that the leading terms, so they can be
safely reabsorbed in the leading ${\cal O}(1)$ coefficients by the
redefinition $c_{i}\to
c_{i}-c_{i}^{\prime}\,\varepsilon_{1}\varepsilon_{2}/(\sqrt{2}\,\varepsilon_{\chi})$.:

$\displaystyle-{\cal L}_{\nu}^{(12)}$ $\displaystyle=$
$\displaystyle\frac{c^{\nu}_{1}}{\Lambda}\left(\Phi_{2}\,\bar{\nu}_{L}\right)\nu_{\mu
R}\,H_{2}\;+\;\frac{c^{\nu}_{2}}{\Lambda^{2}}\left(\widetilde{\Phi}_{2}\,\bar{\nu}_{L}\right)\,\chi\,\nu_{\tau
R}\,H_{2}$ (32) $\displaystyle+$
$\displaystyle\frac{c^{\nu}_{3}}{\Lambda^{2}}\left(\widetilde{\Phi}_{1}\Phi_{2}\right)\bar{\nu}_{\tau
L}\,\nu_{\mu
R}\,H_{2}\;+\;\frac{c^{\nu}_{4}}{\Lambda^{2}}\chi^{2}\,\bar{\nu}_{\tau
L}\,\nu_{\tau R}\,H_{2}$ $\displaystyle+$
$\displaystyle\frac{c^{\nu}_{5}}{\Lambda^{2}}\left(\widetilde{\Phi}_{1}\,\bar{\nu}_{L}\right)\chi^{\dagger}\,\nu_{\mu
R}\,H_{2}\;+\;\frac{c^{\nu}_{6}}{\Lambda^{3}}\left(\bar{\nu}_{L}\,\widetilde{\Phi}_{1}\right)\,\left(\widetilde{\Phi}_{1}\Phi_{2}\right)\nu_{\mu
R}\,H_{2}$ $\displaystyle+$
$\displaystyle\frac{c^{\nu}_{7}}{\Lambda^{3}}\left(\bar{\nu}_{L}\,\Phi_{1}\right)\chi^{2}\,\nu_{\tau
R}\,H_{2}\;+\;{\rm h.c.},$ $\displaystyle{\cal L}_{M}^{(12)}$ $\displaystyle=$
$\displaystyle
M_{0}\;\left[\,\frac{c_{8}^{\nu}}{2\Lambda}\,\chi^{\dagger}\,\bar{\nu}^{c}_{eR}\,\nu_{eR}\;+\;\frac{c^{\nu}_{9}}{2\Lambda}\,\chi^{\dagger}\left(\,\bar{\nu}^{c}_{\mu
R}\,\nu_{\tau R}+\bar{\nu}^{c}_{\tau R}\,\nu_{\mu R}\right)\right.$ (33)
$\displaystyle+$
$\displaystyle\left.\frac{c_{10}^{\nu}}{2\Lambda^{2}}\,\chi^{2}\,\bar{\nu}^{c}_{\mu
R}\,\nu_{\mu
R}\;+\;\frac{c_{11}^{\nu}}{2\Lambda^{3}}\,\left(\widetilde{\Phi}_{2}\Phi_{1}\right)\chi\,\bar{\nu}^{c}_{\tau
R}\,\nu_{\tau R}\,\right]\;+\;{\rm h.c.}\,,$

which produce the following Yukawa and Majorana matrices

$\displaystyle Y_{\nu}^{(12)}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{2}}{\sqrt{2}}\left(\begin{array}[]{ccc}~{}0{}&-c^{\nu}_{1}+c^{\nu}_{6}\,\varepsilon_{1}^{2}&\left(c^{\nu}_{2}+c_{7}^{\nu}\,\varepsilon_{1}\frac{\varepsilon_{\chi}}{\varepsilon_{2}}\right)\varepsilon_{\chi}\\\\[4.0pt]
~{}0{}&c^{\nu}_{1}+c^{\nu}_{5}\,\varepsilon_{1}\frac{\varepsilon_{\chi}}{\varepsilon_{2}}&c^{\nu}_{2}\,\varepsilon_{\chi}\\\\[2.0pt]
~{}0{}&~{}~{}c^{\nu}_{3}\,\varepsilon_{1}&\sqrt{2}\,c^{\nu}_{4}\,\frac{\varepsilon_{\chi}}{\varepsilon_{2}}\varepsilon_{\chi}\end{array}\right)\,,$
(37) $\displaystyle M_{\nu}^{(12)}$ $\displaystyle=$ $\displaystyle
M_{0}\;\varepsilon_{\chi}\begin{pmatrix}c^{\nu}_{8}&0&0\\\
0&c_{10}^{\nu}\,\varepsilon_{\chi}&c_{9}^{\nu}\\\
0&c_{9}^{\nu}&c^{\nu}_{11}\,\frac{\varepsilon_{1}\varepsilon_{2}}{\sqrt{2}}\\\
\end{pmatrix}.$ (38)

No coupling involving the RH electron neutrino is present in eqs. (32) and
(33), except for the $\bar{\nu}^{c}_{e_{R}}\nu_{e_{R}}$ vertex. This is
because the $U(1)_{f}$ charge of this field has been chosen so that it cannot
be compensated by any combination of fields, unless it appears in pairs101010
Unlike $\nu_{\mu_{R}}$ and $\nu_{\tau_{R}}$, the $\nu_{e_{R}}$ charge is an
odd multiple of $q_{\chi}/2$.. As a consequence, the zeros appearing in the
Yukawa and Majorana matrices of eqs.(37) and (38) are exact at all orders and
lead to one massless active neutrino. As in this formulation the first and
second generation of LH neutrinos are in the same doublet, the natural
scenario is having $m_{\nu_{3}}=0$, which implies an inverted hierarchy.

The active neutrino mass matrix exhibits the following structure at leading
order:

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}_{H_{2}}}{2}\,Y_{\nu}\,M^{-1}_{\nu}\,Y^{T}_{\nu}\;=\;-\frac{v^{2}_{H_{2}}}{2}\,\cfrac{\varepsilon^{2}_{2}}{M_{0}}\,\begin{pmatrix}a_{1}&~{}a_{2}\,\cfrac{\varepsilon_{1}}{\varepsilon_{2}}\,\varepsilon_{\chi}~{}&a_{3}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\\\\[5.0pt]
~{}a_{2}\,\cfrac{\varepsilon_{1}}{\varepsilon_{2}}\,\varepsilon_{\chi}~{}&-\,a_{1}&-a_{3}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\\\\[5.0pt]
a_{3}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}&-a_{3}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}&~{}a_{4}\,\cfrac{\varepsilon_{1}}{\varepsilon_{2}}\,\varepsilon_{\chi}~{}\\\
\end{pmatrix}\,.$ (39)

with

$a_{1}~{}=~{}c_{1}^{\nu}c_{2}^{\nu},\hskip
28.45274pta_{2}~{}=~{}-\frac{c_{1}^{\nu}c_{7}^{\nu}+c_{2}^{\nu}c_{5}^{\nu}}{2},\hskip
28.45274pta_{3}~{}=~{}\frac{c_{1}^{\nu}c_{4}^{\nu}}{\sqrt{2}},\hskip
28.45274pta_{4}~{}=~{}-\sqrt{2}\,c_{3}^{\nu}c_{4}^{\nu}.$ (40)

The light active neutrinos are Majorana-like, therefore the rotation to the
mass basis is just given by the orthogonal transformation

$U_{\nu_{L}}^{T}\,m_{\nu}\,U_{\nu}~{}=~{}{\rm
diag}\left(m_{\nu_{1}},\,m_{\nu_{2}},0\right).$ (41)

$\chi^{2}_{3\sigma}$ | $c^{e}_{2}$ | $c^{e}_{3}$ | $c^{e}_{4}$ | $c^{e}_{5}$ | $c^{e}_{6}$ | $c^{e}_{7}$ | $c^{e}_{8}$ | $\epsilon_{1}$ | $\epsilon_{2}$ | $\epsilon_{\chi}$
---|---|---|---|---|---|---|---|---|---|---
2.16 | -1.30 | 0.93 | -1.63 | -1.43 | -0.82 | -2.00 | -1.02 | 0.05 | 0.17 | 0.15
F.T. | $c^{\nu}_{1}$ | $c^{\nu}_{2}$ | $c^{\nu}_{3}$ | $c^{\nu}_{4}$ | $c^{\nu}_{5}$ | $c^{\nu}_{6}$ | $c^{\nu}_{7}$ | $c^{\nu}_{10}$ | $c^{\nu}_{11}$ |
8.30 | -0.51 | -3.50 | -2.36 | -0.50 | -1.02 | 1.29 | -1.46 | 0.54 | -0.43 |

Table 3:  Values of the model parameters at the benchmark point with
$c^{e}_{1},\,c^{\nu}_{8},\,c^{\nu}_{9}=1.00$.

|  $\theta_{12}^{\rm IH}$ (º) |  $\theta_{23}^{\rm IH}$ (º) |  $\theta_{13}^{\rm IH}$ (º) | $m_{e}/m_{\tau}$ | $m_{\mu}/m_{\tau}$ | $\Delta m_{21}^{2}/\Delta m_{23}^{2}$ |  |
---|---|---|---|---|---|---|---|---
b.p. : | 32.00 | 52.1 | 8.61 | $2.88\times 10^{-4}$ | $0.059$ | $-0.030$ |  |
exp. : | 33.45 | 49.5 | 8.60 | $2.88\times 10^{-4}$ | $0.059$ | $-0.030$ | $-$ | $-$
$3\sigma$ : | $31.27-35.87$ | $39.8-52.1$ | $8.24-8.98$ | $\left(2.79-2.96\right)\times 10^{-4}$ | $0.0577-0.0612$ | $-\left(0.027-0.033\right)$ |  |

Table 4:  Value of the measured observables from the NuFIT 5.1 (2021) global
fit [45] and predictions of our model at the benchmark point.

A fit is performed in order to fix the values of the free parameters that best
reproduce the observed masses and mixing angles in the leptonic sector. More
details about the cost function and how the fit is done can be found in ref.
[67]. Tables 3 and 4 show the coefficients of the model and estimations for
the observables at the benchmark point. Notice that the PMNS angles are mostly
given by the rotations of the charged leptons, being the neutrino contribution
totally negligible for the 1-2 and 1-3 sectors and a correction of the $20\%$
in the 2-3 sector. The fit provides the correct ratio of masses between
different generations. From its predictions in table 4, the value of the
$H_{2}$ vev and the Majorana mass scale can be determined by setting the
absolute value of the masses. It renders $v_{H_{2}}=0.17\,v_{H}$, with
$v_{H}=246$ GeV the SM Higgs vev, and $M_{0}=9.04\times 10^{11}$ GeV.

Regarding the ALP phenomenology, the benchmark point of the model produces a
decay constant equal to

$f_{a}=1.3\times 10^{9}~{}{\rm GeV}.$ (42)

In figure 1, the position of our benchmark point in the (scaled) ALP parameter
space is marked with a yellow diamond in the blue area111111Notice that the
blue and purple regions are computed considering $Q=-q=1$ while the charges of
our model are $Q=-q=1/3$. As can be seen, an axial coupling to electrons of
$|A_{11}|=0.3$ is generated while its flavor-violating couplings in the 1-2
sector make feasible to prove its presence through the $\mu\to e\,a$ decay at
Mu3e. However, it is not detectable at MEGII-fwd which can only test up to
$f_{a}/|C_{21}^{e}|=8.9\times 10^{8}$ GeV for couplings like the ones derived
in this scenario ($A_{21}\simeq-0.9\,V_{21}$).

###  4.2 The 23-doublet case

Fields | $H_{1}$ | $H_{2}$ | $\bar{e}_{L},\bar{\nu}_{eL}$ | $e_{R}$ | $\nu_{eR}$ | $\nu_{\mu R}$ | $\nu_{\tau R}$ | $\bar{\ell}_{L},\bar{\nu}_{L}$ | $\ell_{R}$ | $\Phi_{1}$ | $\Phi_{2}$ | $\chi$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\rm SU(2)$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | $2$ | 1
$\rm U(1)$ | 0 | 2/3 | 2/3 | 2/3 | -1/12 | 1/2 | -2/3 | -1/3 | -1/3 | 5/6 | -2/3 | -1/6

Table 5:  Particle content and charge assignment under $U(2)_{f}$ for the
23-doublet case.

This version of the $U(2)_{f}$ model is very similar to the previous
12-doublet case but with $e_{L},\,e_{R}$ the singlets under $SU(2)_{f}$. The
charges of $\Phi_{1}$ and $\chi$ under the $U(1)_{f}$ symmetry are also
slightly modified, as can be seen in table 5.

The leading operators in the charged sector are exactly those in eqs. (29) and
(4.1) but making the replacements:

$\tau_{R}\to e_{R},\qquad\tau_{L}\to e_{L},\qquad\chi\to\chi^{2}.$ (43)

In view of this, the mass matrix for the charged leptons can then be precisely
obtained as
$M^{(12)}_{e}=P^{T}_{231}.\,M^{(23)}_{e}(\varepsilon_{\chi}\rightarrow\varepsilon^{2}_{\chi}).P_{231}$,
where $P_{231}$ is the $231$ column permutation matrix of the identity. The
resulting charged lepton mass matrix and its first order correction read as

$\displaystyle M^{(23)}_{e}$ $\displaystyle=$
$\displaystyle\frac{v_{H_{2}}}{\sqrt{2}}\frac{\varepsilon^{2}_{2}}{2}\begin{pmatrix}2\,c^{e}_{4}\,\frac{\varepsilon_{\chi}^{2}}{\varepsilon_{2}^{2}}\,\varepsilon_{\chi}^{2}&\sqrt{2}\,c^{e}_{2}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\,\varepsilon_{\chi}&\sqrt{2}\,c^{e}_{2}\,\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\,\varepsilon_{\chi}\\\
\sqrt{2}\,c^{e}_{3}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\,\varepsilon_{\chi}&c^{e}_{1}&c^{e}_{1}\\\
\sqrt{2}\,c^{e}_{3}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\,\varepsilon_{\chi}&c^{e}_{1}&c^{e}_{1}\\\
\end{pmatrix},$ (44) $\displaystyle\delta M^{(23)}_{e}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{1}\varepsilon_{2}\varepsilon_{\chi}}{2}\begin{pmatrix}0&\sqrt{2}\,c^{e}_{7}\,\frac{\varepsilon_{\chi}}{\varepsilon_{2}}\,\varepsilon_{\chi}&0\\\\[5.0pt]
\sqrt{2}\,c^{e}_{8}\,\frac{\varepsilon_{\chi}}{\varepsilon_{2}}\,\varepsilon_{\chi}&c^{e}_{5}+c^{e}_{6}+c^{e}_{9}\,\frac{\varepsilon_{1}}{\varepsilon_{2}}\varepsilon_{\chi}\,&c^{e}_{5}\\\\[5.0pt]
0&c^{e}_{6}&0\\\ \end{pmatrix}.$ (45)

At variance with the 12-doublet case, in this model, given that the LH
neutrinos of the second and third generations belong to the same doublet, a
scenario with normal hierarchy for the neutrino mass spectrum, $m_{\nu 1}\ll
m_{\nu 2}<m_{\nu 3}$, is the natural prediction. With the charge assignment in
Table 5, in the neutral sector the following leading order Majorana and Yukawa
terms can be written in the Lagrangian

$\displaystyle-{\cal L}^{(23)}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{c^{\nu}_{1}}{\Lambda}\left(\widetilde{\Phi}_{1}\,\bar{\nu}_{L}\right)\,\nu_{\mu
R}\,H_{2}\;+\;\frac{c^{\nu}_{2}}{\Lambda^{3}}\left(\widetilde{\Phi}_{2}\,\bar{\nu}_{L}\right)\,\chi^{2}\,\nu_{eR}\,H_{2}$
$\displaystyle+$
$\displaystyle\frac{c^{\nu}_{3}}{\Lambda^{4}}\left(\widetilde{\Phi}_{1}\Phi_{2}\right)\chi^{2}\,\bar{\nu}_{eL}\,\nu_{\mu
R}\,H_{2}\;+\;\frac{c^{\nu}_{4}}{\Lambda^{4}}\chi^{4}\,\bar{\nu}_{eL}\,\nu_{eR}\,H_{2}$
$\displaystyle+$
$\displaystyle\frac{c^{\nu}_{5}}{\Lambda^{2}}\left(\Phi_{2}\,\bar{\nu}_{L}\right)\chi\,\nu_{\mu
R}\,H_{2}\;+\;\frac{c^{\nu}_{6}}{\Lambda^{4}}\left(\Phi_{1}\,\bar{\nu}_{L}\right)\,\chi^{3}\,\nu_{eR}\,H_{2}\;+\;{\rm
h.c.},$ $\displaystyle{\cal L}^{(23)}_{M}$ $\displaystyle=$ $\displaystyle
M_{0}\,\chi^{\dagger}\,\left[\cfrac{c^{\nu}_{7}}{\Lambda}\left(\bar{\nu}^{c}_{\mu
R}\,\nu_{eR}+\bar{\nu}^{c}_{eR}\,\nu_{\mu
R}\right)+\cfrac{c^{\nu}_{8}}{\Lambda}\,\bar{\nu}^{c}_{\tau R}\nu_{\tau
R}\right]\;+\;{\rm h.c.}.$ (47)

Similar to the previous case, we chose the $U(1)_{f}$ charge of the RH tau
neutrino to be a fraction of the smallest flavon charge:
$q_{\nu_{eR}}=q_{\chi}/2$, hence it is never compensated by the considered
fields. Thus, by construction, in $\mathcal{L}^{(23)}_{\nu}$ there are no
couplings involving the RH tau neutrino. The resulting Yukawa and Majorana
mass matrices display the following textures

$\displaystyle Y^{(23)}_{\nu}$ $\displaystyle=$
$\displaystyle\cfrac{\varepsilon_{1}}{\sqrt{2}}\begin{pmatrix}\sqrt{2}\,c^{\nu}_{4}\frac{\varepsilon_{\chi}}{\varepsilon_{1}}\,\varepsilon^{3}_{\chi}&c^{\nu}_{3}\,\varepsilon_{2}\,\varepsilon^{2}_{\chi}&0\\\\[5.0pt]
c^{\nu}_{2}\,\frac{\varepsilon_{2}}{\varepsilon_{1}}\,\varepsilon^{2}_{\chi}-c^{\nu}_{6}\,\varepsilon_{\chi}^{3}&c^{\nu}_{5}\frac{\varepsilon_{2}}{\varepsilon_{1}}\,\varepsilon_{\chi}&0\\\\[5.0pt]
c^{\nu}_{2}\,\frac{\varepsilon_{2}}{\varepsilon_{1}}\,\varepsilon^{2}_{\chi}&c^{\nu}_{1}-c^{\nu}_{5}\frac{\varepsilon_{2}}{\varepsilon_{1}}\,\varepsilon_{\chi}&0\\\
\end{pmatrix},$ (48) $\displaystyle M_{\nu}^{(23)}$ $\displaystyle=$
$\displaystyle M_{0}\;\varepsilon_{\chi}\begin{pmatrix}0&c^{\nu}_{7}&0\\\
c^{\nu}_{7}&0&0\\\ 0&0&c^{\nu}_{8}\\\ \end{pmatrix}.$ (49)

Once more, the zeros in the third column of eq. (48) and the third column and
row of eq. (49) are exact at all orders and lead to a simplified scenario with
one massless neutrino. In particular, the simplification of $m_{\nu 1}=0$,
implies $m_{\nu 2}=\sqrt{\Delta m^{2}_{21}}$ and $m_{\nu 3}=\sqrt{\Delta
m^{2}_{31}}$.

The effective neutrino mass matrix after the seesaw exhibits a the following
structure:

$\displaystyle m^{(23)}_{\nu}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}_{H_{2}}}{2}\cfrac{\varepsilon_{1}\,\varepsilon_{2}\,\varepsilon_{\chi}}{M_{0}}\begin{pmatrix}2\,a_{6}\,\varepsilon^{4}_{\chi}&a_{5}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{1}}\,\varepsilon^{2}_{\chi}&a_{4}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\varepsilon_{\chi}\\\\[5.0pt]
a_{5}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{1}}\,\varepsilon^{2}_{\chi}&2\,a_{2}\frac{\varepsilon_{2}}{\varepsilon_{1}}\,\varepsilon_{\chi}&a_{1}-a_{3}\cfrac{\varepsilon_{1}}{\varepsilon_{2}}\,\varepsilon_{\chi}\\\\[5.0pt]
a_{4}\,\cfrac{\varepsilon_{\chi}}{\varepsilon_{2}}\varepsilon_{\chi}&a_{1}-a_{3}\cfrac{\varepsilon_{1}}{\varepsilon_{2}}\,\varepsilon_{\chi}&2\,a_{1}\\\
\end{pmatrix}\,,$ (50)

where the effective parameters are related with the parameters of the
lagrangian as

$\displaystyle a_{1}=c^{\nu}_{1}c^{\nu}_{2},\quad
a_{2}=c^{\nu}_{2}c^{\nu}_{5},\quad a_{3}=c^{\nu}_{1}c^{\nu}_{6},\quad
a_{4}=\sqrt{2}\,c^{\nu}_{1}c^{\nu}_{4},\quad
a_{5}=\sqrt{2}\,c^{\nu}_{4}c^{\nu}_{5},\quad
a_{6}=\sqrt{2}\,c^{\nu}_{3}c^{\nu}_{4}\,.$ (51)

$\chi^{2}_{3\sigma}$ | $c^{e}_{1}$ | $c^{e}_{2}$ | $c^{e}_{3}$ | $c^{e}_{4}$ | $c^{e}_{5}$ | $c^{e}_{6}$ | $c^{e}_{7}$ | $c^{e}_{8}$ | $c^{e}_{9}$
---|---|---|---|---|---|---|---|---|---
3.69 | 1.649 | -0.451 | -1.74761 | -0.728 | -1.883 | 2.152 | 0.863 | -2.343 | 2.914
F.T. | $c^{\nu}_{1}$ | $c^{\nu}_{2}$ | $c^{\nu}_{3}$ | $c^{\nu}_{4}$ | $c^{\nu}_{5}$ | $c^{\nu}_{6}$ | $\varepsilon_{1}$ | $\varepsilon_{2}$ | $\varepsilon_{\chi}$
22 | -0.509 | 0.844 | -1.393 | -1.041 | 2.572 | 2.775 | 0.371 | 0.389 | 0.246

Table 6:  Values of the model parameters at the benchmark point with
$c^{\nu}_{7}=c^{\nu}_{8}=1$.

|  $\theta^{\rm NH}_{12}$ (º) |  $\theta^{\rm NH}_{23}$ (º) |  $\theta^{\rm NH}_{13}$ (º) | $m_{e}/m_{\tau}$ | $m_{\mu}/m_{\tau}$ | $\Delta m^{2}_{21}/\Delta m^{2}_{31}$
---|---|---|---|---|---|---
b.f. : | 33.56 | 48.0 | 8.55 | $2.88\times 10^{-4}$ | 0.059 | 0.029
exp. : | 33.44 | 49.2 | 8.57 | $2.88\times 10^{-4}$ | 0.059 | 0.029
$3\sigma$ : | $31.27-35.86$ | $39.5-52.0$ | $8.20-8.97$ | $2.79-2.96\times 10^{-4}$ | 0.0577-0.0612 | 0.027-0.033

Table 7:  Values of the measured observables from the NuFIT 5.1 (2021) global
fit [45] and values obtained from the 23-model at the benchmark point. The
absolute mass values can be obtained with $v_{H_{2}}=v_{H}/26$ and
$M_{0}\simeq 3.17\times 10^{10}$ GeV.

In Table 6 and 7, the values of the coefficients and observables at the
benchmark point are presented. From Table 7 we see that the PMNS is correctly
reproduced even at the $1\sigma$ level, being the neutrino contribution 17%
for the $\theta_{23}$ angle, 50% for the $\theta_{12}$ angle, and 80% for the
$\theta_{13}$ angle. From the Table 7, the values of the $H_{2}$ vev and the
Majorana mass scale $M_{0}$ can be determined by imposing to reproduce the
absolute values of $m_{\tau}$ and $\Delta m^{2}_{31}$, then we have
$v_{H_{2}}=v_{H}/26$ and $M_{0}=3.17\times 10^{10}$ GeV. The prediction for
the axial ALP coupling to electron is $|A^{e}_{11}|=0.342$. Consequently, from
eq.(13), the prediction for the ALP decay constant compatible with the Xenon
1T result reads

$f^{\text{X1T}}_{a}=1.75\times 10^{9}\,\text{GeV}.$ (52)

From the requirement of large mixings to come from the charged lepton sector,
we obtain large LFV axion couplings with $|A^{e}_{21}|\gg|V^{e}_{21}|$, in
particular $|C^{e}_{21}|=0.409$ and $|V^{e}_{21}+A^{e}_{21}|=0.484$. With
these values the bounds in eq.(10-12) translates into:
$f^{\text{Jodidio}}_{a}\geq 1.10\times 10^{9}$ GeV, $f^{\text{Mu3e}}_{a}\geq
6.5\times 10^{9}$ GeV and $f^{\text{MEGII-fwd}}_{a}\geq 5.8\times 10^{9}$ GeV.
So, while the point evades the current limit imposed by the Jodidio et al.
experiment, it lies within a region that will be fully tested by the future
Mu3e and MEGII-fwd experiments, as it is displayed in Figure 1 (yellow diamond
inside the violet region).

##  5 Conclusions

Motivated by the Xenon1T excess in electron-recoil measurements, we studied
the possible lepton flavor violation effects from an ALP explaining this
excess and originating from the spontaneous breaking of a flavor group. This
flavor group is built to explain the masses and mixing patterns in the lepton
sector.

A priory, we have two options for flavor symmetries with a non-anomalous
$U(1)$ subgroup and flavor dependent charges, $U(1)$ or $U(2)$. We have seen
that a plain $U(1)$ symmetry can not provide sizeable flavor-changing
couplings reachable in near future experiments. Therefore, we have
concentrated in the flavor group $U(2)_{f}=SU(2)_{f}\times U(1)_{f}$. We
considered two scenarios, where the first-second or second-third generations
of leptons are grouped in a doublet representation of $SU(2)_{f}$. In these
models, we require the symmetry to reproduce correctly lepton masses and
mixing with $\mathcal{O}(1)$ coefficients and sizeable mixings in the charged
lepton sector. We find that, in both scenarios, the ALP consistent with the
Xenon1T anomaly could be probed by future LFV experiments. Interestingly, we
find that for the 1-2-doublet case, an inverted hierarchy of neutrino masses
is preferred while for the 2-3-doublet case, a normal hierarchy of the
neutrino masses is more likely. Therefore, if LFV experiments find a positive
signal from ALP interactions, future neutrino experiments could help to
disentangle these two models.

## Acknowledgments

We acknowledge the collaboration of L. Calibbi in the early stages of this
work. C. H. is supported by the Guangzhou Basic and Applied Basic Research
Foundation under Grant No. 202102020885, and the Sun Yat-Sen University
Science Foundation. MLLI, AM and OV acknowledge support from Spanish AEI-
MICINN, PID2020-113334GB- I00/AEI/10.13039/501100011033. OV acknowledges
partial support from the “Generalitat Valenciana” grant PROMETEO2017-033. A.M.
was supported by the Estonian Research Council grant MOBTT86 “Probing the
Higgs sector at the LHC and beyond”. JMY acknowledges funding from the
National Natural Science Foundation of China (NNSFC) under grant Nos.12075300
and 11821505, from Peng-Huan-Wu Theoretical Physics Innovation Center
(12047503), from the CAS Center for Excellence in Particle Physics (CCEPP),
and by a Key R&D Program of Ministry of Science and Technology of China under
number 2017YFA0402204, by the Key Research Program of the Chinese Academy of
Sciences, grant No. XDPB15.

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