# Is the muon a third family lepton?

Giacomo Cacciapaglia<EMAIL_ADDRESS>Aldo Deandrea
<EMAIL_ADDRESS>Institut de Physique des Deux Infinis (IP2I) de Lyon,
IN2P3/CNRS,
4 rue Enrico Fermi, 69622 Villeurbanne, France  Université Claude Bernard Lyon
1, Univ Lyon, 69001 Lyon, France  Shahram Vatani  Galileo Galilei Institute
for Theoretical Physics, Largo Enrico Fermi 2, I-50125 Firenze, Italy  Centre
for Cosmology, Particle Physics and Phenomenology, Université catholique de
Louvain, Louvain-la-Neuve B-1348, Belgium

###### Abstract

We propose a new family structure for the Standard Model fermions, where the
muon is reassigned to the third family. This reveals potentially meaningful
patterns in the masses and mixing, while pointing to precise and testable
predictions for experiments.

††preprint: none

The Standard Model (SM) of particle physics [1, 2, 3] has been formulated in
the 60’s around the principle of gauge invariance [4]. The electroweak sector
of the theory consists of an SU(2)${}_{L}\times$ U(1)Y gauge symmetry, while
an SU(3)c gauge symmetry is added to account for Quantum ChromoDynamics (QCD)
[5]. One peculiarity is the parity-violating nature of the weak interactions,
incarnated in the fact that the SU(2)L gauge bosons only couple to left-handed
leptons [2]. Quarks respect the same rule as they are also charged under the
weak interactions. This forces the fermion content of the SM to be chiral, in
the sense that left and right-handed components of the same particles do
couple differently to the weak gauge bosons. As a consequence, the lightest
states, namely the electron, the electron-neutrino, and the quarks up and
down, come together as a “family” or “generation” of fermions with quantum
numbers chosen to cancel gauge and gravity quantum anomalies [6, 7, 8, 9, 10].
In fact, a SM family is the minimal anomaly-free set of fields [11, 12, 13].
The breaking of the electroweak symmetry down to Quantum ElectroDynamics (QED)
via a Brout-Englert-Higgs (BEH) scalar field [14, 15] has the benefit of both
explaining the Fermi theory of weak interactions and providing mass to all
fermions (except for the neutrino). The latter is obtained via the Yukawa
sector of the theory, i.e. the most general set of renormalisable couplings of
the scalar field with pairs of fermion fields.

Down to this stage, the construction of the SM and of one family of fermions
is only and purely driven by the gauge principle. However, the presence of the
muon discovered in 1936 [16, 17] and of the strange quark postulated in 1953
[18, 19], and later of other heavier leptons and quarks (tau lepton [20],
charm quark [21, 22, 23], etc.) revealed the presence of three families, i.e.
three copies of the same anomaly-free combination of fields. The measurement
of the $Z$ boson width and of the Higgs boson couplings to gluons and photons
has ruled out the possible presence of additional chiral families. It is
remarkable that the three families is the only aspect of the SM that does not
derive from the gauge principle. Via the Yukawa couplings, fermions in
different families receive hierarchical masses and non-trivial mixing in the
$W^{\pm}$ couplings via the Cabibbo-Kobayashi-Maskawa (CKM) [24, 25] and
Pontecorvo-Maki-Nakagawa-Sakata (PNMS) [26, 27] matrices for quarks and
leptons, respectively. The three family structure of the SM, and the patterns
in the Yukawa couplings, have been intensively investigated with the purpose
of understanding its origins, with notable attempts to restore the gauge
principle by gauging family symmetries [28, 29, 30, 31, 32, 33, 34, 35],
however, no fully satisfactory explanation has been found so far.

In this letter we propose a new family structure for the SM fermions, where
the muon is reassigned to the third family. As we will see, this reveals
potentially meaningful patterns in the masses and mixing, while pointing to
precise and testable predictions for experiments.

First, let us reconsider the rationale behind the standard family assignment
in the SM. As one can see from the current knowledge of the fermion sector
[36], charged fermions show a clear mass hierarchy between particles of the
same charge. Furthermore, the mixing angles between quark families are small,
signifying that the $W$ boson couples dominantly to well defined couples of
quarks: $u$–$d$, $c$–$s$ and $t$–$b$, also following the mass hierarchy. It is
therefore natural to associate each pair to a family, with ordering following
the masses. This rationale is, however, less evident for leptons, as the
mixing angles between families are clearly not hierarchical. Furthermore, in
the SM there are no direct connections between quarks and leptons besides the
cancellation of anomalies: no interactions allow to clearly associate a family
of quarks with a charged lepton. Henceforth, the only guiding principle is the
mass, yielding the following family associations: $e$–$u$–$d$, $\mu$–$c$–$s$,
$\tau$–$t$–$b$. We want to stress here that this family association within the
pure SM realm is completely arbitrary and conventional. While the $e$–$u$–$d$
association makes physical sense as those are the only charged fundamental
fermions we observe in ordinary matter, the fermions in the other two families
decay down to the first family states. Hence, considering a different
assignment, where the muon is associated to the third family instead of the
second, is a relevant question to ask.

While within the SM this reassignment is a purely academic exercise, in models
of new physics this reassignment can have physical meaning. This is the case
if particles interacting with both quarks and leptons are present, as for
instance lepto-quarks. They are bosons, of spin zero or one, coupling to one
quark and one lepton, hence carrying both baryon and lepton number global
charges. Lepto-quarks arise, for instance, in models of Grand Unification
[37], and in Pati-Salam models [38]. In the latter, leptons are associated to
quarks by extending the QCD gauge group to SU(4), hence embedding the lepton
number. Pati-Salam can be an intermediate step in SO(10) models [39]. In all
these models, quarks and leptons must be embedded into the same representation
of the larger gauge groups, therefore the association within families becomes
physically meaningful. In this letter we propose the following novel family
assignment:

$\displaystyle 1^{\rm st}\;\rm{family}\;\;$ $\displaystyle\Rightarrow$
$\displaystyle\;\;e-u-d\,,$ $\displaystyle 2^{\rm nd}\;\rm{family}\;\;$
$\displaystyle\Rightarrow$ $\displaystyle\;\;\tau-c-s\,,$ (1) $\displaystyle
3^{\rm rd}\;\rm{family}\;\;$ $\displaystyle\Rightarrow$ $\displaystyle\;\;\mu-
t-b\,.$

We mainly focus on two aspects of this new family structure:

  * a)

A new mass pattern emerges, pointing to a different origin for the Yukawa
couplings of the third generation and the other two.

  * b)

In models with light new physics, sizeable effects are predicted in muon
couplings, without the need for lepton flavour violation.

Figure 1:  Mass ordering for the quarks and leptons in the SM, depending on
their electric charge. Colours label the new family assignment we propose.

We first discuss the new mass pattern appearing in the new family arrangement,
as shown in Fig. 1. First, we observe that the fermions in the first and
second families, in blue and green respectively, have similar masses within
one order of magnitude, pointing to the MeV scale and GeV scale, respectively.
On the other hand, the third family, in red, shows a marked hierarchy between
top, bottom, and muon masses. Numerically, the two mass ratios

$X_{\rm tb}=\frac{m_{\rm t}}{m_{\rm b}}=41.31^{+0.31}_{-0.21}\,,\quad
X_{\rm{b}\mu}=\frac{m_{\rm b}}{m_{\mu}}=39.56^{+0.28}_{-0.19}\,,$ (2)

are surprisingly close. This can be better visualised via the ratio of ratios

$\xi=\frac{X_{\rm{b}\mu}}{X_{\rm tb}}=0.958^{+0.014}_{-0.009}\,,$ (3)

which is equal to one within 5%. This observation suggests that the origin of
the Yukawa couplings for the third generation, and for the other two, may be
very different. The most likely scenario has the following features:

  * •

A family symmetry SU(2) acting on the first two generations. This is broken by
effective Yukawa couplings of the order $y_{2}\approx 10^{-2}$ and
$y_{1}\approx 10^{-5}$.

  * •

The third generation has tree-level Yukawa couplings $y_{3}\approx 1$, which
prevents a larger family symmetry SU(3).

  * •

Only the Yukawa coupling of the top quark is allowed, while some new
interactions communicate the electroweak symmetry breaking to bottom and muon,
explaining their suppression compared to unity.

The mass ratio $X\sim X_{\rm tb}\sim X_{\rm{b}\mu}$ is, therefore, generated
by the latter mechanism. We envision two model-building avenues: the Froggatt-
Nielsen mechanism [40], where lighter fermions inherit the top Yukawa via
mixing to heavy fermions, and loop mass generation. The latter has been first
used in attempts to explain the electron mass from the muon one in the early
70’s [41]. Attempts to generate all quark and lepton masses from the top
Yukawa date back to the 90s [42, 43, 44], but mainly failed to generate the
lightest masses and to pass flavour changing neutral current (FCNC) tests. Our
scenario is free from such issues, as only the third generation masses would
need to be loop generated.

A bottom-up approach reveals that there is a unique minimal model with two
additional scalars that can generate the bottom and muon masses out of one and
two loops, respectively, from the top Yukawa. The proposed scalars are
triplets under QCD, and carry the following weak isospin and hypercharge,
${\bf I}_{Y}$: $S={\bf 1}_{-1/3}$ and $\phi={\bf 2}_{1/6}$. Interactions with
the SM fermions are generated at a high scale $\Lambda$, in the form of the
following Yukawa-like couplings:

$\displaystyle\mathcal{L}_{\rm Yuk}=$ $\displaystyle y_{t}\ \bar{t}_{R}q_{L}\
H+\lambda_{S}\ S\ \Big{(}c_{qq}\ \bar{q}^{c}_{L}q_{L}+c_{tb}\
\bar{b}^{c}_{R}t_{R}$ $\displaystyle+c_{ql}\ \bar{q}_{L}l_{L}^{c}+c_{t\mu}\
\bar{t}_{R}\mu_{R}^{c}+c_{b\nu}\ \bar{b}_{R}\nu_{R}^{c}\Big{)}+$
$\displaystyle\lambda_{\phi}\ \phi\ \Big{(}c_{q\nu}\
\bar{q}_{L}\nu_{R}+c_{bl}\ \bar{b}_{R}l_{L}\Big{)}+\mbox{h.c.}$ (4)

where the coefficients $c_{ij}$ depend on the ultra-violet completion of the
model, and $H$ is the BEH doublet. We have included right-handed neutrinos
$\nu_{R}^{i}$ with a Majorana mass, which will be needed to implement the see-
saw mechanism to suppress neutrino masses. To generate the correct mass
hierarchies, it is crucial that $c_{ql}(\Lambda)=0$ while a non-zero value is
generated by loops of $\phi$ to yield the correct suppression in the muon
mass. For simplicity, we will fix all the other coefficients to unity,
$c_{lk}=1$. Defining

$\epsilon_{S/\phi}=\frac{1}{8\pi^{2}}\ln\frac{\Lambda}{M_{S/\phi}}\,,$ (5)

at the scalar mass scale

$c_{ql}=\lambda_{\phi}^{2}\ n_{R}\ \epsilon_{\phi}\,,$ (6)

where $n_{R}$ is the number of right-handed neutrinos lighter than the
scalars. Both bottom and muon masses are generated at one loop from the top
mass, however $m_{\mu}$ contains an extra suppression via $c_{ql}$. Including
only the logarithmic contribution, corresponding to the renormalisation group
evolution from $\Lambda$ down to the scalar mass scales, we find:

$\frac{m_{\rm b}}{m_{\rm t}}=N_{c}\ \lambda_{S}^{2}\
\epsilon_{S}\,,\quad\frac{m_{\mu}}{m_{\rm b}}=n_{R}\ \lambda_{\phi}^{2}\
\epsilon_{\phi}\,,$ (7)

where $N_{c}=3$ is the number of QCD colours. These values should match the
mass ratios at a scale corresponding to the scalar masses. A Dirac neutrino
mass is also generated from the bottom mass via a loop of $\phi$, giving

$\frac{m_{\nu}^{D}}{m_{\rm b}}=N_{c}\ \lambda_{\phi}^{2}\ \epsilon_{\phi}\,.$
(8)

Hence, the Majorana mass for light neutrinos is given by

$m_{\nu}=\frac{(m_{\nu}^{D})^{2}}{M_{R}}=N_{c}^{2}\lambda_{\phi}^{4}\epsilon_{\phi}^{2}\frac{m_{\rm
b}^{2}}{M_{R}}\,.$ (9)

These estimates are valid for the mass hierarchy $M_{R}\lesssim M_{\phi}\sim
M_{S}$. By matching these estimates with the values of the ratios $X_{\rm tb}$
and $X_{\rm{b}\mu}$ at the scale $M_{R}$ and the neutrino mass, we can
estimate the values of the necessary mass scales. For $m_{\nu}=1$ eV,
$n_{R}=3$ and $\lambda_{S}=\lambda_{\phi}=0.3$, we obtain, for instance,

$M_{R}=1.3\
10^{7}~{}\mbox{GeV}\,,\quad\frac{M_{S}}{M_{\phi}}=1150\,,\quad\frac{\Lambda}{M_{S}}=490\,,$
(10)

where the value of the scalar masses cannot be determined, but are higher than
the right-handed neutrino masses. Note that the scale $\Lambda$ is large
enough to avoid dangerous FCNCs, however the couplings of $S$ violate baryon
and lepton number and will generate proton decay via the process $p\to
K_{0}\mu^{+}$. The effective scale suppressing this process is required to be
$\Lambda_{p}>10^{16}$ GeV. Assuming a CKM-like suppression for the mixing of
the up and down-type quarks with the third generation, we estimate
$M_{S}/\lambda_{S}\gtrsim 10^{12}$ GeV. The bound from proton decay could be
removed altogether by introducing two $S$ scalars: $S_{qq}$ that couples only
to di-quarks and carries baryon number $B=-2/3$, and $S_{lq}$ that couples to
one lepton and one quark and has $B=1/3$ and $L=1$. Note, finally, that the
large scale is mainly due to the neutrino masses and the seesaw mechanism. It
is possible to lower the right-handed neutrino masses $M_{R}$ by employing the
inverse seesaw mechanism [45], hence lowering the scalar masses closer to the
TeV scale.

The second consequence of putting the muon in the third generation could
emerge in the low energy effects of light new physics, around the TeV scale.
This is justified by the inherent link between the electroweak symmetry
breaking and the top quark, who enjoys the largest coupling to the BEH field
in the SM. Henceforth, new physics at the TeV scale may couple preferentially
to third generation fermions. New physics coupling dominantly to the muon has
been hinted at by recent measurements (see the recent review [46]), mainly
stemming from $B$ meson decays and the measurement of the muon anomalous
magnetic moment. For the former, a minimal explanation is offered by a TeV
scale spin-1 leptoquark with electroweak quantum numbers ${\bf 1}_{2/3}$ [47,
48]. If this state were to couple dominantly to third generation, following
the usual assignment, a sizeable mixing between tau and muon would be
required, hence leading to lepton-flavour violation (LFV) [49, 50], which is
strongly constrained by experiments. If the muon is assigned to third
generation, the observed anomalies could be explained without the need for LFV
signatures [51], hence avoiding the most stringent constraints on the models.
The minimal coupling of the leptoquark, named here $U_{1}$, are

$\mathcal{L}_{U}=U_{1}^{\mu}\
\bar{b}\left(\kappa_{L}P_{L}+\kappa_{R}P_{R}\right)\mu+\mbox{h.c.}$ (11)

where a left-handed coupling seems to be preferred by data. The precise value
of the couplings depends on the underlying model. Note that a spin-1
leptoquark with this quantum numbers is naturally generated by a Pati-Salam
(PS) unification.

Figure 2:  Allowed parameter space in the PS3 model with muon in the third
generation. The parameter space is delimited by the blue line, with constrains
from LFV ($\tau\to 3\mu$) in blue [52, 53] and $\Delta F=2$ bounds in red
[53]. Contours for the leptoquark coupling $C_{U}$ are indicated in black.

Henceforth, as a concrete higher energy realisation, in the following we will
consider the PS3 model in Refs [54, 55, 56]. The model is based on a family-
specific PS symmetries, broken at different energy scales. As a consequence,
the PS symmetry on the third generation is broken at the lightest scale, close
to the TeV. This leads to a $U_{1}$ leptoquark that couples dominantly to
third generation. Furthermore, the generic mass matrix for the SM fermions has
the following structure:

$\mathcal{L}_{\rm
mass}=\bar{f}_{R}\left(\begin{array}[]{ccc}\epsilon_{LR}U_{11}&\epsilon_{LR}U_{12}&0\\\
\epsilon_{LR}U_{21}&\epsilon_{LR}U_{22}&\epsilon_{L}\\\
0&\epsilon_{R}&y_{f}v\end{array}\right)f_{L}\,,$ (12)

where the $\epsilon$ terms are generated by higher dimensional operators,
stemming from the breaking of the first and second generation PS symmetries,
while $y_{f}$ comes from a renormalisable coupling of the BEH field to the
third generation. Hence, a natural hierarchy
$\epsilon_{R}<\epsilon_{LR}<\epsilon_{L}<1$ emerges from the mode. The matrix
$U$ parameterises the breaking of the U(2) symmetry and the mixing with the
first generation. In this model, assigning the muon to the third generation is
equivalent to assuming that $\epsilon_{L}\sim m_{\tau}$, while $y_{f}v\sim
m_{\mu}$. Note that $\epsilon_{L}\sim$ GeV is inconsistent with the masses of
the second generation quarks. However, the assumption that
$y_{f}v\ll\epsilon_{L}$ implies that: a) The left-handed muon belongs to the
third generation; b) The right-handed muon belongs to the second generation.
The PS leptoquark $U_{1}$ that couples to the third generation, therefore,
will dominantly couple to the left-handed muons and the right-handed taus. We
re-analysed the current low energy limits in the new configuration, which
naturally predicts sizeable effects on the muon couplings. On Fig. 2 we show
the allowed parameter space of the model, once the muon is assigned to third
generation. We show a benchmark in the parameter space characterised by the
$b\to s\mu^{+}\mu^{-}$ processes, incarnated in the ratios

$R_{K^{(*)}}=\frac{\mbox{Br}[B\to K^{(*)}\mu^{+}\mu^{-}]}{\mbox{Br}[B\to
K^{(*)}e^{+}e^{-}]}\,,$ (13)

and $\mbox{Br}[B_{S}\to\mu^{+}\mu^{-}]$ [36] normalised to the SM central
value

$\left.\mbox{Br}[B_{S}\to\mu^{+}\mu^{-}]\right|_{\rm SM}=(3.65\pm 0.23)\cdot
10^{-9}\,.$ (14)

We recomputed the bounds on the PS3 model under the assumption of third
generation muon, i.e. $\epsilon_{L}\sim m_{\tau}$, showing the allowed region
for a specific benchmark model in Fig. 2. The black lines contours indicate
the predicted region, while the red and blue areas indicate the regions
preferred by $\Delta F=2$ constraints (mainly $B_{s}-\bar{B}_{s}$ mixing) [53]
and LFV ones (mainly $\tau\rightarrow 3\mu$)[52, 53], respectively, while the
bounds from Lepton Flavor Universality do not constrain the allowed region.
The intersection of the two regions indicate the prediction of the model to be
compared to the experimental determinations of $R_{K}$ [57] and
$\mbox{Br}[B_{S}\to\mu^{+}\mu^{-}]$ [36], indicated by the orange points in
the plot (the error bands take into account the experimental errors only).
Compared to the original model, in our set up the new physics effects are
driven by the coupling $g_{U}$ of the leptoquark, which is expressed in terms
of the couplings

$C_{U}=\frac{g_{U}^{2}v^{2}}{M_{U}^{2}}\,,$ (15)

where $v=246$ GeV is the SM vacuum expectation value, $M_{U}$ and $g_{U}$ are
the leptoquark mass and coupling, respectively. For natural values of
$g_{U}\sim 3$ [54], this leads to $M_{U}\sim 10$ TeV, hence well above the LHC
reach but within reach of future hadron colliders.

In conclusions, we have shown that the assignment of the muon to second
generation is arbitrary in the SM. We have listed the potential benefits of
assigning the muon to third generation, leading to a new understanding of the
mass patterns, and also predicting large deviations in the couplings of muons
to the SM. The latter effects, mainly due to light new physics around the TeV
scale, is in agreement with recent deviation in low energy properties of
muons.

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