# Muon ${g-2}$ Anomaly and Neutrino Magnetic Moments

K.S. Babu<EMAIL_ADDRESS>Department of Physics, Oklahoma State University,
Stillwater, OK, 74078, USA  Sudip Jana<EMAIL_ADDRESS>Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Manfred
Lindner<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Vishnu P.K<EMAIL_ADDRESS>Department of Physics, Oklahoma State University, Stillwater, OK, 74078, USA

###### Abstract

We show that a unified framework based on an $SU(2)_{H}$ horizontal symmetry
which generates a naturally large neutrino transition magnetic moment and
explains the XENON1T electron recoil excess also predicts a positive shift in
the muon anomalous magnetic moment. This shift is of the right magnitude to be
consistent with the Brookhaven measurement as well as the recent Fermilab
measurement of the muon $g-2$. A relatively light neutral scalar from a Higgs
doublet with mass near 100 GeV contributes to muon $g-2$, while its charged
partner induces the neutrino magnetic moment. In contrast to other multi-
scalar theories, in the model presented here there is no freedom to control
the sign and strength of the muon $g-2$ contribution. We analyze the collider
tests of this framework and find that the HL-LHC can probe the entire
parameter space of these models.

_Introduction_.– There has been considerable interest in understanding the
long-standing discrepancy between the measured and predicted values of the
anomalous magnetic moment of the muon, $a_{\mu}$. The Brookhaven Muon g-2
collaboration has measured it to be $a_{\mu}({\rm BNL})=116592089(63)\times
10^{-11}$ two decades ago Bennett:2006fi , while theoretical predictions find
it to be $a_{\mu}({\rm theory})=116591810(43)\times 10^{-11}$ Aoyama:2020ynm .
Taken at face value the difference, $\Delta a_{\mu}=a_{\mu}({\rm
experiment})-a_{\mu}({\rm theory})\simeq 279\times 10^{-11}$, is a 3.7 sigma
discrepancy, which may indicate new physics lurking around or below the TeV
scale. Very recently the Fermilab Muon g-2 collaboration Abi:2021gix has
announced their findings, which measures it to be $a_{\mu}({\rm
FNAL})=116592040(54)\times 10^{-11}$ which confirms the Brookhaven measurement
and increases the significance of the discrepancy to the level of 4.2 sigma.
These results make the motivations for new physics explanation more
compelling.

The purpose of this paper is draw a connection between new physics
contributions to $a_{\mu}$ and a possible neutrino transition magnetic moment
$\mu_{\nu_{\mu}\nu_{e}}$. A sizable neutrino magnetic moment has been
suggested as a possible explanation for the excess in electron recoil events
observed in the $(1-7)$ keV recoil energy range by the XENON1T collaboration
recently Aprile:2020tmw . The neutrino magnetic moment needed to explain this
excess lies in the range of $(1.6-2.4)\times 10^{-11}\mu_{B}$, where $\mu_{B}$
stands for the electron Bohr magneton Babu:2020ivd . Such a value would
require new physics to exist around the TeV scale. As we shall show in this
paper, models that induce neutrino magnetic moments, while maintaining their
small masses naturally, also predict observable shifts in the muon anomalous
magnetic moment. We focus on a specific class of models based on an
$SU(2)_{H}$ horizontal symmetry (or family symmetry) acting on the electron
and muon families that naturally leads to a large neutrino magnetic moment
Voloshin:1987qy ; Babu:1989wn . We find that within this class of models, an
explanation of the XENON1T excess will necessarily lead to a positive
contribution to $\Delta a_{\mu}$, which lies neatly within the Brookhaven and
the recent Fermilab measurements of $a_{\mu}$ Abi:2021gix . This class of
models is thus in accordance with Occam’s razor, explaining both anomalies in
terms of the same new physics.

_Model_.– We focus on models that can generate sizable transition magnetic
moments for the neutrinos while keeping their masses naturally small. Models
which generate a neutrino transition magnetic moment of order
$10^{-11}\mu_{B}$ would typically also induce an unacceptably large neutrino
mass of order 0.1 MeV. An $SU(2)_{H}$ family symmetry acting on the electron
and muon families can resolve this conundrum as it decouples the neutrino
magnetic moment from the neutrino mass Babu:1989wn . This follows from the
different Lorentz structures of the transition magnetic moment and mass
operators. The magnetic moment operator
$(\nu_{e}^{T}C\sigma_{\mu\nu}\nu_{\mu})F^{\mu\nu}$, being antisymmetric in
flavor is a singlet of $SU(2)_{H}$, while the mass operator
$(\nu_{e}^{T}C\nu_{\mu})$, being symmetric in flavor, belongs to $SU(2)_{H}$
triplet and vanishes in the $SU(2)_{H}$ symmetric limit Voloshin:1987qy . The
class of models we study makes use of this symmetry argument. New physics at
the TeV scale is required to generate the magnetic moment of the desired
magnitude. This should violate lepton number, since the transition magnetic
moment operator is a $\Delta L=2$ operator. This requirement naturally leads
to the model of Ref. Babu:2020ivd . It is based on an $SU(2)_{H}$ extension of
the Zee model of neutrino mass Zee:1980ai involving a second Higgs doublet and
a charged singlet $\eta^{\pm}$. The extension incorporates the $SU(2)_{H}$
symmetry.

The class of models under study is based on the SM gauge group and an
$approximate$ $SU(2)_{H}$ horizontal symmetry acting on the electron and muon
families Babu:2020ivd ; Babu:1989wn ; Babu:1990wv . The lepton fields
transform under $SU(2)_{L}\times U(1)_{Y}\times SU(2)_{H}$ as follows:

$\psi_{L}=\begin{pmatrix}\nu_{e}&\nu_{\mu}\\\
e&\mu\end{pmatrix}_{L}\sim(2,-\dfrac{1}{2})(2),\quad\psi_{R}=\left(e\quad\mu\right)_{R}\sim(1,-1)(2),\quad\psi_{3L}=\begin{pmatrix}\nu_{\tau}\\\
\tau\end{pmatrix}_{L}\sim(2,-\dfrac{1}{2})(1),\quad\tau_{R}\sim(1,-1)(1).$ (1)

All quark fields transform as singlets of $SU(2)_{H}$ symmetry and will not be
discussed further here. The scalar sector of the model contains the following
fields:

$\phi_{S}=\begin{pmatrix}\phi_{S}^{+}\\\
\phi_{S}^{0}\end{pmatrix}\sim(2,\dfrac{1}{2})(1),\quad{\Phi=\begin{pmatrix}\phi_{1}^{+}&\phi_{2}^{+}\\\
\phi_{1}^{0}&\phi_{2}^{0}\end{pmatrix}}\sim(2,\dfrac{1}{2})(2),\quad\eta=\left(\eta_{1}^{+}\quad\eta_{2}^{+}\right)\sim(1,1)(2).$
(2)

Here the $\phi_{S}$ field is the SM Higgs doublet which acquires a vacuum
expectation value (VEV) $\langle\phi_{S}^{0}\rangle=v/\sqrt{2}$, with $v=246$
GeV. The $\Phi$ and $\eta$ fields, which are doublets of $SU(2)_{H}$, are
needed to generate transition magnetic moments for the neutrino. Both fields
are necessary in order to break lepton number. The neutral components of the
$\Phi$ field are assumed to not acquire any VEV. With this particle content,
in the $SU(2)_{H}$ symmetric limit, the leptonic Yukawa couplings of the model
are given by the Lagrangian Babu:2020ivd

$\displaystyle\mathcal{L}_{\rm Yuk}$ $\displaystyle=$ $\displaystyle
h_{1}\operatorname{Tr}\left(\bar{\psi}_{L}\phi_{S}\psi_{R}\right)+h_{2}\bar{\psi}_{3L}\phi_{S}\tau_{R}+h_{3}\bar{\psi}_{3L}\Phi
i\tau_{2}\psi_{R}^{T}$ (3)
$\displaystyle+f\eta\tau_{2}\psi_{L}^{T}\tau_{2}C\psi_{3L}+f^{\prime}\operatorname{Tr}\left(\bar{\psi}_{L}\Phi\right)\tau_{R}+H.c.$

Eq. (3) would lead to the relation $m_{e}=m_{\mu}$, which of course is
unacceptable. This is evaded by allowing for explicit but small $SU(2)_{H}$
breaking terms in the Lagrangian. In particular, the following Yukawa
couplings are necessary:

$\displaystyle{\cal L}_{\rm Yuk}^{\prime}=\delta
h_{1}[(\overline{L}_{e}\phi_{S}e_{R})-(\overline{L}_{\mu}\phi_{S}\mu_{R})]+H.c.$
(4)

Here $L_{e}^{T}=(e\leavevmode\nobreak\ \nu_{e})_{L}$, etc. With this term
included, $m_{e}\neq m_{\mu}$ is realized, since now we have
$m_{e}=(h_{1}+\delta h_{1})v/\sqrt{2}$ and $m_{\mu}=(h_{1}-\delta
h_{1})v/\sqrt{2}$. While the violation of $SU(2)_{H}$ is small, $\delta
h_{1}=(m_{e}-m_{\mu})/(\sqrt{2}v)\simeq 3\times 10^{-4}$, it is nevertheless
significant in constraining the parameter space of the model, as we shall see.
There are also other $SU(2)_{H}$ breaking terms in the Lagrangian of similar
order, but these terms will not play any significant role in our discussions.

The scalar potential of the model contains $SU(2)_{H}$ symmetric terms of the
form Babu:2020ivd :

$\displaystyle V$ $\displaystyle\supset$ $\displaystyle
m_{\eta}^{2}(|\eta_{1}|^{2}+|\eta_{2}|^{2})+m_{\phi^{+}}^{2}(|\phi_{1}^{+}|^{2}+|\phi_{2}^{+}|^{2})$
(5) $\displaystyle+$ $\displaystyle
m_{\phi^{0}}^{2}(|\phi_{1}^{0}|^{2}+|\phi_{2}^{0}|^{2})+\\{\mu\,\eta\,\Phi^{\dagger}i\tau_{2}\phi_{S}^{*}+H.c.\\}$

The cubic term in the Eq. (5) would lead to mixing between the charged scalars
$\eta_{i}^{+}$ and $\phi_{i}^{+}$ (with $i=1,2$). The corresponding mass
eigenstates are denoted as $h_{i}^{+}$ (with mass $m_{h^{+}}^{2}$) and
$H_{i}^{+}$ (with mass $m_{H^{+}}^{2}$) and are given by

$\displaystyle
h_{i}^{+}=\cos{\alpha}\;\eta_{i}^{+}+\sin{\alpha}\;\phi_{i}^{+},$
$\displaystyle
H_{i}^{+}=-\sin{\alpha}\;\eta_{i}^{+}+\cos{\alpha}\;\phi_{i}^{+},$ (6)

where the mixing angle $\alpha$ is defined as

$\displaystyle\tan 2\alpha=\frac{\sqrt{2}\mu
v}{m_{\eta}^{2}-m_{\phi^{+}}^{2}}.$ (7)

This mixing is essential for lepton number violation and the generation of
magnetic moment of the neutrino. Note that the two copies of $h_{i}^{+}$ are
degenerate, as are the two $H_{i}^{+}$ owing to $SU(2)_{H}$ symmetry.

_Muon anomalous magnetic moment and neutrino transition magnetic moment_.–

Figure 1: New physics affecting muon anomalous magnetic moment as well as
$\nu_{\mu}$-magnetic moment.  Figure 2: The dominant contributions to $\Delta
a_{\mu}$ (top) and neutrino magnetic moment (bottom) are shown. For the
neutrino magnetic moment, the outgoing photon can be emitted also from the
$\tau$ lepton line. There are other diagrams, which are subleading.

Fig. 1 shows the connection between $\Delta a_{\mu}$ and
$\mu_{\nu_{\mu}\nu_{e}}$ in a generic model. Fig. 2 has the explicit diagrams
in the $SU(2)_{H}$ symmetric models. We first focus on the new contributions
to the muon anomalous magnetic moment within our framework. Both the neutral
scalars and charged scalars present in the model contribute to $\Delta
a_{\mu}$ via one-loop diagrams shown in Fig. 2. Since chirality flip occurs on
the external legs in these diagrams, the loop corrections mediated by the
neutral scalars $\phi_{1}^{0}$ and $\phi_{2}^{0}$ contribute positively to
$a_{\mu}$, whereas the corrections from the charged scalars result in negative
$a_{\mu}$. The full one-loop contributions to $\Delta a_{\mu}$ are given by
Leveille:1977rc

$\displaystyle\Delta a^{\varphi^{0}}_{\mu}$
$\displaystyle=\frac{m_{\mu}^{2}}{16\pi^{2}}\left(|f^{\prime}|^{2}+|h_{3}|^{2}\right)F_{\varphi^{0}}[m_{\phi^{0}}],$
(8) $\displaystyle\Delta a^{\varphi^{+}}_{\mu}$
$\displaystyle=\frac{m_{\mu}^{2}}{16\pi^{2}}\left(|f\cos{\alpha}|^{2}+|h_{3}\sin{\alpha}|^{2}\right)F_{\varphi^{+}}[m_{h^{+}}]$
$\displaystyle+\frac{m_{\mu}^{2}}{16\pi^{2}}\left(|\text{--}f\sin{\alpha}|^{2}+|h_{3}\cos{\alpha}|^{2}\right)F_{\varphi^{+}}[m_{H^{+}}],$
(9)

where the loop functions are given by

$\displaystyle
F_{\varphi^{0}}[m_{\varphi^{0}}]=\int_{0}^{1}dx\frac{x^{2}(1-x)}{m_{\mu}^{2}x^{2}+m^{2}_{\varphi^{0}}(1-x)+x(m^{2}_{\tau}-m^{2}_{\mu})},$
(10) $\displaystyle
F_{\varphi^{+}}[m_{\varphi^{+}}]=\int_{0}^{1}dx\frac{x^{2}(x-1)}{m_{\mu}^{2}x^{2}+x(m^{2}_{\varphi^{+}}-m_{\mu}^{2})}.$
(11)

Figure 3: The $2\sigma$ allowed range (in green) for the muon anomalous
magnetic moment measurement Abi:2021gix at the Fermilab in the
$|f^{\prime}|$-$m_{\phi^{0}}$ plane. The region labelled XENON1T explains the
electron recoil excess at $90\%$ C.L Aprile:2020tmw . The dashed line indicate
the limit on neutrino magnetic moment from the BOREXINO experiment
Borexino:2017fbd . The shaded regions denote the excluded parameter space from
various experiments: the light pink shaded region from the LEP experiment
LEP:2003aa and the yellow shaded region from the $Z$-decay width measurements
LEP:2003aa ; Tanabashi:2018oca . Here we have fixed $f$ = $10^{-2}$,
$\sin\alpha$ = 0.35, $m_{h^{+}}=m_{\phi^{0}}+50$ GeV and
$m_{H^{+}}=m_{h^{+}}+350$ GeV.

Here $m_{\phi^{0}}$ is the common mass $m_{\phi_{1}^{0}}=m_{\phi_{2}^{0}}$. In
Eq. (8), the two terms proportional to $|f^{\prime}|^{2}$ and $|h_{3}|^{2}$
arise from diagrams with $\phi_{2}^{0}$ and $\phi_{1}^{0}$ exchange
respectively. The two contributions in Eq. (9) arise from the exchange of
$h^{+}$ and $H^{+}$.

We now turn to the transition magnetic moment of the neutrino in our framework
and show its correlation with $\Delta a_{\mu}$. The diagrams generating
sizeable $\mu_{\nu_{\mu}\nu_{e}}$ are shown in Fig. 2, bottom panel. Owing to
the $SU(2)_{H}$ symmetry of the model, the two diagrams add in their
contributions to the magnetic moment, while they subtract in their
contributions to neutrino mass when the photon line is removed from these
diagrams (for details, see Ref. Babu:2020ivd ; Babu:1990wv ). The resulting
neutrino magnetic moment is given by Babu:2020ivd ; Babu:1990wv

$\mu_{\nu_{\mu}\nu_{e}}=\frac{ff^{\prime}}{8\pi^{2}}m_{\tau}\sin
2\alpha\left[\frac{1}{m_{h^{+}}^{2}}\left\\{\ln\frac{m_{h^{+}}^{2}}{m_{\tau}^{2}}-1\right\\}-\frac{1}{m_{H^{+}}^{2}}\left\\{\ln\frac{m_{H^{+}}^{2}}{m_{\tau}^{2}}-1\right\\}\right]\leavevmode\nobreak\
.$ (12)

Comparing Eq. (12) with Eqs. (8) and (9) we see that $\Delta a_{\mu}$ and
$\mu_{\nu_{\mu}\nu_{e}}$ are dependent on the same set of parameters. To see
their correlation quantitatively, we have to take into account the various
constraints that exist on these parameters, which we now address.

First of all, for $\mu_{\nu_{\mu}\nu_{e}}$ to be in the range of
$(1.6-3.4)\times 10^{-11}\mu_{B}$ so that the XENON1T excess is explained, one
needs $|ff^{\prime}\sin 2\alpha|\geq(1-8)\times 10^{-3}$, corresponding to
$m_{h^{+}}=(100-300)$ GeV. The coupling $|f|$ can be constrained from the
induced neutrino masses from the $SU(2)_{H}$ breaking effects. The
cancellation that occurs among the two diagrams in the bottom panel of Fig. 2
in neutrino mass is valid only in the strict $SU(2)_{H}$ limit. Since
$m_{e}\neq m_{\mu}$ is necessary, one should examine neutrino masses including
$SU(2)_{H}$ breaking effects. The resulting neutrino mass is given by
Babu:2020ivd

$\displaystyle m_{\nu}\simeq\frac{ff^{\prime}m_{\tau}\sin
2\alpha}{16\pi^{2}}\left[\frac{\delta m_{\eta}^{2}}{m_{\eta}^{2}}-\frac{\delta
m_{\phi}^{2}}{m_{\phi}^{2}}+2(\delta\alpha)\cot{2\alpha}\text{ln}\frac{m_{\phi}^{2}}{m_{\eta}^{2}}\right].$
(13)

Here $\delta m_{\eta}^{2}$ is the squared mass splitting between the two
charged scalars $\eta_{1}^{+}$ and $\eta_{2}^{+}$ arising from $SU(2)_{H}$
breaking effects. Similarly, $\delta\alpha$ is the shift in the common mixing
angle $\alpha$ that parametrizes $\eta_{1}^{+}-\phi_{1}^{+}$ mixing and
$\eta_{2}^{+}-\phi_{2}^{+}$ mixing. The mass splitting among charged scalars
arising from $m_{e}\neq m_{\mu}$ is found to be

$\displaystyle\delta
m_{\eta}^{2}\simeq\frac{|f|^{2}(m_{\mu}^{2}-m_{e}^{2})}{16\pi^{2}}$ (14)

which does not induce large neutrino mass, even for $|f|\sim 1$. (Eq. (14)
should replace Eq. (4.35) of Ref. Babu:2020ivd , which is incorrect.) However,
the coupling $|f|$ is constrained from the universality limits from $\tau$
decay. In Ref. Babu:2020ivd it was shown that the leptonic decay rate of the
$\tau$ will receive additional contributions from the charged scalars leading
to the modification $\Gamma(\tau\rightarrow e)=\Gamma_{\rm
SM}\times(1+\epsilon_{\tau})^{2}$, where

$\epsilon_{\tau}=\frac{f^{2}}{g^{2}}m_{W}^{2}\left(\frac{\cos^{2}\phi}{m_{h^{+}}^{2}}+\frac{\sin^{2}\phi}{m^{2}_{H^{+}}}\right)\leavevmode\nobreak\
.$ (15)

A limit of $\epsilon_{\tau}\leq 0.004$ can be derived at 2 $\sigma$ from the
constraint $g_{\tau}/g_{\mu}=1.0011\pm 0.0015$ Pich:2013lsa obtained by
comparing $\tau$ decay rate into muon with muon decay rate. This leads to the
constraint $|f|\leq(0.05-0.25)$ for $h^{+}$ mass in the range of $(100-500)$
GeV. As a result, the charged scalar contributions to the muon anomalous
magnetic moment turn out to be not significant.

The coupling $h_{3}$ is constrained from the induced tau neutrino mass within
the model, which is given by

$m_{\nu_{\tau}}=\frac{h_{3}f\sin 2\alpha}{32\pi^{2}}m_{\mu}{\rm
ln}\left(\frac{m_{h^{+}}^{2}}{m_{H^{+}}^{2}}\right)\leavevmode\nobreak\ .$
(16)

Demanding $m_{\nu_{\tau}}\leq 0.05$ eV, while also requiring
$\mu_{\nu_{\mu}\nu_{e}}\sim 1.6\times 10^{-11}\mu_{B}$ one obtains a limit
$|h_{3}|<10^{-2}$. Consequently, the neutral $\phi_{1}^{0}$ exchange of Fig. 2
to the muon anomalous magnetic moment is negligible. Thus we arrive at the
model prediction that $\Delta a_{\mu}$ receives significant contributions only
from $\phi_{2}^{0}$ exchange diagram.

In Fig. 3 we show the parameter space in the Yukawa coupling $|f^{\prime}|$
versus the mass of the neutral scalar ($m_{\phi^{0}}$) plane consistent with
the observed muon anomalous magnetic moment measurement $\Delta
a_{\mu}=(251\pm 59)\times 10^{-11}$ at Fermilab Abi:2021gix as indicated by
green band. Here we have fixed the charged scalar masses at
$m_{h^{+}}=m_{\phi^{0}}+50$ GeV and $m_{H^{+}}=m_{h^{+}}+350$ GeV, a choice
consistent with electroweak $T$ parameter constraint. A lower bound on the
neutral scalar mass is obtained from the decay width measurements of the $Z$
boson: $m_{\phi^{0}}\gtrsim 45$ GeV LEP:2003aa ; Tanabashi:2018oca , which is
indicated by the yellow shaded region. The Yukawa coupling $f^{\prime}$ leads
to additional contributions to the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ process at
the LEP experiment LEP:2003aa via $t-$channel neutral scalar exchange. This
leads to an upper limit on $f^{\prime}$ which is depicted by the pink shaded
region. Moreover, our model leads to the most promising signal
$pp\rightarrow\phi_{i}^{0}\phi_{i}^{*0}\to\mu^{-}\mu^{+}\tau^{-}\tau^{+}$ at
the LHC where different flavored leptons can be easily reconstructed as a
resonance. The collider phenomenology of this model has been studied in detail
in Ref. Babu:2020ivd . The blue dashed line represents the 5$\sigma$
sensitivity of the neutral scalar of mass 398 GeV Babu:2020ivd at the HL-LHC
with an integrated luminosity of $\mathcal{L}=1$ ab-1. This can be a promising
test of the model for neutrino magnetic moment and muon $g-2$. We also note
that the model predicts a deviation in the anomalous magnetic moment of the
electron, $a_{e}$, which is calculable owing to the $SU(2)_{H}$ symmetry. We
find it to be $\Delta a_{e}=\Delta a_{\mu}(m_{e}/m_{\mu})^{2}\approx 6\times
10^{-14}$. This predicted deviation is consistent with the current
measurements Parker_2018 ; Morel:2020dww , but may serve as a future test of
the model.

In Fig. 4 we have shown a direct correlation between the muon anomalous
magnetic moment and neutrino magnetic moment within our framework. As noted
before, the dominant contribution to $\Delta a_{\mu}$ solely depends on the
Yukawa coupling $f^{\prime}$ and the neutral scalar mass $m_{\phi^{0}}$. The
transition magnetic moment of the neutrino $\mu_{\nu_{\mu}\nu_{e}}$ depends on
the same Yukawa coupling $f^{\prime}$ as well as the charged scalar masses.
The measurement of the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ process at the LEP
experiment LEP:2003aa imposes a strong limit on the Yukawa coupling
$f^{\prime}$ as a function of the neutral scalar mass $m_{\phi_{0}}$.
Therefore, in Fig. 4, we vary the Yukawa coupling $f^{\prime}$ in such a way
that the parameters are consistent with these constraints. An optimal scenario
is realized when the neutral scalar is light, with its mass not lower than 45
GeV so as to be consistent with $Z$ decay constraint Tanabashi:2018oca . The
mass splitting between the lightest charged scalar and neutral scalar is set
to be 50 GeV in order to satisfy the charged Higgs mass limit of $\sim$95 GeV
(for detail see Ref. Babu:2019mfe ) from collider searches. As for the charged
scalar masses which have a strong impact on the neutrino magnetic moment, but
little effect on the muon $g-2$, we note that these masses cannot be
arbitrary, as the mass splittings between the charged and neutral scalars are
tightly bounded from the electroweak precision data Tanabashi:2018oca ;
Babu:2019mfe . Therefore, it follows that there is no extra room to control
the strength and the sign of the muon $g-2$ in our setup. The parameter space
which predicts large neutrino magnetic moment leads to a positive contribution
to muon $g-2$. Thus, the measurement of muon $g-2$ by the Fermilab experiment
can be an indirect and novel test of the neutrino magnetic moment hypothesis,
which can be as sensitive as other ongoing neutrino/dark matter experiments.

Figure 4: Theoretical predictions and experimental measurements of the muon
anomalous magnetic moment and the neutrino transition magnetic moment. Purple,
cyan, light blue, blue, green, yellow, orange and red scattered points depict
the correlated predictions for the muon magnetic moment ($a_{\mu}$) and
neutrino magnetic moment ($\mu_{\nu_{\mu}\nu_{e}}$) in our framework for
different ranges of mass splittings between the charged scalars:
$m_{H^{+}}-m_{h^{+}}=20,50,100,200,300,500,800$ and 1000 GeV. Here we choose
different ranges of parameters as
$\\{f^{\prime}\in(0.001-1),\leavevmode\nobreak\ m_{\phi^{0}}\in(45-500)$ GeV,
and $m_{H^{+}}-m_{h^{+}}\in(20$ GeV - 1 TeV)$\\}$ consistent with the
theoretical and experimental limits discussed in text. The lower black dotted
point with error bar indicates the updated SM prediction Aoyama:2020ynm for
$a_{\mu}$ (see other previous analyses (in chronological order) as: HMNT06
Hagiwara:2006jt , DHMZ10 Davier:2010nc , JS11 Jegerlehner:2011ti , HLMNT11
Hagiwara:2011af , DHMZ17 Davier:2017zfy , KNT18 Keshavarzi:2018mgv ) , and
others Colangelo:2018mtw ; Hoferichter:2019mqg ; Davier:2019can ;
Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ; Masjuan:2017tvw ;
Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ;
Colangelo:2019uex ; Colangelo:2014qya ; Blum:2019ugy ; Aoyama:2012wk ;
Czarnecki:2002nt ; Gnendiger:2013pva ). The upper dotted data points with
error bars represent different experimental measurements: BNLE821
Bennett:2006fi , Fermilab Abi:2021gix , and the the combined experimental
average of the BNL Bennett:2006fi and Fermilab Abi:2021gix results. These
evaluations and measurements of $a_{\mu}$ are independent of neutrino magnetic
moment values. The current uncertainty on the measurement of $a_{\mu}$ at the
Fermilab Abi:2021gix is given by the light brown band at $1\sigma$ level. The
purple band represents the SM prediction from Ref. Aoyama:2020ynm with
$1\sigma$ value and the light green band indicates the combined experimental
average of the BNL Bennett:2006fi and Fermilab Abi:2021gix results at
$1\sigma$ level. Here we choose the reference value $a_{ref}=11659000.$ The
orange band labelled with XENON1T denotes the preferred range
$\mu_{\nu_{e}\nu_{\mu}}\in(1.65-3.42)\times 10^{-11}\mu_{B}$ to explain the
electron recoil excess at $90\%$ C.L Aprile:2020tmw . The vertical lines
indicate the limits on neutrino magnetic moment from various measurements:
blue solid line from TEXONO Deniz:2009mu , dark green solid line from BBN
Vassh:2015yza , light green solid line from BOREXINO Borexino:2017fbd , and
cyan dashed line from globular clusters Viaux:2013lha . The astrophysical
limits on neutrino magnetic moment shown by the dashed cyan line can be evaded
by utilizing a neutrino trapping mechanism Babu:2020ivd (see discussions in
text).

There are strong astrophysical constraints on neutrino magnetic moments from
red giants and horizontal branch stars, since photons in the plasma of these
environments can decay into neutrino pairs Bernstein:1963qh ; Raffelt:1999tx .
A limit $|\mu_{\nu}|\leq 1.5\times 10^{-12}\mu_{B}$ (95% CL) has been derived
from the evolutionary studies of horizontal branch stars Viaux:2013lha ;
Viaux:2013hca ; Capozzi:2020cbu . This would be in contradiction with the
$\mu_{\nu_{e}\nu_{\mu}}$ needed to explain the XENON1T anomaly. However, this
constraint can be evaded if $\nu_{\mu}$ has a matter-dependent mass, which can
arise from its weak coupling to a new light scalar, as shown in Ref.
Babu:2019iml and adopted in Ref. Babu:2020ivd . Note that the transition
magnetic moment $\mu_{\nu_{e}\nu_{\mu}}$ would lead to plasmon decay into
$\nu_{e}+\nu_{\mu}$, which would be kinematically suppressed inside red giants
and horizontal branch stars if the matter-dependent mass of $\nu_{\mu}$ is of
order few keV. This would not suppress the standard $\nu_{e}$ emission, which
could have modified the evolution of these stars significantly Ahlgren:2013wba
; Straniero:2020iyi . The modifications to neutrino emissivity will only
affect $\nu_{\mu}\overline{\nu}_{\mu}$ emissions mediated through neutral
currents. Following Ref. Braaten:1993jw , we estimate that the neutrino
emissivity changes by only $\sim 0.3\%$ in our scenario compared to the
standard case. This shift is small owing to the dominance of the vector-
couplings of the electrons to the $Z$ boson, which is however suppressed by a
factor $(1-4\sin^{2}\theta_{W})$. The contribution of the axial-vector current
is always negligible for all conditions of astrophysical interest
1986ApJ…310..815K ; Raffelt:1996wa .

Conclusions.– In this paper we have analyzed new contributions to the muon
anomalous magnetic moment in a class of models that generates naturally large
transition magnetic moment for the neutrino needed to explain the XENON1T
electron recoil excess. These models are based on an approximate $SU(2)_{H}$
symmetry that suppresses the neutrino mass while allowing for a large neutrino
transition magnetic moment. We have shown that the new scalars present in the
theory with masses around 100 GeV can yield the right sign and magnitude for
the muon $g-2$ which has been confirmed recently by the Fermilab
collaboration. Such a correlation between muon $g-2$ and the neutrino magnetic
moment is generic in models employing leptonic family symmetry to explain a
naturally large $\mu_{\nu_{\mu}\nu_{e}}$. We have also outlined various other
experimental tests of these models at colliders. The entire parameter space of
the model can be explored at the HL-LHC through the pair production of neutral
scalars and their subsequent decays into $e\tau$ and $\mu\tau$ final states.

###### Acknowledgements.

Acknowledgments.– We thank Evgeny Akhmedov for discussions. The work of KSB
and VPK is in part supported by US Department of Energy Grant Number DE-SC
0016013 and by a Fermilab Neutrino Theory Network grant.

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