Large Neutrino Magnetic Moments in the Light of Recent Experiments

The excess in electron recoil events reported recently by the XENON1T experiment may be interpreted as evidence for a sizable transition magnetic moment $\mu_{\nu_e\nu_\mu}$ of Majorana neutrinos. We show the consistency of this scenario when a single component transition magnetic moment takes values $\mu_{\nu_e\nu_\mu} \in(1.65 - 3.42) \times 10^{-11} \mu_B$. Such a large value typically leads to unacceptably large neutrino masses. In this paper we show that new leptonic symmetries can solve this problem and demonstrate this with several examples. We first revive and then propose a simplified model based on $SU(2)_H$ horizontal symmetry. Owing to the difference in their Lorentz structures, in the $SU(2)_H$ symmetric limit, $m_\nu$ vanishes while $\mu_{\nu_e\nu_\mu}$ is nonzero. Our simplified model is based on an approximate $SU(2)_H$, which we also generalize to a three family $SU(3)_H$-symmetry. Collider and low energy tests of these models are analyzed. We have also analyzed implications of the XENON1T data for the Zee model and its extensions which naturally generate a large $\mu_{\nu_e\nu_\mu}$ with suppressed $m_\nu$ via a spin symmetry mechanism, but found that the induced $\mu_{\nu_e\nu_\mu}$ is not large enough to explain recent data. Finally, we suggest a mechanism to evade stringent astrophysical limits on neutrino magnetic moments arising from stellar evolution by inducing a medium-dependent mass for the neutrino.


Introduction
The XENON collaboration has recently performed a search for new physics with low-energy electronic recoil data recorded with the XENON1T detector and reported an excess of events over the known backgrounds in the recoil energy range (1−7) keV, peaked around 2.5 keV [1].
This excess, observed with an unprecedented low background rate of (76 ± 2) events/(tonne × year × keV) between (1 − 30) keV, and an exposure of 1042 kg × 226.9 days, is quite intriguing. One possible explanation of these anomalous events would be the presence of a sizable neutrino magnetic moment. Within this interpretation, the signal is favored over background at 3.2σ significance. The preferred range of an effective neutrino magnetic moment is µ ν ∈ (1.4, 2.9) × 10 −11 µ B at 90 % C.L. [1]. This excess is also consistent with a solar axion signal, and more conservatively, with a Tritium background in the detector that is unaccounted for [1].
Here we wish to explore the large magnetic moment interpretation of the observed XENON1T excess. We interpret the anomaly in terms of a transition magnetic moment In this paper we show that new symmetries acting on the lepton sector can render neutrino magnetic moment of order 10 −11 µ B compatible with the known neutrino masses.
In the absence of additional symmetries (and without severe fine-tuning) one would expect neutrino masses several orders of magnitude larger than their measured values. The main reason for this expectation is that the magnetic moment and the mass operators are both chirality flipping, which implies that by removing the photon line from the loop diagram that induces µ ν one would generate a neutrino mass term. This would lead to the naive estimate of m ν originating from such diagrams given by where M represents the mass of a heavy particle circling inside the loop diagram. Since the photon is emitted from an internal line to induce a magnetic moment operator, at least some of the particles inside the loop must be electrically charged. Experimental limits show that any such charged particle should be heavier than about 100 GeV, in which case Eq.
(1.1) would lead to m ν ∼ 0.1 MeV, some six orders of magnitude larger than the observed masses. 1 1 There is an exception for M being large: If the internal particles are milli-charged, direct experimental limits won't exclude them from being light. Even in this case, owing to other experimental constraints on milli-charged particles, the maximum µ ν that can be induced is µ ν ∼ 10 −15 µ B [2].
This magnetic moment-mass conundrum was well recognized three decades ago when there was great interest in explaining the apparent time variation of solar neutrino flux detected by the Chlorine experiment in anti-correlation with the Sun-spot activity [3,4].
Such a time variation could be explained if the neutrino has a magnetic moment of order (10 −11 − 10 −10 )µ B which would lead to spin-flip transition inside the solar magnetic field [5,6]. Such transitions could even undergo a matter enhanced resonance [7,8]. While this explanation of the solar neutrino data has faded with the advent of other experiments, in the late 1980's and early 1990's there were significant theoretical activities that addressed the compatibility of a large neutrino magnetic moment with a small mass. These discussions become relevant today, if the XENON1T anomaly is indeed a signal of neutrino magnetic moment.
In this paper we revive and extend mechanisms for enhancing neutrino magnetic moments based on a horizontal SU (2) H symmetry [9,10]. In the limit of this symmetry, owing to differences in the Lorentz structures in the operators, the neutrino mass vanishes while the magnetic moment does not [11][12][13][14][15][16][17][18]. We propose a simplified model based on approximate SU (2) H symmetry that induces sufficiently large neutrino magnetic moment to explain the XENON1T excess. While the old models almost always relied on exact symmetries, here we show that an approximate SU (2) H is sufficient, with explicit breaking of the symmetry provided by the electron and muon masses. Thus, new models can be realized with fewer particles, making them simpler. We also propose an extension of the SU (2) H symmetry to a three-family SU (3) H which has the desired property of suppressing neutrino mass while generating large magnetic moment. Collider and low energy constraints of these models will be analyzed and future tests outlined. A distinct signature of these models for the LHC is the presence of neutral scalars decaying into ± τ ∓ with masses not exceeding a TeV. We also revisit models with a spin symmetry argument [17,19] that enhances magnetic moment, but find that these models do not generate large enough µ ν in order to explain the XENON1T data.
Large neutrino magnetic moments are strongly constrained by stellar evolution, since the photon, which has a plasma mass in these surroundings can decay to neutrinos. The most stringent limit arises from the energy loss of red giant branch in globluar clusters, which require µ ν < 4.5 × 10 −12 µ B [20]. This value is a factor of 5 below what is needed to explain the XENON1T anomaly. We show here that by invoking interactions of the neutrinos with a light scalar, such plasmon decays may be kinematically suppressed, as the neutrino acquires a medium-dependent mass which is larger than the plasmon mass.
This paper is organized as follows. In Sec. 2 we provide our fit to the XENON1T data in terms of a single component transition magnetic moment µ νeνµ and we give a short overview other experimental information. In Sec. 3 we discuss general theoretical aspects of neutrino magnetic moments and point out how new symmetries can explain a large value. Here we review the SU (2) H symmetric mechanism and the spin symmetry argument to suppress neutrino mass relative to its magnetic moment. In Sec. 4 we present a concrete and simplified model based on an approximate SU (2) H . We carry out a detailed phenomenological analysis of the model in Sec. 5. In Sec. 6 we extend the symmetry to an approximate SU (3) H . In Sec. 7 we analyze magnetic moments in the Zee model and its extensions, where we show their inadequacy to explain XENON1T data. In Sec. 8, we suggest a mechanism to evade the astrophysical limits on neutrino magnetic moments. Finally, we conclude in Sec. 9.
2 Neutrino magnetic moments: the experimental situation In this section we briefly summarize the current status of neutrino magnetic moment searches.
First we show the consistency of interpreting the XENON1T excess in terms of a single component transition magnetic moment of ν e . Then we summarize the experimental status on neutrino magnetic moments from reactor and accelerator neutrinos as well as from astrophysics.

XENON1T
The excess in electron recoil events observed by XENON1T collaboration [1] may be explained by solar neutrinos which have nonzero magnetic moments. With its low threshold, XENON1T detector is very sensitive to magnetic moments of Dirac neutrinos or to transition moments of Majorana neutrinos, since in either case the neutrino-electron scattering cross-section at low energies will increase. Here we focus on the transition magnetic moment, which is what the models discussed later predict. The differential cross section for the neutrino-electron scattering process ν α e → ν α e in the presence of a magnetic moment is given by where µ ef f is an effective neutrino magnetic moment (defined in Eq. (2.7) below), T is the recoil kinetic energy of the electron and E ν the energy of the neutrino. The Standard Model cross section for ν α e → ν α e is given by where α represents the neutrino flavor, m e denotes the electron mass and G F is the Fermi constant. The flavor-dependent (since ν e undergoes charged current scattering, while ν µ,τ do not) vector and axial vector couplings are given by The solar neutrinos flux at low energies is primarily composed of the continuous pp-flux and a discrete 7 Be-flux with values given by [21] φ pp = 5.94 × 10 10 cm −2 s −1 , It is clear that the pp flux is dominant with the 7 Be flux an order of magnitude smaller.
Flux from 8 B and other sources are even smaller at low energies. It is sufficient then to keep only the pp flux in the calculation of electron recoil excess. ν e s produced in the solar core oscillate into ν α with α = µ, τ , with the flavor transition being adiabatic inside the Sun. Since solar neutrinos arriving at earth are a mixture of incoherent states, the effective magnetic moment relevant for the neutrino-electron scattering can be defined as [22]: Here λ ij = µ ij − id ij , which contain the transition magnetic and electric dipole moment operators of the physical neutrino states ν 1,2,3 . These quantities are related to the transition moments in the flavor basis denoted as λ αβ , with α, β = e, µ, τ via the relationλ = U T λU , where U is the PMNS matrix, withλ denoting λ ij in the mass eigenstate basis, and λ denoting λ αβ in the flavor basis. In Eq. (2.7) P 2ν e1 denotes the probability of observing the mass eigenstate ν 1 at the scattering point for an initial electron flavor in the two-neutrino oscillation scenario. It is clear from Eq. (2.7) that CP violating phases of the PMNS matrix do not affect µ 2 ef f . We shall be interested in a scenario where only the µ νeνµ component of the magnetic moment matrix, expressed in the flavor basis, is nonzero. When converting this into the mass eigenbasis so that Eq. (2.7) can be used, all the neutrino oscillation parameters come into play, including the Dirac CP phase δ. We use central values of the oscillation parameters given in Ref. [23], viz., {sin 2 θ 12 = 0.310, sin 2 θ 23 = 0.580, sin 2 θ 13 = 0.0224, δ CP = 215 • }.
For the effective 2-neutrino oscillation probability, we use the best fit value P 2ν e1 = 0.667 [24]. With these, we can express the effective neutrino magnetic moment in terms of single component transition neutrino magnetic moment (in flavor basis) as: We shall use this value in our numerical analysis. 2  In order to compute XENON1T signal prediction and analyze the recoiled electron spectrum for a single component transition magnetic moment µ νeνµ , one can define the differential event rate in terms of the reconstructed recoiled energy (T ) as where dφ/dE ν denotes the solar neutrino flux spectrum [21], (T ) indicates the detector efficiency [1,25], n e is the number of target electrons in fiducial volume of one ton Xenon [1] and G (T, T r ) represents a normalized Gaussian smearing function in order to account for the detector finite energy resolution [1,25]. The detector threshold and the maximum recoil energy are respectively given by T th = 1 keV and T max = 30 keV, while the other integration limits are E min ν = T + √ 2m e T + T 2 /2 and E max ν = 420 keV. 2 If we use the coefficient on the right hand side of Eq. (2.8) to be 1, we have verified that the XENON1T [1] analysis can be reproduced.
By folding the expected solar neutrino flux [21] and imposing a step-function approximation to account for the electron binding energies, we analyze the recoiled energy spectrum for different values of neutrino transition magnetic moment µ νeνµ = {1.4 × 10 −11 , 2.9 × 10 −11 , 5 × 10 −11 }µ B in Fig. 1. For this analysis, we adopt the background model spectrum from Ref. [1]. The preferred values of neutrino transition magnetic moment µ νeνµ for the excess observed at XENON1T experiment [1] at 90% confidence interval corresponds to µ νeνµ ∈ (1.65, 3.42) × 10 −11 µ B . In the right panel of Fig. 1 we show results of our analysis of the signal and background spectrum where we also compare ths with the observed data [1].
The green shaded region indicates the background spectra and purple shaded zone shows the expected combined spectrum for signal and background. One sees that owing to the presence of sizable neutrino magnetic moment, and the resulting 1/T enhancement in the cross section, the signal spectrum gives a good fit to the observed data in the electron recoil energy range between (1 − 7) keV peaking around 2.5 keV. This shows the consistency of a single component transition magnetic moment interpretation of the Xenon data.

Experimental searches for neutrino magnetic moments
The quest for measuring a possible magnetic moment of the neutrino was begun even before the discovery of the neutrino. Cowan, Reines and Harrison set an upper limit of µ ν < 10 −7 µ B in the process of measuring background for a free neutrino search experiment [26] with reactor antineutrinos. This limit is obtained by studying ν e e elastic scattering process and observing a possible excess in the electron recoil events. This Cowan-Reines-Harrison limit was subsequently improved by several orders of magnitude by a variety of reactor antineutrino experiments. KRASNOYARSK reactor experiment obtained a limit of µ ν < 2.7 × 10 −10 µ B [27], with subsequent improvements by ROVNO (µ ν < 1.9 × 10 −10 µ B ) [28], MUNU (µ ν < 1.2 × 10 −10 µ B ) [29] and TEXONO (µ ν < 2 × 10 −10 µ B ) [30]. The GEMMA collaboration reports a more stringent limit on ν e magnetic moment of µ ν < 2.9 × 10 −11 µ B [31]. These limits apply specifically to either a Dirac magnetic moment or a Majorana transition magnetic moment of ν e .
Accelerator based experiments have also searched for neutrino magnetic moments via low energy ν e , ν µ and ν µ scattering off electrons. By studying ν e − e scattering, a bound on an effective magnetic moment has been obtained at LAPMF which translates into a muon neutrino magnetic moment limit of µ νµ < 7.4 × 10 −10 µ B [32]. LSND experiment has obtained a limit of µ νµ < 6.4 × 10 −10 µ B , also by studying ν e − e scattering.
The Borexino experiment has studied the shape of the electron recoil spectrum from solar neutrino interactions and found no significant deviations from expectations. A limit on an effective neutrino magnetic moment µ eff ν < 2.8 × 10 −11 µ B was obtained [33]. When interpreted as a single component Majorana neutrino transition magnetic moment, this would translate into µ νeνµ < 3.29 × 10 −11 µ B . This limit, which is more directly related to the XENON1T excess, is consistent with the needed value to explain the excess. For a global fit including all the experimental limits on neutrino magnetic moments, see Ref. [24,34] and also Ref. [35].

Limits on µ ν from astrophysics and cosmology
Evolution of stars can provide indirect constraints on the magnetic moments of either Dirac or Majorana neutrinos. Photons in the plasma of stellar environments can decay either into νν for the case of Dirac neutrinos or into ν α ν β for the case of Majorana neutrinos [36,37].
Such decays are kinematically allowed in a plasma since the photon acquires a mass. If such decays occur too rapidly, that would drain energy of the star, in conflict with standard stellar evolution models which appear to be on strong footing. Limits on µ ν have been derived by requiring the energy loss in such decays to be not more than via standard processes.
The best limit on µ ν from this argument arises from red giant branch of globular clusters, resulting in a limit of µ ν < 4.5 × 10 −12 µ B [20]. Validity of this limit would make the neutrino magnetic moment interpretation of the XENON1T excess questionable. We note that these indirect constraints from astrophysics may be evaded if the plasmon decay to neutrinos is kinematically forbidden. As we show in Sec. 8, this can indeed be achieved by invoking interactions of the neutrino with a light scalar. The neutrino will then acquire a medium-dependent mass greater than the plasmon mass, while being consistent with other observations, and thus forbidding plasmon decays. There are also cosmological limits on µ ν arising from big bang nucleosynthesis. However, these limits are less severe, of order 10 −10 µ B [38].
We now turn to theoretical interpretation of the suggested transition magnetic moment µ νeνµ .

New symmetries and a large neutrino magnetic moment
In this section we recall theoretical expectations for neutrino magnetic moment and revive symmetry based mechanisms to generate sizable µ ν .

Neutrino magnetic moment in the Standard Model and beyond
The magnetic moment and mass operators for the neutrino have the same chiral structure, which for a Dirac neutrino has the form: As a result, µ ν typically becomes proportional to m ν . For example, in the Standard Model when right-handed neutrinos are introduced so that the neutrino has a small Dirac mass, its magnetic moment is given by [39] If neutrinos are Majorana particles, their transition magnetic moments resulting from Standard Model interactions is given by [40] where m i stands for mass of neutrino i, m is the charged lepton mass, and U i denotes the PMNS matrix element. The resulting transition magnetic moment is even smaller than the value given in Eq. (3.12), at most of order 10 −23 µ B . Clearly, these values are well below the sensitivity of current experiments.
Nonstandard interactions of the neutrinos can lead to enhanced magnetic moments, esepcially when the new physics lies near the TeV scale. For example, in left-right symmetric models, the right-handed neutrino couples to a W ± R gauge boson, which also has mixing with the W boson. For the case of a Dirac neutrino the magnetic moment now becomes proportional to the charged lepton mass, rather than m ν , and is given by where ξ is the mixing angle between W ± R and W ± , which is of order (M 2 W /M 2 W R ). This mixing angle is constrained by muon decay asymmetry parameters [41], as well as by b → sγ decay rate [42], leading to a limit µ νe < 10 −14 µ B [43]. While significantly enhanced compared to the Standard Model value of Eq. (3.12), this is still well below experimental sensitivity.
In supersymmetric extensions of the Standard Model, lepton number may be violated by R-parity breaking interactions. In such contexts, without relying on additional symmetries, the neutrino transition magnetic moment will be of the order where λ is an R-parity breaking coupling, A is the SUSY breaking trilinear coupling, and M˜ is the slepton mass. Imposing experimental constraints on the SUSY parameters, this would yield a value of µ ν at most about 10 −13 µ B , which is too small to be relevant for XENON1T. Transition magnetic moments can be larger in presence of new vector-like leptons [44].
It is possible to induce µ ν ∼ few × 10 −11 µ B via charged scalar loops, which are less constrained by other processes. An SU (2) L singlet charged scalar η + can induce significant µ ν for a Dirac neutrino or to a Majorana neutrino of the desired order to explain the XENON1T anomaly [45,46]. However, even in this case, the neutrino mass -magnetic moment problem shown in Eq. (1.1) would prevail. While µ ν can be large as desired, m ν will become unacceptably large, unless it is strongly fine-tuned to about one part in 10 6 .

SU (2) H symmetry for enhanced neutrino magnetic moment
While the neutrino mass operator and the magnetic moment operator both are chirality flipping, there is one important difference in their Lorentz structures. The mass operator, being a Lorentz scalar, is symmetric, while the magnetic moment, being a Lorentz tensor operator is antisymmetric in the two fermion fields. Voloshin suggested to exploit this property to suppress neutrino mass while enhancing its magnetic moment [9]. He proposed a new SU (2) ν symmetry that transforms ν into ν c , the left-handed antiparticle of the right-handed neutrino. A neutrino mass term, being symmetric under the exchange of ν and ν c , would then be forbidden by the SU (2) ν symmetry, which requires such an invariant to be antisymmetric, since a singlet made out of two SU (2) ν doublets is in the antisymmetric combination.
On the other hand, the magnetic moment operator, ν T Cσ µν ν c F µν is antisymmetric under ν ↔ ν c interchange, and thus is allowed in the SU (2) ν symmetric limit.
As it turns out, since the ν c field does not feel weak charged current or neutral current interactions, the SU (2) ν symmetry operating on (ν, ν c ) fields is not easy to implement [47].
It was suggested in Ref. [10] that a horizontal SU (2) H symmetry acting on the electron and the muon families can serve the same purpose, which is easier to implement as such a symmetry commutes with the weak interactions. This would lead to a transition magnetic moments for Majorana neutrinos. Models based on such SU (2) H symmetries were built, which we shall revive and simplify in the next section. The main point of the SU (2) H symmetry is that the neutrino transition magnetic moment interaction given by where C is the charge conjugation matrix, is invariant under any SU (2) H transformations as U T iτ 2 U = iτ 2 for any 2 × 2 unitary matrix U that rotates ν e and ν µ . On the other hand, the Majorana neutrino mass term given by is not invariant under a unitary rotation involving (ν e , ν µ ). Thus, in the SU (2) H symmetric limit, neutrino mass is forbidden, while a transition magnetic moment µ νeνµ is permitted.
Explicit realization of this idea was given in Ref. [10,11].
It has been also realized that the full SU (2) H symmetry is not essential to realize a large µ ν with a suppressed mass, a non-Abelian subgroup of SU (2) H would suffice [12][13][14][15][16].
To see this, note that an iτ 2 rotation would prevent an off-diagonal neutrino mass, while an iτ 3 rotation would prevent any diagonal masses. The magnetic moment operator is invariant under both rotations. Since iτ 2 rotation does not commute with iτ 3 rotation, the full symmetry group should contain at least eight elements: The quaternion group of order 8 is an example of such a symmetry. For a review of these developments see Ref. [18]; for a recent update see Ref. [2].
In the next section we shall present a model based on approximate SU (2) H symmetry to induce a large µ νeνµ . Since the symmetry is only approximate, there is no significant difference between models based on SU (2) H or one of its non-Abelian subgroups. By requiring only an approximate SU (2) H , as opposed to exact symmetry of Ref. [10,11], the model becomes simpler. It should be noted that the mass splitting between the electron and the muon breaks the approximate SU (2) H symmetry. If all violations of SU (2) H are of the order of (m 2 µ − m 2 e )/M 2 , where M is a heavy mass scale of order 100 GeV or more, then the naive estimate of Eq. (1.1) would be modified to This estimate will give m ν ∼ 0.1 eV for µ ν ∼ 10 −11 µ B , which is just about acceptable. In fact, we shall see that there is in addition, a loop suppression factor, which would make m ν related to the magnetic moment operator smaller by another two orders of magnitude.

Large magnetic moment from spin symmetry
A somewhat independent mechanism is known for generating enhanced µ ν with a suppressed m ν . This relies on a spin symmetry argument. In renormalizable gauge theories there are no direct couplings of the type γW + S − where S − is a charged scalar field. However, such a coupling could be generated via loops. Barr, Friere and Zee [19] used this induced vertex to construct models of large µ ν . At the two loop level, this vertex will contribute to µ ν . As for its contribution to m ν , it is well known that for transversely polarized vector bosons, the transition from spin 1 to spin 0 cannot occur. Only the longitudianl mode, the Goldstone mode, would contribute to such transitions. This implies that in the two loop diagram utilizing the γW + S − for generating µ ν , if the photon line is removed, only the longitudinal W ± bosons will contribute, leading to a suppression factor of m 2 /m 2 W in the neutrino mass, compared to the naive estimate of Eq. (1.1).
This idea of utilizing spin symmetry has a simple realization in the popular Zee model of neutrino masses [48], as was shown in Ref. [17]. We have investigated the current status of neutrino magnetic moment in this class of models. We found that while these models typically induce large µ ν , after taking account of low energy constraints as well as LHC constraints on new particles, the maximun µ ν that can be generated here is about an order of magnitude smaller than the value needed to explain XENON1T anomaly.

Dirac vs Majorana neutrino magnetic moments
It has been argued, based on effective field theory (EFT) calculations, that Dirac neutrino magnetic moments exceeding about 10 −15 µ B would not be natural, as that would induce at higher loops unacceptably large neutrino masses [49]. For Majorana neutrinos, the transition magnetic moments are allowed to be much larger from EFT naturalness arguments [50,51]. The SU (2) H symmetry based models, as well as the spin symmetry based models, fit well within this categorization.

SU (2) H model for large neutrino magnetic moment
In this section we present a simplified model for large transition magnetic moment µ νeνµ based on an approximate SU (2) H horizontal symmetry acting on the electron and the muon families. A full SU (2) H symmetric model was presented in Ref. [10,11], which is our starting point. Our simplification is that the symmetry is only approximate, broken explicitly by electron and muon masses. Fewer new particles would then suffice to complete the model.
The explicit breaking of SU (2) H by the lepton masses is analogous to chiral symmetry breaking in the strong interaction sector by masses of the light quarks. Such breaking will have to be included in the neutrino sector as well. We have computed the one-loop corrections to the neutrino mass from these explicit breaking terms and found them to small enough so as to not upset the large magnetic moment solution.
The only violation of SU (2) H acting on the electron and muon fields arises from their unequal masses. This mass splitting, normalized to the weak scale, is indeed a small param-eter: (m 2 µ − m 2 e )/m 2 W = 1.7 × 10 −6 . Violation of SU (2) H symmetry in the neutrino masses can be of this order, which from Eq. (1.1) suggests that large µ νeνµ can be realized without inducing large m ν . In fact, the effect of the SU (2) H breaking parameter (m 2 µ − m 2 e )/m 2 W in the neutrino sector will be accompanied by a loop suppression factor of order 10 −2 , which would make m ν even smaller.
Our model is a simple extension of the Zee model [48] of neutrino mass that accommodates an SU (2) Here SU (2) H acts horizontally, while SU (2) L acts vertically. The first two families of leptons form a doublet of SU (2) H while the τ family is a singlet. All quark fields are assumed to be SU (2) H singlets.
The Higgs sector of the model consists of the following multiplets: The φ S filed is the Standard Model Higgs doublet, which has its usual Yukawa couplings with the quarks. The φ S field is also responsible for electroweak symmetry breaking. The The φ fields are assumed to acquire no VEVs. This is a consistent assumption, which is valid even after the explicit breaking of SU (2) H symmetry.
Under SU (2) L × SU (2) H , the transformation of various fields is as follows: Here U L and U H are unitary matrices associated with SU (2) L and SU (2) H transformations.
The Yukawa Lagrangian in the lepton sector that is invariant under the gauge symmetry as well as SU (2) H is then Expanded in component form, this reads as: It becomes clear that the h 1 term gives equal mass for the electron and the muon once The h 2 term generates a mass for the τ lepton. If h 3 = 0, τ lepton number would be a good symmetry of the Lagrangian. The h 3 term induces a nonzero ν τ mass in conjunction with the f term, which is allowed in the limit of exact SU (2) H . The terms f and f are crucial for the generation of the neutrino transition magnetic moment.
We shall introduce explicit breaking of the SU (2) H symmetry, so that the relation m e = m µ which follows from Eq. (4.22) can be corrected. Since this breaking is small, we first discuss the model in the SU (2) H symmetric limit.
The scalar potential contains, among other terms, the following terms: 3 Here m 2 η includes the bare mass term as well as a contribution from the quartic coupling Similarly, m 2 φ + includes the bare mass of φ as well as the contribution from the quartic coupling The mass of φ 0 is split from that of φ + through the interaction term Tr|Φ † φ S | 2 . All terms in Eq. (4.23) respect SU (2) H symmetry. In component form the cubic coupling reads as: Once the VEV of φ 0 S is inserted, this term would lead to the mixing of η + 1 with φ + 1 and η + 2 with φ + 2 . These mass matrices are identical, owing to the unbroken SU (2) H and are given by The two mass eigenstates will be denoted as h + i and H + i wtih i = 1, 2 and their masses will be denoted as (m 2 h + , m 2 H + ). These states are related to the original states via the relations where the mixing angle is given by (4.27)

Neutrino transition magnetic moment
The Lagrangian of the model given in Eqs.  Since the masses of the particles inside the loop are the same in the two diagrams of Fig.   2, and since all the couplings are identical, the magnitudes of the two graphs are identical.
However, they have a relative minus sign at one of the vertices. When the induced neutrino mass is computed from here, the two diagrams cancel. On the other hand, with the photon attached to the loop, the two diagrams add (note that the direction of electric charge flow is opposite in the two diagrams). As a result, the two graphs add to give a finite magnetic moment: Here (m h + , m H + ) denote the common masses of the two charged scalars (h + i , H + i ). We have plotted contours of constant magnetic moment in the plane of m h + and |f f sin 2α| in Fig. 3. The different contours represent different values of the heavier charged Higgs mass m H + . Also shown in the figure are the exclusion limit from Borexino on µ νeνµ as well as the limit on the parameters from τ decay asymmetry, discussed later. It can be seen that for h + masses below about 1 TeV, and for couplings less than one, the model can generate sufficiently large µ νeνµ to explain the XENON1T anomaly.   Since the breaking we introduce is hard, via dimension four terms in the Lagrangian, to be consistent, other renormalizable SU (2) H breaking terms should also be included. Neutrino masses will then be induced proportional to these SU (2) H breaking terms. Here we show that such breaking terms of the order needed to achieve e − µ mass splitting induce m ν of the right order of magnitude to explain neutrino oscillation data.
The Yukawa couplings of Eq. (4.22) will receive explicit SU (2) H breaking corrections given by The δh 1 term would split the masses of the electron and the muon, since their masses are now given by We therefore can express h 1 and ∆h 1 in terms of the Standard Model Yukawa couplings h e and h µ as which can be used to explicitly show SU (2) H breaking. Note that Eq.  4 If m µ = m e is realized by the VEVs of φ 0 1 = φ 0 2 , the mixing angles amongst µ L − τ L , as well as e R − τ R will become relatively large, leading to unacceptably large m ν when µ ν is demanded to be large.
Similarly, the scalar potential of Eq. (4.23) should include the following terms that break SU (2) H explicitly: Together with Eq. (4.23), this becomes the most general potential relevant for neutrino mass/magnetic moment discussions. We now proceed to compute the shifts in the couplings and masses induced by the electron and muon masses and Yukawa couplings. The diagrams shown in Fig. 5 would lead to a splitting in the masses of η + 1 and η + 2 , once the Yukawa interactions of Eq. (4.30) are included. We evaluate these diagrams in dimensional regularization, adopt a zero momentum subtraction scheme to determine the mass and wave function counter-terms, and obtain the finite shift in the mass of η + 1 as A similar expression is obtained for the renormalized mass correction of η + 2 with m µ replaced by m e , resulting in a shift in the two masses given by A similar calculation for the mass splitting of φ + 1 and φ + 2 shows Since the dependence on the leptons masses in these mass splittings is only logarithmic, in order to achieve δm 2 /m 2 ∼ 10 −5 needed to ensure the smallness of neutrino masses, it is necessary to take |f | ∼ few × 10 −3 , which is consistent with the needed µ νeνµ , see Fig. 3.
The first set of Feynman diagrams of Fig. 6 would result in a shift in the coupling δf , which is however proportional to the electron and the muon Yukawa couplings: This shift is also not excessive, for |f | 10 −2 .
Once the masses of (η + 1 , η + 2 ) split, and similarly with (φ + 1 , φ + 2 ), nonzero neutrino masses will be induced. Keeping the largest splittings into account the induced m νeνµ is given by Here δα is the shift in the mixing angle arising primarily from the shift in the µ term. For f ∼ 10 −2 , f ∼ 1, δm 2 /m 2 ∼ 10 −5 , one obtains m νeνµ ∼ 0.1 eV, which is roughly the value needed to explain neutrino oscillation data. As for other entries in the neutrino mass matrix, one could introduce SU (2) H violating small couplings in the f and f matrices, so that the full mass matrix explains the entirety of neutrino oscillation data. It is sufficient to work in the SU (2) H symmetric limit, since violations of SU (2) H will be very small. There is a similar diagram mediated by η + 2 leading to τ → µν τ ν µ with identical strength. The diagram on the right contributes to a new decay channel, τ → µν e ν τ , with an analogous diagram mediated by φ + 2 = η + 2 for τ → eν µ ν τ .
The charged scalar η + 2 mediates the decay τ → eν τ ν e , as shown in Fig. 8, left panel. There is an identical diagram for the decay τ → µν τ ν µ mediated by η + 1 . These two amplitudes being the same, there is no lepton universality violation in τ decays. The leptonic decay rates will be modified as The τ lifetime has been measured accurately to be τ exp = (290.75 ± 0.36) × 10 −15 sec., while the Standard Model prediction for the lifetime is τ SM = (290.39 ± 2.17) × 10 −15 sec. Adding the two errors in quadrature we find new contributions should be limited to 1.64%. Noting that the new contributions only affect the leptonic modes that has a branching ratio of 17.8%, we obtain τ < 0.0227. This constraint is rather easy to satisfy within the model, and has been imposed in our calculation of µ νeνµ .
The right panel of Fig, 8 shows new lepton number violating decays of the τ , τ → µν e ν τ .
There is a similar diagram for the decay τ → eν µ ν τ mediated by η + 2 − φ + 2 exchange. Note that these processes conserve L e −L µ , which is an apprixmate symmetry of the model. Thee new decays do not interfere with the standard decay, but can modify the lifetime and decay asymmetry parameters. The effective Lagrangian for the decay is found to be we see that Γ new τ = Γ SM τ (1 + 2 τ ). This would lead to a limit of | τ | < 0.205 from τ lifetime. Comparing with muon decay formalism of Ref. [52], we see that for the lepton number violating τ decays, g S LR = 2 τ . This leads to the following modifications of the asymmetry parameter in τ decay [53]: Using the experimental value δ ξ = 0.746 ± 0.021 [54], and using 2 σ error bar, we obtain | τ | < 0.175. The constraint from the measurement ξ = 0.985 ± 0.030, | τ | < 0.194 is somewhat weaker.
The parameters that enter the new τ decay are the same as for the neutrino magnetic moment. We have indicated the most stringent constraint, arising from δ ξ in Fig. 3.
While this does provide a useful constraint, large magnetic moment of the neutrino is still permitted by this constraint.

LEP constraints
At LEP, the t-channel exchange of the neutral component of the Φ multiplet (φ 0 1 ) can contribute significantly to the process e + e − → τ + τ − as shown in Fig. 9. Contact interactions involving e + e − and a pair of fermions [55] are tightly constrained by the LEP experiments.
For heavy mass of φ 0 1 , one can integrate it out and express its effect via a dimension-6 operator 5 . Therefore, the LEP constraint on the scale Λ of the contact interaction for the process e + e − → τ + τ − , viz., Λ > 2.2 TeV, can be translated to a limit as m φ 0 1 /|f | > 0.44 TeV. However, if the φ 0 1 scalar is lighter than about 300 GeV, the LEP contact interaction limit is not applicable. For lighter φ 0 1 scalar we compute the cross-section at the parton-level for the process e + e − → τ + τ − using MadGraph5aMC@NLO event generator [57] and compare it with the measured cross sections [55,58], imposing identical acceptance criteria [58] and obtain limits on the Yukawa coupling f as a function of the mass m φ 0 1 . At

√ s = 207
GeV, with an integrated luminosity of 134.5 pb −1 , LEP observed a total of 206 events [58] for the process e + e − → τ + τ − , which can be translated into a limit on the cross-section of 7.21 ± 0.57 ± 0.19 pb (the first error shown is statistical and the second one is systematic).
The SM predicted cross-section is 7.12 pb. Comparing this cross-section bound at 2σ level, we find that for the benchmark values of the φ 0 1 mass m φ 0 1 = 100 and 200 GeV, the Yukawa coupling f can be as big as 0.675 and 0.925 respectively. It should be mentioned that the other new neutral scalar field φ 0 2 has no impact on the LEP experiment as, it couples to µ and τ leptons.
these charged scalars can be pair-produced via Drell-Yan process with the exchange of γ or Z boson in the s channel. η + 2 can also be pair-produced through the t-channel ν τ exchange. Due to the absence of flavor-diagonal Yukawa couplings of the charged scalars, they cannot be produced in association with W boson, which relaxes various LEP search limits such as csτ ν searches. Once produced on-shell, the charged scalar and the neutral scalar would decay into various leptonic final states with the dominant decay modes given by For our numerical analysis, we compute the cross-sections using MadGraph5aMC@NLO event generator [57] at the parton level. There are several supersymmetirc slepton searches [59] at LEP, which we reinterpret [59] as limits on charged Higgs particles since they mimic these signatures. In our model, the decay branching ratios of η + 1 to µν τ and τ ν µ modes are equal and 50%. Hence, η + 1 needs to obey both the smuon and stau search limits. The smuon searches impose a limit on η + 1 wich allowed to be as low as 91 GeV. Similarly, η + 2 decays to eν τ and τ ν e modes equally. Selectron and stau search impose limit on the mass of η + 2 , and we find the more severe limit of m η + 2 > 96 GeV arising from the selectron searches. We also find lower limits on the masses of φ + 1 and φ + 2 to be 88 GeV, originating mainly from stau searches since both decay to τ ν e and τ ν µ 100% times. We conclude that in SU (2) H model, a light charged scalar with mass as low as ∼88 GeV is consistent with all LEP search limits.  Figure 10: Di-Higgs production cross-section at the LHC in the SU (2) H model.

LHC prospects
In our SU (2) H model, the neutral scalars φ 0 1 and φ 0 2 do not couple to quarks, and hence, cannot be directly produced via the gluon fusion process. The CP − even neutral scalars Re(φ 0 1 ) and Re(φ 0 2 ) will be dominantly produced in association with their CP − odd partners Im(φ 0 1 ) and Im(φ 0 2 ) via s− channel Z boson exchange in quark fusion processes at the LHC. The dominant production mechanism for the charged scalars (φ + i ) at the LHC will be via s− channel Z/γ exchange. In addition, the neutral scalars φ 0 1 and φ 0 2 can be produced in association with their charged partners φ + 1 and φ + 2 via s− channel W boson exchange. The production cross-sections at the 14 TeV LHC for the processes pp → φ 0 Fig. 10. Searches for a heavy neutral at the LHC in the context of either 2HDM (two Higgs doublet model) or MSSM (minimal supersymmetric standard mode) are not directly applicable to our scenario since φ 0 1 and φ 0 2 do not couple to quarks. There has been searches for sleptons produced in pairs at √ s = 13 TeV LHC in the mass range above a 100 GeV. We found that the current limits [60] on these cross section are larger than the φ + i φ − i Drell-Yan pair-production rate, and hence there are no stringent limits for these leptophilic charged scalars from the LHC. The most promising signal of the model is pp → e − e + τ − τ + , µ − µ + τ − τ + at the LHC as shown in Fig. 11. Once produced, φ 0 i would decay into a pair of different flavored leptons; i.e., φ 0 1 decays to eτ and φ 0 2 decays to µτ . Since final state leptons with large transverse momenta can be identified cleanly with good resolution at the LHC, this signal will be a good test for this model. Although there are several experimental multi-lepton searches [61,62], most of them assume a heavy ZZ ( ) resonance [61,62], which are not applicable to our scenario. In the context of supersymmetric models, there are inclusive multilepton searches, mostly with large missing transverse energy [63,64]. There are no dedicated searches for the l + l − τ + τ − process. However, in the context of sneutrino searches, this type of signal could arise for specific scenarios explored in Ref. [65]. The SM background for the pp → l + l − τ + τ − process is principally from the pair production of gauge bosons ZZ, W W and ZW ; from the top quark production through the channelstt andtttt; the associated productionttV, tV    and tV V ; and the Higgs production in association with top quark pair. We have analyzed all these processes and summarized the results in Table I. We note several distinguishing characteristics of the signal of the SU (2) H model: (a) the invariant mass distributions for different flavor lepton pair from the φ 0 i decay would peak at a value different from the Z boson mass; (b) if it is originated from Higgs signal, h → ZZ * should be accompanied by h → W W * , with a ratio of about 1 to 2; (c) Z decay also leads to neutrino decay modes that are absent in our scenario; (d) since the process does not involve quarks, the signal events suffer from significantly smaller hadronic activity than the associated background events including a leptonically decaying top t l ; (e) since the leptons are produced from heavy particle φ 0 i , they are expected to be more energetic than the ones produced in the decay of gauge bosons. We analyze the signal and show that there is a huge potential to unravel this multilepton signal above the SM background.
Our results are summarized in Table. I for five different masses m φ 0 i = 300, 400, 500, 600 and 800 GeV. After passing through all these acceptance criteria, for a 500 GeV massive φ 0 i , we estimate a total of 57 events at √ s=14 TeV LHC with 300 fb −1 integrated luminosity.
This corresponds to a significance of S √ S+B = 3.8σ. In Fig. 12, we estimate the significance of the signal pp → l + l − τ + τ − in our model at 14 TeV LHC with an integrated luminosity of L = 100, 300, 500 and 1000 fb −1 . We find that at 5σ level, the scalars φ 0 i can be probed up to masses of 538, 498, 466 and 398 GeV respectively for the integrated luminosities L = 100, 300, 500 and 1000 fb −1 , whereas at 3σ level, this can be probed upto 598, 560, 528 and 458 GeV respectively. The singly-charged scalar η + 2 can induce non-standard neutrino interaction (NSI) at the tree level via the Yukawa coupling f , given by (we use the standard notation, see [75])

Non-Standard interactions and IceCube
While significant τ τ could have been induced within the model, the τ lifetime constraint restricts τ τ < 2%. Also, there are direct constraints on NSI from neutrino experiments.
In Fig. 13, we have shown these constraints. The best experimental constraint on the NSI parameter τ τ arises from IceCube atmospheric neutrino data [73], which is shown as light-green shaded region in Fig. 13. We have also included constraints from global fit to neutrino oscillation + COHERENT data [71] (dark-green shaded region) and neutrinoelectron scattering experiments such as Borexino [72] (orange shaded region). We have also shown future DUNE sensitivity for 300 kt.MW.yr exposure [74] by blue solid line.
The constraint from LEP on charged scalar searches discussed in Sec. 5 is indicated by blue shaded region. At LEP there are new contributions to the monophoton process e + e − → ννγ mediated by η + 2 in the t-channel, which is bounded from LEP data [76]. This limit is shown by the purple shaded region. A light charged scalar η + 2 /φ + 2 of the model could potentially give rise to a Glashow-like resonance feature [77] in the ultra-high energy neutrino event spectrum at the IceCube; this future IceCube sensitivity is shown by solid brown curve corresponding toan exposure time of 50 ×T 0 . The gray shaded region indicates the current exclusion limit on neutrino magnetic moment from Borexino experiment. One can see that there is a large parameter space in between the two solid red curves which could explain XENON1T electron recoil excess at 90% C.L. while consistent with all the experimental constraints.  The Higgs sector consists of triplet fields Φ and η which induce large transition magnetic moments for the neutrino, the SM doublet φ S , and a flavon field to break SU (3) H down to SU (2) H : The τ lepton mass induced by this Lagrangian can be read off from Fig. 14: There are also lepton number violating interactions in the model. The following additional Yukawa couplings are permitted: The scalar potential contains a term Here ijk refers to the invariant symbol of SU (3) H , and in the second line, we inserted the VEV of the flavon field ϕ = u, which breaks SU (3) H down to SU (2) H . Owing to this unbroken symmetry, the masses of (φ + 1 , φ + 2 ) are identical, as are the masses of (η + 1 , η + 2 ) fields. The interactions of Eq. (6.55) would mix (φ + 1 , η + 2 ) and (φ + 2 , η + 1 ) leading to mass matrices given as The resulting mass eigenstates (h + i , H + i ) for i = 1, 2 have the same mass. However, the mixing angle sin 2α in the two sectors now have an opposite sign.
The diagrams shown in Fig. 15 will induce an SU (2) H -invariant transition magnetic moment µ νeνµ . Owing to the relative minus sign in the mixing angle of Eq. (6.56), when the photon line is removed in Fig. 15, the two diagram add to yield zero neutrino mass.
(All other couplings in the two diagrams are identica.) For the magnetic moment, the two diagrams add, since ν T µ Cσ µν ν e = −ν T e Cσ µν ν µ . The resulting µ νµνe is given as in Eq. (4.28), but with m τ replaced by M E ξ, where ξ is a mixing parameter in the E − E sector, which could be of order 0.1. Clearly, large magnetic moment can arise, consistent with neutrino mass as well as other experimental constraints. As for the breaking of the remaining SU (2) H , we adopt the same explicit breaking mechanism of Sec. 3.

Collider signals
The vector-like leptons (N, E, E ) present in the SU (3) H model can be searched for at colliders. Here we briefly highlight their novel signatures and discovery potential at the LHC. We also outline the existing bounds on the masses of these leptons. Although there is mixing between the SU (2) L doublet lepton E and the singlet lepton E , this mixing is of order 0.1, which we shall ignore for the present discussion. Being a singlet, E can only be pair-produced (pp → E ± E ∓ ) via s− channel Z/γ exchange, whereas the doublet vector-like lepton E ± can also be produced via s− channel W boson exchange: The discovery potential for the doublet vector-like lepton is much brighter than the singlet case, since the largest production rate pp → E + N/E −N arises from the s− channel W − exchange process. After being produced on-shell, the doublet charged lepton E ± mostly decays to Zl ± and hl ± , while the neutral lepton N mostly decays to W ± l ∓ . On the other hand, the singlet charged lepton E ± has three decay modes to Zl ± , hl ± and W ν. Assuming Similar phenomenological implications of vector-like leptons can be found in Ref. [81,82], although in a different context.

Neutrino Magnetic Moment in the Zee model
In 1990, Barr, Freire, and Zee (BFZ) proposed a spin symmetry mechanism [19] (reviewed briefly in Sec. 3) which provides for a large neutrino transitional magnetic moment µ ν , with a relatively small neutrino mass. To illustrate the mechanism, they extended the scalar sector of the popular Zee model of neutrino mass [48] with an additional Higgs doublet.
Subsequently it was shown in Ref. [17] that this mechanism can be realized within the Zee The simplest realization of the spin-symmetry mechanism is the Zee model [48], which contains an SU (2) L -doublet scalar H 2 and an SU (2) L -singlet charged scalar η ± , in addition to the SM-like Higgs doublet H 1 . The Wolfenstein version of the model [83], which is more predictive by virtue of a Z 2 symmetry is ruled out by oscillation data [84,85]. However, it has been shown that the original version of the Zee model [48] is fully consistent with neutrino oscillation data with interesting phenomenology [86,87].
Here, we mainly concentrate on the prediction of the neutrino transition magnetic moment in the Zee model. We adopt the scalar potential and the resulting scalar mass spectrum and the conventions of Ref. [87]. We choose a rotated basis for the Higgs doublets [88] in which only one neutral Higgs H 1 has a nonzero vacuum expectation value. Specifically, (7.60) The Yukawa Lagrangian can be written as: The scalar potential contains a cubic coupling given by (7.63) which leads to mixing between H + 2 and η + , with a mixing angle denoted as ϕ. Neutrino masses are generated at one-loop level and are given by [48] (7.64) Here we will analyze the neutrino transition magnetic moment and its relation to the neutrino mass in the Zee model. In Fig. 16, we show all the leading Barr-Zee [89] diagrams that contribute to a large neutrino magnetic moment. It is these diagrams that enhance µ νeνµ , as the mass contributions obtained from the same diagrams with the photon lines removed would have an additional suppression factor of (m 2 µ /m 2 W ). As noted in Sec. 3, with this suppression, m ν ∼ 0.1 eV, when µ ν ∼ 10 −11 µ B is realized. Note that f eτ cannot contribute to a large µ νeντ via such diagrams, as the resulting mass contribution would be 0.1 eV(m 2 τ /m 2 µ ) ∼ 25 eV, which is excessive. We thus focus on µ νeνµ induced via f eµ . We have generalized the calculation of muon g − 2 in two Higgs doublet model of Ref. [90,91] to the magnetic moment of the muon in the Zee model. µ νµνe arising from the three sets of diagrams in Fig. 16 are found to be: In Zee model, the cubic scalar coupling λ ϕ 0 can be written as By analyzing the contributions from the diagrams of Fig. 16, we find that one can achieve neutrino transition magnetic moment as big as µ νeνµ 3 × 10 −12 µ B , which is not sufficient to explain the observed XENON1T electron recoil excess [1]. Now we shall explain the strategy we adopted for the optimization of µ νeνµ in the model. There is stringent constraint on the Yukawa coupling f eµ from lepton/hadron universality [92], which can be translated to The top-bottom loop contribution is numerically larger than the scalar loop contribution in Fig. 16, owing to a color factor and an extra factor of 2 arising from Dirac trace, so we focus on this contribution first. From Eq. (7.65), we see that the contribution proportional to the top mass will dominate, which has a linear dependence on Y t . Thus, one has to set Y t as large as possible, while being consistent with perturbativity and other experimental constraints. Now it turns out that Y t is tightly constrained from the searches of SM Higgs observables [93] at the LHC as well as from heavy Higgs searches [94][95][96][97][98][99]. We summarize all these existing collider bounds inỸ t − m H plane in Fig. 17  are excluded from current di-Higgs limit looking at different final states bbγγ [94], bbbb [95] and bbτ + τ − [96] respectively; Blue and green shaded zones are excluded from the resonant ZZ and W + W − searches [97,98].  Fig. 18 while fixing m h + and sin ϕ. This is because the mixing angle ϕ and the two charged scalar masses will fix the µ term. From Fig. 18 we see that the contribution from the scalar loops is suppressed compared to the top-bottom loop contribution by a factor of 4 or so. In this optimized setup, one can achieve neutrino transition magnetic moment as big as µ νeνµ 3 × 10 −12 µ B , which is insufficient to explain the observed XENON1T electron recoil excess [1]. We also observe that our analysis is equally applicable to predictions of neutrino transition magnetic moment in extensions of the Zee model making use of the spin-symmetry mechanism. We have extended our analysis of µ νeνµ to the BFZ model [19]. Due to the presence of an extra scalar doublet there, the cubic scalar coupling is free compared to the Zee model, see Eq. (7.69). However this cubic coupling in the BFZ model is bounded from unitarity constraints [100] and we can gain a factor 2 to 3 from here compared to the scalar loop contribution of the Zee model, so that µ νeνµ ∼ 3 × 10 −12 µ B may be obtained. This is however not sufficient to achieve the desired values to explain the observed XENON1T electron recoil excess.

Mechanism to evade astrophysical limits on neutrino magnetic moments
As noted in Sec. 3, Majorana neutrino transition magnetic moments may be severely constrained by stellar energy loss arguments [36,37]. Photons inside the stars, which has a plasma mass, can decay into neutrinos that would escape, thus contradicting the successful stellar evolution models. The red giant branch of globular clusters provides the most stringent limits, µ ν < 4.5 × 10 −12 µ B [20], which is in conflict with the value of µ νeνµ ∈ (1.65 − 3.42) × 10 −11 µ B that is needed to explain the XENON1T excess. Here we provide a mechanism that evades this astrophysical bound on µ ν by invoking new interactions of the neutrino with a light scalar. In the presence of such interactions, neutrinos would acquire a medium-dependent mass, which may exceed the core temperature of the star, thus preventing plasmon decay kinematically.
We shall closely follow the recent field theoretic evaluation of the medium-dependent mass of the neutrino in the presence of a light scalar that also couples to ordinary matter [101] in illustrating our mechanism. This work follows the observation that such interactions would provide the neutrino with a matter-dependent mass [102]. Phenomenological implications of this scenario, including long-range force effects, were studied in Ref. [103].
Ref. [101] analyzed phenomenological constraints from laboratory experiments, fifth force experiments, astrophysics and cosmology. We shall make use of these constraints here in providing a neutrino trapping mechanism. These interactions induce a finite density neutrino mass through the diagram shown in Fig.   19. Using quantum field theory at finite temperature and density the induced neutrino mass arising from such diagrams has been computed to be [101] ∆m ν,αβ = y αβ y f m f Here n f and nf are the occupations numbers of the background fermions and antifermions.
This integral can be evaluated analytically in several interesting regimes: The nonrelativistic low temperature expansion is valid for both the electron and the nucleon in red giant stars (T 10 keV), while the high chemical potential expansion is valid for electron background in supernovae which has µ 150 MeV >> m e . The last expansion in Eq. (8.72) will be valid in early universe cosmology.
It should be noted that when the mediator mass m φ becomes smaller than the inverse size of the star, R −1 , in Eq. (8.72) m 2 φ in the denominator should be replaced by R −2 . Thus, increasing the effective mass of the neutrino by going to extremely low mass of φ is not possible. We shall be interested in m φ ∼ 10 −14 eV, which is roughly the inverse size of red giant stars.
We recall that horizontal branch stars have core temperature of order 10 keV, radius of 5×10 4 km and density of 10 4 g/cc. Red giants have core temperature of order 10 keV, radius of 10 4 km and density of 10 6 g/cc. Thus, R −1 = 2 × 10 −14 eV for the case of red giants.
The effective number of neutrino species for big bang nucleosynthesis will increase by 0.57, which appears to be not excluded by the Planck data [105].
Since the induced mass of the neutrino inside red giants can be as large as 12 MeV, plasmon decays would be highly suppressed. We could also consider interactions of φ with the nucleon instead of the electron. In this case, the supernova limit on the coupling is y N < 10 −32 , which would lower the induced mass of the neutrinos to about 23 keV, which may still be sufficient to suppress plasmon decays into neutrinos in red giants.
With the choice of parameters that induces an in-medium mass of order 12 MeV inside red giants, the neutrino would acquire keV mass inside the Sun. Since the solar core temperature is about a keV, and since solar neutrinos have been detected, it is necessary to require m eff ν < keV in the Sun. We note that the parameters can be chosen such that m eff ν inside red giants is a 1000 times smaller, say around 12 keV, in which case m eff ν inside sun would be about 2 eV. This may affect neutrino signals from the sun, but if the new couplings are flavor universal, the medium induced mass would provide an overall phase and not affect oscillations. The derived value of ∆m 2 21 may be interpreted as ∆m 2 21 + 2m 0 (m 2 − m 1 ), where m 0 is the flavor universal medium-induced mass. If the two neutrino masses m 1 and m 2 are sufficiently close, there would be no significant departure in the determination of ∆m 2 21 from solar neutrino and terrestrial neutrinos.

Summary and Conclusions
We have revived and proposed a simplified model based on SU (2) H horizontal symmetry that can generate large neutrino transition magnetic moment without inducing unacceptably large neutrino masses. In the SU (2) H symmetric limit, the transition magnetic moment is nonzero, while the neutrino mass vanishes. The simplification we suggest is based on the symmetry being approximate.
The model presented can explain the recently reported excess of electron recoil events by the XENON1T collaboration [1]. We have explored other phenomenological consequences of the model relevant for the LHC. We found that the prospects for detecting neutral scalar bosons decaying to + τ − are high in the high luminosity LHC. The model also predicts charged scalar bosons which could lead to trilepton signatures.
We also investigated a spin symmetry mechanism that can generate large µ ν while keeping m ν small. An example of such models is the Zee model of neutrino masses. However, we found that the value of µ ν induced in these models turns out be about (2 − 4) × 10 −12 µ B , which is insufficient to explain the XENON1T anomaly.
A neutrino transition magnetic moment of order 3×10 −11 µ B , as needed for the XENON1T excess, would be in apparent conflict with astrophysical arguments on stellar cooling, which sets a constraint on µ ν < 4.5 × 10 −12 µ B . We have proposed a mechanism to evade this constraint based on interactions of neutrinos with a light scalar. Such interactions can induce a medium dependent mass for the neutrino in the interior of stars, which could prevent kinematically energy loss by plasmon decay into neutrinos.