Neutrino magnetic moment and inert doublet dark matter in a Type-III radiative scenario

We narrate dark matter, neutrino magnetic moment and mass in a Type-III radiative scenario. The Standard Model is enriched with three vector-like fermion triplets and two inert doublets to provide a suitable platform for the above phenomenological aspects. The inert scalars contribute to total relic density of dark matter in the Universe. Neutrino aspects are realized at one-loop with magnetic moment obtained through charged scalars, while neutrino mass gets contribution from charged and neutral scalars. Taking inert scalars up to $2$ TeV and triplet fermion in few hundred TeV range, we obtain a common parameter space, compatible with experimental limits associated with both neutrino and dark matter sectors. Using a specific region for transition magnetic moment (${\cal O} (10^{-11}\mu_B$)), we explain the excess recoil events, reported by the XENON1T collaboration. Finally, we demonstrate that the model is able to provide neutrino magnetic moments in a wide range from $10^{-12}\mu_B$ to $10^{-10}\mu_B$, meeting the bounds of various experiments such as Super-K, TEXONO, Borexino and XENONnT.


I. INTRODUCTION
The fascinating description of elementary particle physics is elegantly portrayed by the Standard Model (SM) in the low energy regime.The locally gauge invariant Lagrangian is able to describe how the interactions proceed at the most fundamental level.This gauge theory has provided a pathway to understand the behavior of nature at very tiny length scale and serves as a theoretical torch for exploring several unknown things beyond, a never ending tale of theorists and experimentalists to unravel the mysteries of the Universe.A small chunk of puzzles include neutrino masses and mixing [1][2][3][4][5][6][7][8][9][10], nature and identity of dark matter (DM) [11][12][13][14], matter anti-matter asymmetry [15][16][17][18] and the recently observed anomalies in the flavor sector [19].
Several neutrino experiments have unambiguously proved that oscillation of neutrino flavor occurs during propagation and neutrinos possess small but non-zero masses.With this extra degree of freedom, many new possibilities are opened up, and one such possibility amongst them is that neutrinos can possess electromagnetic properties like electric and magnetic dipole moments.Solar, accelerator and reactor experiments help us in the direct measurement of magnetic moments and eventually put the limits on them.One possible mode of measurement involves the study of neutrino/anti-neutrino electron scattering at low energy limits.In minimally extended SM, one can have neutrino magnetic moment (νMM) of the order 10 −19 µ B for Dirac type of neutrinos.However, these values are beyond the sensitivity reach of any experimental measurement.On the other hand, for Majorana type neutrinos, one can have a very high transition magnetic moment, fitting the experimental observations.That's why the study of νMM becomes important for the distinction of Dirac and Majorana type of neutrinos.
In recent past, XENON collaboration performed a search for new physics with its 1 ton detector and reported an excess of events over the known backgrounds in the recoil energy range 1 − 7 keV, peaked around 2.5 keV [20].It turns out that such excess can be explained with large transition magnetic moment of neutrinos.With new data from its successor XENONnT [21], no visibly bold excess events were seen in the low energy region creating an anomalous situation between these two experiments.The collaboration is suspecting this excess in XENON1T was due to uncounted tritium whose presence or absence they can't corroborate.In this scenario, we cannot completely ignore the possible implication of new physics effects at XENON1T and that is why it is very interesting to explore such possibilities.Several works explaining this excess and neutrino electromagnetic properties can be found in the literature [22][23][24][25][26][27][28][29][30][31] .
Zwicky made a proposal in 1933 for the existence of dark matter through observations of spiral galaxy rotation curves, however the physics of this mysterious particle is still unsettled.
Freeze-out scenario has been the one that has fascinated theoretical physicists, a paradigm that is able to provide proper relic density as per Planck satellite by a weakly interacting massive particle (WIMP).Now, we raise a question, whether a dark matter particle running in the loop, forming an electromagnetic vertex can provide neutrino magnetic moment.
With this view point, we provide a simple model that can accommodate non-zero magnetic moment for neutrino and also discuss dark matter phenomenology in a correlative manner.
The paper is organized as follows.In section-II, we describe the model along with the particle content and interaction terms to address neutrino magnetic moment, neutrino mass and dark matter.The mass spectrum of scalar sector due to mixing is also discussed in this section.Section-III narrates neutrino magnetic moment and neutrino masses at one-loop.
Section-IV describes the dark matter relic density and its detection prospects.Section-V provides the detailed analysis, showing common parameter space to obtain observables related to the aspects of neutrino and dark matter sectors.We also emphasize more specific constraints on Yukawa couplings from current neutrino oscillation data.Section-VI gives the signature of magnetic moment in the light of electron recoil event excess at XENON1T and also the overall obtained range of magnetic moment in the concerned model.Finally, concluding remarks are provided in section-VII.

II. MODEL DESCRIPTION
To address the neutrino mass, magnetic moment and dark matter in a common platform, we extend the SM framework with three vector-like fermion triplets Σ k , with k = 1, 2, 3 and two inert scalar doublets η j , with j = 1, 2. We impose an additional Z 2 symmetry to realize neutrino phenomenology at one-loop and also for the stability of the dark matter candidate.
The particle content along with their charges are displayed in Here, σ a 's represent Pauli matrices and Σ 0 Lagrangian terms of the model is given by In the above, Σ +,0 = Σ +,0 L + Σ +,0 R and Σ = (Σ 1 , Σ 2 , Σ 3 ) T .The covariant derivative for Σ is given by The Lagrangian for the scalar sector takes the form where, the inert doublets are denoted by and the scalar potential is expressed as [36,37] V

B. Mass spectrum
The mass matrices of the charged and neural scalar components are given by Here, One can diagonalize the above mass matrices using The flavor and mass eigen states can be related as The invisible decays of Z and W ± at LEP, limit the masses of inert scalars as [38,39] Moving on to fermion sector, electroweak radiative corrections provide a mass splitting of 166 MeV [40] between the charged and neutral component of triplet.We work in the high scale regime, this small splitting does not effect the phenomenology.

III. NEUTRINO PHENOMENOLOGY A. Neutrino Magnetic moment
Though neutrino is electrically neutral, it can have electromagnetic interaction at loop level, as shown in Fig. 1, where ψ(p) and ψ(p ′ ) denote the incoming and outgoing neutrino states.The effective interaction Lagrangian takes the form [41] L EM = ψΓ µ ψA µ .
In the above, the electromagnetic vertex function varies with the type of neutrinos, i.e., Dirac or Majorana.In case of Dirac neutrino, Γ µ takes the form where f Q (q 2 ), f M (q 2 ), f E (q 2 ) and f A (q 2 ) represent the form factors of charge, magnetic dipole, electric dipole and anapole respectively.
In the non-relativisitic regime, f Q (0) = Q stands for the charge, f M (0) = µ represents magnetic dipole moment, f E (0) denotes electric dipole moment and f A (0) stands for the Zeldovich anapole moment of the particle.All the four form factors remain finite in Dirac type neutrino.For Majorana case, using the property of charge conjugation ψ c = Cψ T , we get which results f Q (q 2 ) = f M (q 2 ) = f E (q 2 ) = 0 for a Majorana neutrino.However, if the electromagnetic current is between two different neutrino flavors in the initial and final states i.e., ψ i Γ µ ψ j A µ with i ̸ = j, Majorana neutrinos can have non-zero transition dipole moments.
In the present model, the magnetic moment arises from one-loop diagram shown in Fig. 2, and the expression of corresponding contribution takes the form [42] (µ ν where 2: One-loop Feynman diagram for transition magnetic moment.

B. Neutrino mass
In the present model, contribution to neutrino mass can arise at one-loop from two diagrams, one with charged scalars and fermion triplet in the loop while the other with neutral scalars and fermion triplets.The relevant diagrams are provided in Fig. 3 and the corresponding contribution takes the form [43][44][45] FIG. 3: One-loop diagram that generates light neutrino mass.
the dark matter density of the Universe through annihilations and co-annihilations.With the mediation of scalar Higgs, ϕ R i ϕ R j can annihilate to f f , W + W − , ZZ, hh and via Z boson, ϕ R i ϕ I j can co-annihilate to f f , W + W − , Zh.The charged and neutral components can co-annihilate to f ′ f ′′ , AW ± , ZW ± , hW ± through W ± .Here, f ′ = u, c, t, ν e , ν µ , ν τ and f ′′ = d, s, b, e, µ, τ [46][47][48].The abundance of dark matter can be computed by where, M Pl = 1.22 × 10 19 GeV and g * = 106.75denote the Planck mass and total number of effective relativistic degrees of freedom respectively.The function J is In the above, the thermally averaged cross section ⟨σv⟩ reads as Here K 1 , K 2 are the modified Bessel functions, x = M DM /T , with T being the temperature, M DM is dark matter mass, σ is the dark matter cross section and x f stands for the freeze-out parameter.

B. Direct searches
Moving to direct searches, the scalar dark matter can scatter off the nucleus via the Higgs and the Z boson.Mass splitting between real and imaginary components above 100 KeV can forbid gauge kinematics [48].Thus the DM-nucleon cross section in Higgs portal can provide a spin-independent (SI) cross section, whose sensitivity can be checked with stringent upper bound of LZ-ZEPLIN experiment.The effective interaction Lagrangian in Higgs portal takes the form The corresponding cross section is given by [46][47][48] σ SI = 1 4π where, M n denotes the nucleon mass, nucleonic matrix element f ∼ 0.3 [49].We have implemented the model in LanHEP [50] package and used micrOMEGAs [51][52][53] to compute relic density and also DM-nucleon cross section.The detailed analysis of neutrino and dark matter observables and their viability through a common parameter space will be discussed in the upcoming section.

V. ANALYSIS
In the present framework, we consider ϕ R 1 to be the lightest inert scalar eigen state and there are five other heavier scalars.To make the analysis simpler, we consider the mass parameters related to the scalar masses as follows: one parameter M R1 corresponding to the mass of ϕ R 1 and three mass splittings namely δ, δ IR and δ CR .The masses of the rest of the inert scalars can be derived using the following relations: where, i = 1, 2. In the above set up, the scalar mixing angles can be related as follows We have performed the scan over model parameters as given below, in order to obtain the region, consistent with experimental bounds associated with both dark matter and neutrino sectors: We filter out the parameter space by providing Planck constraint on relic density [54] in 3σ and then compute DM-nucleon SI cross section for the available parameter space.We project the cross section as a function of M R1 in the left panel of Fig. 4 with cyan data points, where the dashed brown line corresponds to LZ-ZEPLIN upper limit [55].Choosing a set of values for the Yukawa and fermion triplet mass, with the obtained parameter space, one can satisfy the discussed aspects of neutrino phenomenology.The blue, green and red data points corresponding to 25, 80 and 420 TeV of triplet mass and suitable Yukawa satisfy the neutrino magnetic moment and light neutrino mass in the desired range simultaneously, as projected in the right panel.We notice that a wide region of dark matter mass is favoured as we move towards high scale (triplet mass) and moreover the favourable region shifts towards larger values with scale.The suitable region of Yukawa and fermion triplet mass is depicted in the left panel of Fig.    FIG.4: Left panel projects SI WIMP-nucleon cross section as a function M R1 , with dashed brown line of LZ-ZEPLIN upper limit [55].Cyan data points satisfy Planck limit [54] on relic abundance in 3σ.Blue, green and red data points satisfy neutrino mass and magnetic moment for a specific set of values for fermion triplet and Yukawa, visible in the right panel.
favourable to explain both neutrino and dark matter aspects discussed so far, we project relic abundance scalar dark matter in Fig. 6.More specific constraints on the Yukawa couplings can be obtained from the neutrino oscillation parameters.For this purpose, we consider the neutrino mixing matrix as the product of a tri-bimaximal (TBM) matrix with a rotation matrix U 13 , given by From eqns.16,17, the matrices associated with neutrino magnetic moment and mass can be written in a compact form as where, Diagonalizing the matrices in Eq. ( 28) using U ν , we obtain three unique solutions where the Yukawa couplings corresponding to different flavors become linearly dependent.The relations take the form Thus, the obtained diagonalized matrices associated with neutrino magnetic moment and mass in the basis of active neutrinos are [56][57][58] µ where, The matrix U ν replicates the standard Pontecorvo-MakiNakagawa-Sakata (PMNS) matrix, where the mixing angles, Θ and φ can be fixed using the observed neutrino oscillation Furthermore, the Yukawa matrix turns out to be Yukawa FIG.7: Allowed region of Yukawa (colored data points), satisfying the 3σ limit on mass squared differences [59] and cosmological bounds on sum of active neutrino masses [60] (colored vertical lines).

VI. IMPLICATIONS
In the experimental perspective, non-zero neutrino magnetic moment of solar neutrinos can provide explanation for the excess in electron recoil events at XENON1T collaboration [20].In other words, the neutrino transition magnetic moment can provide additional contribution to the neutrino-electron scattering process.In this section, we utilize non-zero transition neutrino magnetic moment to explain the excess in electron recoil events.
In the presence of magnetic moment, the total differential cross section can be written as [61] dσ dT e TOT = dσ dT e SM + dσ dT e EM , where T e is the electron recoil energy.The first contribution in eq.37 is due to standard weak interactions, given by In the above, G F stands for the Fermi constant and The second contribution comes from the effective electromagnetic vertex of the neutrinos, i.e., magnetic moment contribution, which is expressed as where, α is the electromagnetic coupling, E ν is the initial neutrino energy, µ νeµ is the neutrino magnetic moment and µ B is the Bohr magneton.For high T e value, weak cross-section dominates and for low T e value, the electromagnetic cross-section dominates and hence, we search for the signature of neutrino magnetic moment in the low energy region.For simplicity, we take one transition magnetic moment µ νeµ to explain the XENON1T excess.
In the above, ε(T e ) denotes the efficiency of detector [20], n te is the count of number of target electrons in the fiducial volume of one ton Xenon [62], dϕ s /dE ν represents the solar neutrino flux spectrum [63], and the function G(T e , T r ) reflects the normalised Gaussian smearing function that takes into account the detector's limited energy resolution [20].The limits T th = 1 KeV and T max = 30 KeV stand for the threshold and maximum recoil energy of detector respectively.The extremes of neutrino energy for the integral are given by And the disappearance probability can be taken as P eµ/τ = 1 − P ee [64,65].The oscillation parameters are taken from [59].In Fig. 8, we project the event rate as a function of recoil energy T r , for two set of values for magnetic moment, i.e., µ νeµ = 2.6 × 10 −11 µ B and 3.2 × 10 −11 µ B (red curves).Adding with the background (green curve), we are able to meet the observed recoil event excess in the low energy region near 2.5 KeV as of XENON1T experiment [20].
In Fig. 9, we project neutrino magnetic moment as a function of dark matter mass, choosing specific set of values assigned to triplet fermion.As seen earlier in the left panel of Fig. 4, a specific range of dark matter mass is favoured with the scale of triplet mass.It is transparent that the model parameters are able to provide neutrino magnetic moment in the range 10 −12 µ B to 10 −10 µ B , sensitive to the upper limits of Super-K [34], TEXONO [33], Borexino [32], XENON1T [20], XENONnT [21] and white dwarfs [35] (colored horizontal lines).Thus, from all the above discussions made, it is evident that this simple framework can provide a consistent phenomenological platform for a correlative study of neutrino magnetic moment (especially), mass and dark matter physics.this simple model provides a suitable platform to study neutrino phenomenology, especially the neutrino magnetic moment and also dark matter aspects.

FIG. 5 :
FIG. 5: Left panel displays the suitable region for triplet mass and Yukawa to explain neutrino phenomenology.Right panel shows the allowed region for scalar mass splittings, thick (thin) bands correspond to δ IR (δ CR ) respectively.

TABLE I :
Fields and their charges in the present model.

TABLE II :
Set of benchmarks from the consistent parameter space.

TABLE III :
Neutrino and dark matter observables for the given benchmarks.
The differential event rate to estimate the XENON1T signal is given by e ) × G(T e , T r ).