Dark matter assisted lepton anomalous magnetic moments and neutrino masses

We propose a framework that addresses the origin of neutrino mass, explains the observed discrepancies in the electron and the muon anomalous magnetic moments (AMMs) data and incorporates the dark matter (DM) relic abundance. Both the neutrino mass and the lepton AMMs are generated at one-loop level mediated by a common set of beyond the Standard Model (SM) states. In this class of models, the SM is extended with vector-like charged fermion and scalar multiplets, all odd under an imposed $\mathcal{Z}_2$ symmetry, which stabilizes the fermionic or scalar DM candidate residing in one of them. Two scalar multiplets appear in the AMM loops, thus allowing for different signs of their contributions, in agreement with the observed discrepancies which are of opposite sign for electron and muon. The vector-like fermions give rise to large new physics contributions to the lepton AMMs via chirally enhanced terms that are proportional to their mass. To demonstrate the viability of this framework, we perform a detailed study of a particular model for which a fit to the neutrino masses and mixing together with lepton AMMs are provided. Furthermore, DM phenomenology and collider signatures are explored.


Introduction
The origin of the neutrino mass is among the most crucial problems of the Standard Model (SM) of particle physics. On the other hand, the almost a century old dark matter (DM) problem is another tremendous puzzle yet to be solved. The most straightforward approach to this issue is the particle nature of the DM (for a review see Ref. [1]). One of the many popular mechanisms for neutrino mass is the radiative one (for a recent review see Ref. [2,3]) due to the natural accessibility of the involved particles at colliders and low energy experiments.
There have been lots of attempts in the literature to combine these two seemingly uncorrelated issues, one of the most prominent example being the scotogenic model [4]. In such models, the particles mediating the loop(s) that generate neutrino mass are dark matter.
Typically, new symmetries beyond the SM are required to stabilize the DM and in some cases to forbid the tree-level neutrino mass contributions, for systematic studies along this line, see for example Refs. [5][6][7][8][9][10]. The details of these models largely depend on the nature of the imposed symmetries and the needed particle content. However, common features of these models are: (i) neutrino mass is generated via quantum corrections at a given loop order, (ii) DM candidates naturally arise due to symmetry reasons, (iii) owing to the loop suppression, the new physics (NP) scale can be around the TeV scale without making the Yukawa couplings unnecessarily small, which provides a way to test these models at low energies.
Aside from neutrino mass and DM, there has been a longstanding tension between the SM prediction [11] and the experimental measured value [12] of the muon anomalous magnetic moment (AMM). Additionally, the recently measured fine-structure constant α using Caesium atoms with unprecedented precision [13] implies a deviation of the electron AMM from the SM value [14] of opposite sign compared to the muon AMM. The experimental measurements point towards about 2.5σ and 3.7σ tensions for the electron and the muon AMMs, respectively. More precisely, the corresponding discrepancies are given as with outstanding precision, these observed tensions strongly point towards physics beyond the SM. Therefore, these results recently have entertained a lot of interest in the particle physics community, for attempts to simultaneously explain these discrepancies see Refs.
To address both (g − 2) e and (g − 2) µ , NP may appear at low scale, see for example Ref. [33]. Models of these types are highly constrained from beam dump experiments, Belle and BaBar, which may eventually rule out such scenarios in the near future. We, on the other hand, are interested in scenarios where NP emerges at heavy scale. 1 To incorporate large deviations for ∆a given in Eqs. (1.1) and (1.2) from heavy NP, a chirality flip of a heavy state must take place inside the loop. This can be achieved with TeV scale scalar leptoquarks [32,41] or vector-like fermions [17]. These studies however, made no connection with either neutrino mass or DM issues.
In this work, we bring the issues of the origin of neutrino mass, the DM problem, and the electron and muon AMM puzzles under the same umbrella, and propose a framework for their explanations in a minimalistic approach. In our proposed setup, the particle content of the SM is extended by three generations of vector-like fermions and three scalar multiplets. Furthermore, the model is supplemented with a Z 2 symmetry, under which only the BSM particles are assumed to be odd. Via the propagation of these BSM multiplets, neutrino mass generation as well as new physics contributions to the lepton anomalous magnetic moments of the correct order appear at one-loop level. Two scalar multiplets, and thus two sets of Yukawa couplings, appear in the AMM loops, thus allowing for different signs of their contributions, in agreement with the observed discrepancies which are of opposite sign for electron and muon. The lightest of the neutral BSM particles is stabilized by the imposed Z 2 symmetry, which serves as the DM candidate.
The paper is build up as follows: In Section 2 we address which model classes may solve the AMM discrepancies and at the same time generate neutrino mass radiatively with the same set of new multiplets. From the list of models, in Section 3 we perform a detailed analysis of one of them, analyzing the scalar sector, performing a fit to the AMM and neutrino mass observables, discussing the dark matter phenomenology, and outlining collider phenomenology. We conclude in Section 4.

Framework
Due to its simplicity, we start our discussion with the scotogenic model [4], which employs three generations of singlet Majorana fermions 2 N (1, 1, 0) and an inert Higgs doublet φ(1, 2, 1/2), both odd under an imposed Z 2 symmetry. The neutrino mass is generated at the one-loop level via the generic diagram shown in Fig. 1, with S 1 = S 2 ≡ φ and Majorana fermion). However, a combined explanation of lepton AMMs along with reproducing realistic neutrino masses and mixings cannot be accommodated, since the proportionality relation |a µ | ∝ m 2 µ requires large Yukawa couplings, which would generate too large rates for charged lepton flavor violating (cLVF) processes like µ → eγ.
Quantifying this tension very roughly is possible as follows. In the scotogenic model neutrino mass is given by (setting loop functions to one and assuming that all new particle masses are of order TeV) where flavor indices are ignored and y is the Yukawa coupling of the singlet fermion N with lepton doublets and the inert scalar doublet. This shows that to get the correct order of neutrino mass one requires y = O(10 −5 ). On the other hand, the contribution to the magnetic moment is (2.4) Apart from the fact that the sign of ∆a µ is actually not correct in the scotogenic model, these estimates show that a simultaneous explanation of neutrino mass and the anomalous magnetic moment is not possible. Moreover, the branching ratio for µ → eγ provides additional constraints, namely (2.5) with S a scalar particle of the model. Too large rates would appear for order one Yukawas.
The sign of the muon AMM could be changed by a minimal addition of one more scalar, which provides a freedom to choose the sign of the product of the Yukawa coupling in the AMM contributions. This would utilize either of the two one-loop diagrams presented in . This is precisely what we try to achieve in an economical fashion within our framework. pursue. In this work we strictly stick to Z 2 symmetry for the fixed topology as in Fig. 1 to generate neutrino mass. For general analyses of various topologies of neutrino mass arising by utilizing exotic vector-like fermions, see for example [64][65][66].
From these Feynman diagrams and the above discussion, one sees that a common set  Having stated our criteria, the exercise is now to find a set of vector-like fermions and scalars that allow for the topologies in Figs. 1 and 2. This leads to the models summarized in Table I. Here we have listed only the viable models up to SU (2) L triplets that satisfy our above-mentioned criteria. Multiplets that contain a neutral component and thus a potential DM candidate are shown in red. By following our methodology, models involving higher dimensional representations can be constructed trivially. It is beyond the scope of this work to study each of these models in detail. Instead, in the next section we perform a detailed analysis of the first model (Model-I) in the list.

Details of Model-I
In this section we perform a detailed analysis of Model-I. In this model, the SM particle content is extended by three singly charged vector-like fermions F L,R , and three scalars: a singlet and two doublets under the SU (2) L group. One of the doublets has hypercharge 1/2 and the other has 3/2. As already mentioned, under the imposed Z 2 symmetry, the SM particles are even, whereas all the BSM states are odd. The full quantum numbers of the BSM multiplets are summarized in Table II. With this particle content, the most general Yukawa Lagrangian consistent with all symmetries is given as Here, L L is a left-handed lepton doublet, R is a right-handed lepton, and H is the SM Higgs doublet. In the above Eq. (3.7), for simplicity we have suppressed generation indices.
Note that the third term violates lepton number. Yukawa couplings of the quarks remain unchanged compared to the SM, hence we only focus on the leptonic sector. We work in the basis where the Yukawa coupling y H and the vector-like fermion mass matrix M F are diagonal. The three new Yukawa couplings matrices y 1,2,3 are in general arbitrary.

Scalar sector
The scalar sector of the full model consists of three neutral CP-even states h, S 0 1,2 , one neutral CP-odd A 0 , two singly charged S + 1,2 , and a doubly charged S ++ . Here h is identified with the SM Higgs, which does not mix with the rest of the two states S 0 1,2 due to the imposed Z 2 symmetry. The lightest between these two states S 0 1,2 is identified as the DM. Moreover, we assume the BSM multiplets η and φ 1 do not accrue any VEV, hence, the Goldstone bosons G 0 , G ± originate entirely from the SM Higgs doublet H. The complete scalar potential for Model-I is given as We now derive the masses of the physical Higgs particles from the above potential. The mass-squared matrix M 2 S 0 for the two CP-even states, written in the {η 0 , Re(φ 0 1 )} basis, is The scalars that do not mix with any other fields are the SM Higgs, the CP-odd scalar, and the doubly charged scalar. The corresponding squared masses are where φ A is Im(φ 0 1 ). Finally, the mass-squared matrix for the singly charged scalars in a basis of (φ ± 1 , φ ± 2 ) reads Moreover, the mixing angle α (γ) between the two mixed CP-even (singly charged) states can be calculated from . (3.14) We note that the presence of non-zero α is crucial for generating the AMMs and for the dark matter phenomenology. Non-zero γ is required to generate neutrino mass. Moreover, between the two neutral physical states S 0 1,2 , we will assume S 0 1 to be the ligther one and identify it as the DM candidate. Its decomposition in terms of the original fields is given by

Lepton anomalous magnetic moments
In the present set-up, we assume the vector-like fermions to reside around the TeV scale.
In contrast to the scotogenic case, having such a heavy mass does not require large Yukawa couplings to incorporate the ∆a data given in Eqs. (1.1) and (1.2). Large enough corrections to the lepton AMMs naturally arise due to a chirality flip of the vector-like fermions on the internal line, as can be seen from Fig. 2. Moreover, the sign difference for ∆a e and ∆a µ is obtained by appropriately choosing the sign of the product of the Yukawa couplings that enter in this chirality enhanced AMM term. We derive the complete NP contributions towards (g − 2) that is given by [67] The expressions for the loop functions are The re-scaled Yukawa couplings appearing in Eq. (3.15) are defined by It should be pointed out that the very first term (chirality flip term) in Eq. (3.15) dominates; the remaining contributions can be safely ignored for our case, which we have confirmed numerically. In our set-up, the contribution from the SM Higgs h remains unchanged, which is already part of a SM . We stress here that for α = 0 the dominating first contribution to the AMMs would vanish. This can be understood from the expressions in (3.8) and (3.14).
Vanishing α would correspond to vanishing µ 5 , and thus no triple-scalar coupling of φ 1 with η and the SM Higgs. This in turn would correspond to the absence of the AMM diagram in Fig. 2.
The off-diagonal elements in the Yukawa couplings y 1,3 will lead to cLFV processes such as → γ. Due to the same chirality enhancement effects via the vector-like fermions, these processes impose severe constraints on these off-diagonal Yukawa couplings. Amplitudes of these cLFV processes can be straightforwardly computed for our scenario, however, for the simplicity of our work, we assume the two Yukawa coupling matrices y 1,3 to be diagonal (meaning, small off-diagonal entries are omitted for our analysis). However, non-zero but small off-diagonal entries have no impact on the results obtained in this work. There are also very stringent constraints that arise from the lepton dipole moments (for a review see Ref. [68]) measurements for complex couplings. We avoid these constraints by demanding (g-2) e ± 1σ (g-2) μ ± 2σ these y 1,3 couplings to be real. For completeness, here we present the generic expressions for the cLFV process → γ for our model

Neutrino mass
The same vector-like fermions play a major role in generating radiative neutrino mass and the corresponding Feynman diagram is presented in Fig. 1. The loop is completed via the propagation of the singly charged scalars S + 1,2 , and we obtain the following expression for the neutrino mass matrix for Model-I Here the mixing angle γ between the singly charged physical particles is defined in Eq. (3.14).
We stress here that for γ = 0 the neutrino mass would vanish. This can be understood from the expressions in (3.8) and (3.14). Vanishing γ would correspond to vanishing λ φ 1 φ 2 , and thus no quartic scalar coupling of φ 1 with φ 2 and a SM Higgs pair. This in turn would correspond to the absence of the neutrino mass diagram in Fig. 1.
Note that to reproduce correct lepton mixing, one must have a non-trivial structure for the Yukawa coupling matrix y 2 , since y 1 is taken to be diagonal. This however, does not conflict with lepton flavor violating → γ processes mediated by the doubly charged scalars, since (y 2 ) ij ∼ 10 −5 in order to generate the correct neutrino mass scale. In the next subsection, we provide a realistic fit to neutrino mass spectrum.

A combined fit to data
To demonstrate the viability of our proposed framework, here we present a combined fit to reproduce the experimental results. The expressions for (g − 2) e and (g − 2) µ are given in Eq.  Table III. The neutrino mass matrix can be parameterized as follows   where m i are real eigenvalues and we have defined c ij = cos θ ij , s ij = sin θ ij . In the PMNS mixing matrix there exist three physical phases, one Dirac phase δ ≡ δ CP and two Majorana phases α 21,31 , where we have used the particle data group (PDG) parametrization. In this work, we assume a normal ordering for neutrino masses that corresponds to m 1 < m 2 < m 3 , which still is favored by oscillation data [69,70].
With all these in hand, we perform a combined numerical analysis and provide a benchmark point in the following  Table IV. Since an explanation of the lepton AMMs demands Yukawa couplings of order unity (as can be seen from Eq. (3.29)), and the same Yukawa couplings enter in neutrino mass generation, it can be easily understood that y 2 λ φ 1 φ 2 ∼ 10 −10 (instead of y 2 2 ∼ 10 −10 as in Eq. (2.3)) must be satisfied to reproduce the correct neutrino mass scale. Regarding the smallness of λ φ 1 φ 2 , we recall that it is the coefficient of the quartic coupling (H φ 1 )(φ † 2 H) responsible for mixing the two singly charged states, as defined in Eq. (3.14). In the limit of λ φ 1 φ 2 → 0 neutrino masses are zero, because the theory regains the accidental lepton number conservation of the SM. The chosen DM (S 0 1 ) mass of 120 GeV and the associated mixing angle sin α for this benchmark point will be shown to be consistent with both DM detection bounds as well as DM relic abundance as detailed in the next section. It should be pointed out that in the DM analysis more parameters such as the Higgs portal coupling and lepton coupling portal play role, which are not fixed by the fit performed above. Moreover, the mass of the doubly charged scalar is not determined from the fit, which for simplicity, we choose to be degenerate in mass with its singly charged partner to be consistent with T parameter constraints. However a splitting of order O(100) GeV is still allowed [71].

Dark matter phenomenology
In this subsection, we analyze the Dark Matter (DM) phenomenology in Model-I, where lepton anomalous magnetic moments, neutrino masses and mixings are successfully generated. As aforementioned, in this model the presence of a discrete symmetry Z 2 stabilizes the DM particle. The newly introduced scalars (φ 1 , φ 2 and η) and vector-like leptons F L,R are odd under this discrete symmetry, whereas the SM particles are even. The lightest neutral particle among the new ones qualifies to be a DM candidate. In our setup for Model-I, the dark matter candidate will be an admixture of neutral components of the doublet φ 1 and the singlet η. As one can see from Eq. (3.15), one needs to introduce mixing between these two fields to successfully address electron and muon g −2 anomalies. Hence, the dark matter can be neither pure singlet type [72][73][74][75][76][77][78][79][80][81][82][83][84] nor pure inert doublet type [4,[85][86][87][88][89][90][91][92]. Rather, it will be singlet-doublet scalar dark matter [93]. While scalar singlet dark matter is tightly constrained from DM direct detection experiments [81,83,[94][95][96], inclusion of mixing with an additional doublet can introduce new additional interactions producing the right amount of relic density, and which can potentially allow for evasion of direct detection bounds [94][95][96] for a large region of parameter space.
The dominant processes that contribute to the annihilation of the DM particle are shown in Fig. 4. In our case, the DM can annihilate to SM particles through s-channel Higgsmediated processes (Higgs-portal). These Higss-portal processes can be particularly important when the DM mass is close to half of the Higgs boson mass. Above this mass regime, the DM annihilation to gauge bosons (possible because it is partly a doublet) contributes dominantly to the annihilation processes. In this mass region, the DM annihilation through the t-channel exchange is usually smaller than the contribution from the 4-point vertex shown in Fig. 4. Finally, the presence of the vector-like leptons opens up new annihilation modes for DM via the t-channel processes (lepton-portal) as shown in Fig. 4. In the low mass region, these leptonic portal processes become significant in addition to the Higgs-portal processes.
For the DM analysis, we will denote the lepton portal coupling by λ LP . Since the DM is identified to be the state S 0 1 , its lepton portal couplings with ) which can be read off from Eqs. (3.18) and (3.19). On the other hand, λ HP represents the Higgs portal coupling, which is defined by λ HP = (λ Hφ 1 +λ Hφ 1 +2λ Hφ 1 ) sin 2 α +2λ Hη cos 2 α.
That is, our DM particle S 0 1 couples via  In addition to the DM relic density study, we also consider the constraints from various DM direct detection experiments. In our model, the DM can interact with nuclei dominantly via t-channel Higgs boson exchange. The corresponding spin independent DMnucleon scattering cross-section is estimated in [73,83]. Using this, we recast the limits from TeV LHC data [103], cf. Section 3.6. As we can see from Fig. 7, there is a significant region of parameter space (intersection zones) which can accommodate the correct experimental values of (g − 2) e,µ as well as the DM relic abundance.

Collider implications
Here we discuss the collider phenomenology associated with the dark matter in our model.
Especially, the presence of doubly charged scalar S ±± and vector like lepton F ± can give rise to rich phenomenological implications at the LHC. Generically, DM is searched for at the LHC in mono-X searches, e.g. in association with one or more additional SM particles, preferably a high momentum object (jet, photon, vector boson etc.) radiated by the initial state quarks. Here, we want to highlight a few non-standard collider aspects of DM which naturally arise in our framework. 3 The relevant Feynman diagram for this collider signal of DM are shown in Fig. 8. The charged vector like fermion F ± , which is responsible for lepton anomalous magnetic moments, will be pair-produced at the LHC via s-channel Z/γ exchange and it will further decay back to DM and SM charged leptons. This will lead to DM production in association with two charged leptons (pp → + − + E / T ) at the LHC. This process is somehow similar to the standard slepton searches [104][105][106]. If kinematically allowed, the DM can also be produced in association with same-sign dileptons from the decay of doubly charged scalar S ±± as shown in right panel of the Fig. 8. The dominant production mechanism of the doubly charged scalar S ±± at the LHC is the standard Drell-Yan process via s-channel Z/γ exchange. It will further dominantly decay to S ±± → F ± ± and the vector-like leptons F ± decay dominantly to DM and SM charged leptons. This will lead to DM production in association with four charged leptons (pp → 2 + 2 − + E / T ) at the LHC. Recently, prospects of this type of DM signal with multi-lepton signature were analyzed in detail [107]. On the other hand, if the Z 2 odd charged scalars S ±± and fermions F ± are not kinematically allowed to decay to DM promptly, they will be longlived. In this case the track originating from long-lived charged particles can disappear at a point inside the detector. There are dedicated searches for these stable charged particles at the LHC [103] using signatures of long time-of-flight measurements and anomalously high energy deposits in the silicon tracker. Non-observation of any signal impose severe constraints on these stable charged particles. Using the 13 TeV LHC data [103], we find that the mass of a (long-lived) charged vector-like fermion F ± is constrained up to 550 GeV, whereas the mass limit on (long-lived) doubly charged scalar S ±± is 660 GeV. Recently, displaced vertex and disappearing track signature for long-lived singly-charged lepton were analyzed, see [108,109]. Also, the prospect of discovery of long-lived doubly charged scalars was analyzed [110]. Thus, this model predicts several unique signals like displaced vertex signature, disappearing tracks at the collider and non-standard DM signals with multilepton signature. All these signals have unique discovery prospects which can be tested in the upcoming run of the LHC or other colliders. The investigation of these collider signals is beyond the scope of this article and shall be presented in a future work.
In this work, we have proposed a class of models that intercorrelates and offers a simultaneous explanation of neutrino mass, dark matter, the long-standing puzzle of the muon anomalous magnetic moment, and the recently observed tension in the electron anomalous magnetic moment. In each of these models, the Standard Model is extended with a vectorlike fermion and a set of scalars, which are odd under an added Z 2 symmetry. A common set of these BSM states run through the loops and generates neutrino mass as well as lepton AMMs at one-loop order. If the vector-like fermions are around the TeV scale, they provide large chirality enhanced contributions required to resolve the lepton AMMs. Different signs for the muon and electron magnetic moments are arranged easily because different sets of Yukawa couplings are involved. The lightest of the neutral members of our new multiplets, either fermionic or bosonic in nature, plays the role of the dark matter, which is stabilized by the unbroken Z 2 symmetry. Models belonging to this class are simple in their constructions and provide a framework to unify a number of various seemly uncorrelated issues that cannot be solved with the Standard Model. After a generic discussion, we have focused on a particular model, and performed a detailed analysis that includes a fit to neutrino oscillation parameters as well as electron and muon AMMs, followed by a discussion of DM and collider phenomenology.