Muon $g-2$ and $B$-anomalies from Dark Matter

In the light of the recent result of the Muon g-2 experiment and the update on the test of lepton flavour universality $R_K$ published by the LHCb collaboration, we systematically build and discuss a set of models with minimal field content that can simultaneously give: (i) a thermal Dark Matter candidate; (ii) large loop contributions to $b\to s\ell\ell$ processes able to address $R_K$ and the other $B$ anomalies; (iii) a natural solution to the muon $g-2$ discrepancy through chirally-enhanced contributions.

Class F − Fermion mediator Class S − Scalar mediator Figure 1: Basic diagrams providing a contribution to b → sµµ involving only left-handed SM fields, i.e. of the kind δC 9 µ = −δC 10 µ . Based on this, we classify the models according to the spin of the flavour mediator, the field that couple to both quarks and leptons.

Setup
Recent global analyses of the b → s data [31][32][33][34][35][36][37][38][39], including the new measurement of R K [40,41], show that a satisfactory fit of the observed B-anomalies favours solutions featuring effects in δC 9 µ and δC 10 µ , where these quantities are defined as the NP contributions to the following operators: In particular, the simplest ways to improve the fit to the data is to introduce an exotic contribution to δC 9 µ alone or one of the kind δC 9 µ = −δC 10 µ . Hence substantial (or exclusive) interactions involving left-handed (LH) muons are favoured. Furthermore, global fits require that non-standard contributions from hadronic right-handed (RH) currents (if present at all) be subdominant. In other words, a minimal ingredient of our models will be given by the 1-loop contributions shown in Figure 1, that is, the three fields appearing in either diagram need to be present and one of them will constitute the DM candidate, as discussed in our previous work [68].
Our previous analysis showed that the most satisfactory solutions of the B anomalies (that is, the only viable ones in wide regions of the parameter space without relying on tuning) that provide in addition a natural thermal DM candidate require: (i) DM to be an SU (2) L × U (1) Y singlet; (ii) the DM field to couple to muons (since the large couplings to muons required by the fit to the B-anomalies induce efficient DM annihilation evading the problem of DM overproduction); (iii) DM to be a Majorana fermion, a real scalar, or one of the two components of a complex scalar with a mass splitting > O(100) keV (such that the most dangerous contributions to DM direct detection are suppressed). The above considerations restrict the set of viable possibilities to cases where the fields Ψ/Φ or Φ /Ψ in Figure 1 are (or mix with) a DM singlet.
The subset of NP fields coupling to muons in Figure 1 also contributes to the dipole operator relevant for the muon g − 2: where v is the SM Higgs vacuum expectation value (vev) 246 GeV. The normalisation ∝ v of the above operator highlights that, following from gauge invariance, a flip of the chirality of the muon, hence a Higgs vev insertion, is necessary to induce such effect. On the other hand, the fields Ψ−Φ or Φ−Ψ do not couple to RH muons, hence such a chirality flip can only occur through a muon mass insertion in the external leg, leading to a suppression of the effect by the small muon Yukawa coupling, C µµ ∝ y µ . Minimal models where DM couples only to LH muons therefore can not provide a sizeable contribution to a µ , besides very tuned regions of the parameter space [52]. Thus, a natural explanation of the muon g − 2 anomaly requires a chiral enhancement, i.e. a chirality flip occurring inside the loop through a coupling to the SM Higgs field y µ , see e.g. [46,50,52,71]. The minimal way to achieve this is to add a 4th field to our minimal models: either Ψ /Φ mixing with Ψ/Φ through a Higgs vev, or Φ /Ψ mixing with Φ /Ψ . Illustrative diagrams providing an enhanced contribution to the muon g − 2 are shown in Figure 2. Notice that these mixing fields also induce additional contributions to b → sµµ involving RH muons (thus deviating from the δC 9 µ = −δC 10 µ pattern), as shown in Figure 3. The only combinations of the quantum numbers of the new fields that fulfils the above conditions are displayed in Table 1. A unique choice of the transformation properties under SU (3) c and only three under SU (2) L are possible. For each of these three choices a minimal model would comprise four fields: Considering the two possible choices of the mixing field, as well as the possible hypercharge assignments delivering at least an absolute singlet coupling to leptons, we end up with only 9 options (times the two spin alternatives). These are listed in Table 2. The models highlighted in cyan feature pure singlet DM We highlight in green the combinations that provide a viable simultaneous fit to DM and B-anomalies. Minimal models includes the first three fields plus one chosen from the last two columns.
(scalar or fermion); for the models highlighted in red, DM is in general a mixture of a singlet and an SU (2) L doublet (again scalar or fermion).

Minimal models
In the previous section, we showed how our set of phenomenological requirements lead to a limited number of minimal models featuring four NP fields, which are displayed in Table 2. Here we provide more details about the interactions and the field mixing occurring in the different cases. We classify the models according to the nature of the "flavour mediator" field appearing in the diagrams of Figure 1.
Class F. These models feature a vector-like fermion Ψ as flavour mediator and two extra scalars Φ q and Φ coupling to the SM left-handed fermions. For models augmented with an additional scalar Φ , the interactions are described by the following Lagrangian: where a H is a parameter with a dimension of a mass. In case of Φ being a doublet, one may need to substitute Φ → Φ = iσ 2 Φ . Notice that, depending on the hypercharge, either the charged or the neutral components in Φ and Φ mix upon EW-symmetry breaking (EWSB). The resulting mass eigenstates and the corresponding mass eigenvalues will be given by diagonalising a matrix of the following schematic form: Label For models where instead the fourth field is the additional fermion Ψ mixing with the flavour mediator Ψ, the Lagrangian schematically reads: For illustration here we show the case labelled F IIc (or equivalently F Vc ) in Table 2, where Ψ is a doublet and Ψ a Majorana or Dirac singlet (we recall all the extra fermions, unless they are Majorana, come in vectorlike pairs). We have also omitted couplings to RH quarks, possibly allowed by gauge invariance, of the kind Γ D iD i P R Ψ Φ q and Γ U iŪ i P R Ψ Φ q (that we are assuming to be suppressed in the following). 3 For this kind of models the singlet-doublet mass matrix has the schematic forms: for, respectively, a Majorana and a Dirac singlet field (Ψ in this illustrative examples).
Class S. In these models, we introduce a scalar flavour mediator Φ and two fermions Ψ q and Ψ in vectorlike representations of the SM gauge group, plus either an additional Ψ or a Φ . The Lagrangians and the mass matrices are as those given above mutatis mutandis: 3 Notice that models in the categories FV/SV are identical to those in FII/SII besides the field Φq/Ψq being an SU (2)L triplet instead of a singlet. This forbids coupling to RH quarks but we expect that otherwise it is not modifying DM and flavour phenomenology to large extent. Thus we can omit a detailed analysis of this class models and focus on those belonging to FI/SI and FII/SII.
To the best of our knowledge, the model that here we call S Ib , belonging to this class, is the only example of this kind of models addressing DM, muon g − 2 and B anomalies which has been already discussed in the literature in Ref. [61].

Results
In this section we will illustrate the phenomenology of the minimal models introduced above, choosing two examples with distinctive DM candidates: fermionic singlet-doublet DM (model F Ib ), and real scalar DM (model F IIb ).
Furthermore we will assume all the parameters to be real. In terms of the above rotations and mass eigenvalues, the Wilson coefficients for b → s transitions are: Notice that, in the absence of the coupling to RH muons or of the singlet-doublet mixing (i.e. if V 1i V 1j V 2i V 2j = 0 for any i, j), the above contributions give δC 9 µ = −δC 10 µ , as expected. The effect of our fields on the muon g − 2 reads where we have shown the dominant chirally-enhanced term only. Subdominant contributions can be found e.g. in Ref. [46,52]. Concerning DM phenomenology, the model discussed here has strong similarities with the model F IA;0 illustrated in Ref. [68]; we will hence refer to the latter work for a detailed discussion. The addition of an SU (2) L doublet Ψ causes a notable difference in DM Direct Detection though. In this scenario, indeed, the DM can couple at the tree level with the SM Higgs and Z bosons. These couplings are responsible, respectively, for Spin Independent (SI) and Spin Dependent (SD) interactions between the DM and nucleons, with the former being the most constrained. For a review see e.g. Ref. [73]. Furthermore, the coupling of DM pairs with the Higgs and the Z bosons would lead to 'invisible' decay for the latter bosons, provided that the DM is light enough. Invisible branching fractions for the Z and Higgs bosons are, however strongly constrained.
The results of our analysis are shown in Figure 4, in the (M Ψ , M Ψ ) two-dimensional plane, for two sample assignations of the parameters of the model. In both panels the green bands represent the regions fitting the muon g − 2 anomaly at 1σ while the regions of the parameter space corresponding to a viable fit of the B-anomalies have been marked in orange. Throughout our analyses we set the value of the product of the quark couplings Γ Q * s Γ Q b = 0.15, in accordance with the constraints imposed by B s −B s oscillations as found in Ref. [68]. The correct DM relic density, if conventional freeze-out is assumed, is achieved only in the narrow red strips. The blue-hatched regions are excluded by constraints from XENON1T [74] on DM SI interactions, while the gray regions corresponds to an invisible branching fraction of the Higgs above 11 % [75], or an invisible width of the Z boson above 2.3 MeV [76]. No analogous constrains from LHC, as the ones considered in [68], have been shown in the plot since we have chosen benchmark assignations for m Φ , m Φq beyond current experimental sensitivity. Additional bounds from the production of the charged and neutral partners of the DM should be considered though, being responsible of 2-3 lepton + missing energy signatures (see e.g. [77]). Corresponding limits are not competitive as the ones from Higgs invisible decays and DM direct detection and, hence, have not been shown.
As illustrated by the figure, a combined fit of the g − 2 and of the B-anomalies can be easily achieved, together with the correct DM relic density and without conflicts with experimental exclusions, for M Ψ M Ψ . This corresponds to a mostly singlet-like DM achieving its relic density mostly through annihilations into muon pairs mediated by Φ . For this reason the isocontours of the correct relic density appear as vertical lines since the mass of Φ and the couplings Γ L,E µ have been kept fixed in the plots. 4 It is very promising that both anomalies can be accounted for with a standard thermal DM candidate.

F IIb : Real scalar DM
As a second example, we choose a very different DM candidate. Here DM is a real singlet (Φ ): what mix are (the charged components of) the fermion mediator fields Ψ − Ψ (SU (2) L doublet and singlet, respectively). Notice that both fields have non-zero hypercharge hence they are both Dirac fermions. The Lagrangian of the model reads where Φ q = (3, 1, 2/3), Φ = (1, 1, 0), Ψ = (1, 2, −1/2) = (Ψ 0 , Ψ − ) and Ψ = (1, 1, −1). Due to EWSB, Ψ mixes with the charged state in Ψ and the corresponding mass matrix reads We diagonalise this matrix by performing the field rotations The Wilson coefficients of the operators inducing b → s transitions read: Again, the details of the DM phenomenology have been fully illustrated in Ref. [68] and, hence, we will not discuss them in depth here. In Figure 5, we show the results of our combined analysis are shown in the (M Ψ , m Φ ) two-dimensional plane employing the same color coding as in Figure 4. Since, in the figure, the mass of F − 1 (through M Ψ ) is varied, the region of parameter space shown might be impacted by LHC constraints on the production of F − 1 F + 1 and subsequent decay into µ + µ − pair and DM, thus missing energy. The corresponding exclusion, obtained by recasting the search in Ref. [77], is shown as a hatched-purple region in the figure. Being the DM a SM singlet real scalar, not interacting with coloured new physics states, the only relevant DM constraint comes from the relic density. Again we found that it is possible to achieve the correct relic density compatibly with a fit of B− and g − 2 anomalies. Again we found that it is possible to achieve the correct relic density compatibly with a fit of B− and g − 2 anomalies. It is, in particular, worth remarking a special connection between g − 2 and the DM relic density. Indeed, a real scalar DM interaction, through a fermion mediator, only with left-handed or right-handed fermion would have a dwave suppressed annihilation cross-section, being the s-wave and p-wave contribution helicity suppressed. In our scenario, the mass mixing between the two fermionic mediators, interacting with left-handed and right-handed muons, removes, at least in part, such helicity suppression [78]. As a consequence, a combined explanation of the anomalies and a thermal DM candidate (not in conflict with any experimental constraint) can be achieved in wide regions of the parameter space.

Conclusions
The new results presented by the Muon g-2 collaboration could represent the first departure from the prediction of the Standard Model observed in a particle physics experiment. This nicely combines with the growing significance of R K announced by the LHCb collaboration and the highly-significant deviation from the SM prediction obtained by global fits of all b → s observables. It is very suggestive that both anomalies requires non-standard contributions to operators involving muons. This makes it crucial to find compelling theoretical frameworks where both experimental results can be naturally explained. In this paper, we have presented a particularly simple setup where this is possible within the context of models that also provide a viable thermal Dark Matter candidate. We have discussed what are the minimal ingredients and properties that allows to explain the anomalies through loop effects due to the DM particles and few other BSM fields, being four the minimum number of fields that need to be added to the SM. The general characteristic of this class of models is that the DM phenomenology is controlled by the same parameters that enter the flavour observables. As a consequence, they feature a high degree of correlation among DM, flavour and collider searches and thus an enhanced testability. After a systematic classification of these minimal models according to the quantum numbers of their field content, we have presented two examples with very different DM candidates (a mixed SU (2) L singlet-doublet fermion in one case, a real scalar in the second case). In both examples, a thermal DM candidate can naturally provide a simultaneous explanation of the muon g − 2 (through chirally-enhanced contributions controlled by EWSB effects) and the B-physics anomalies, while evading present bounds from collider and DM searches. Despite the simplicity, their rich phenomenology makes it possible to test them, at least in part, at future runs of the LHC and/or DM direct detection experiments. In the long term, as large interactions involving muons are necessary, these minimal DM models would be an ideal target for a multi-TeV muon collider [9][10][11][79][80][81].