Simple model for large CP violation in charm decays, B-physics anomalies, muon g-2, and Dark Matter

We present a minimal extension of the Standard Model that can simultaneously account for the anomalies in semi-leptonic B meson decays and the muon g-2, give large CP violation in charm decays (up to the value recently measured by LHCb), and provide thermal-relic dark matter, while evading all constraints set by other flavour observables, LHC searches, and dark matter esperiments. This is achieved by introducing only four new fields: a vectorlike quark, a vectorlike lepton, and two scalar fields (a singlet and a doublet) that mix due to the electroweak symmetry breaking and provide the dark matter candidate. The singlet-doublet mixing induces chirally-enhanced dipole transitions, which are crucial for the explanation of the muon g-2 discrepancy and the large charm CP violation, and allows to achieve the observed dark matter density in wide regions of the parameter space.


Introduction
Instead of focusing on UV-complete, theoretically-motivated, new physics (NP) scenarios (e.g. addressing the hierarchy problem, grand unification, etc.), we adopt here a bottomup approach to NP beyond the Standard Model (SM) of particle physics, and just concern ourselves with a simplified model that can accommodate a number of observational hints for NP at (or not far above) the TeV scale. In fact, although the LHC experiments could not establish the existence of new particles beyond the SM, we have been witnessing in recent years to several persisting discrepancies between observations and SM predictions, especially in the flavour sector. One is the muon anomalous magnetic moment, muon g − 2, which features a long-standing disagreement between theoretical predictions and experiments at the level of more than 3σ. If confirmed, possibly by the results of the new Muon g-2 experiment at Fermilab [1], this discrepancy would unambiguously require new particles interacting with muons at the TeV scale or below: cf. [2] for a review. The physics of the B mesons provides other examples. The LHCb and the B-factory experiments have observed hints of Lepton Flavour Universality (LFU) violation in semi-leptonic B decays, especially in the observables R K ( * ) ≡ BR(B → K ( * ) µµ)/BR(B → K ( * ) ee) that are theoretically very clean in the SM (and whose values are predicted to be practically one). In addition to this, semi-leptonic B decay data (again from b → sµµ processes) exhibit a coherent pattern of observables in tension with the SM, namely a general deficit in the differential branching fractions as well as discrepancies in angular observables. Reviews of such 'B-physics anomalies' can be found in [3][4][5]. Besides flavour observables, the evidence of cold dark matter (DM) in the universe could be a further hint for a low-energy NP sector. This follows from the 'WIMP miracle', the remarkable observation that, assuming a standard thermal history of the universe, the DM relic density measured from observations of the Cosmic Microwave Background (CMB) can be accounted for by particles in the mass range of the electroweak-breaking scale, annihilating with a cross section of the typical electroweak size, cf. the reviews [6,7]. This motivates the possibility that DM is a so-called Weakly Interacting Massive Particle (WIMP) and can thus be produced and observed (possibly in association with other new particles) at colliders.
As mentioned above, following a bottom-up approach we want to build and study a minimal model that can simultaneously account for the above hints of new physics. We regard this as a useful exercise to highlight the building blocks that a fully-fledged theory (possibly addressing other major shortcomings of the SM, such as the generation of neutrino masses, the origin of the fermion mass hierarchies, baryogenesis etc.) may incorporate if the above observations will be proven to be indeed due to beyond the SM dynamics. We build on previous attempts [8][9][10][11][12][13][14][15][16] to address (some of) the above experimental results by adding to the SM a limited number of new fields, focusing on heavy scalars and heavy quarks and leptons in vectorlike representations of the SM gauge group (for general discussions of this kind of 1-loop solutions of the B-physics anomalies anomalies see [17][18][19]). In particular, we extend the model discussed in [10,15] by adding a scalar SU (2) L doublet. The mixing of such field with a scalar singlet (via a coupling with the Higgs) introduces chirally-enhanced dipole transitions that allow to account for the muon g − 2 with a heavy enough NP spectrum that can be compatible with LHC constraints and the observed DM abundance without the need of tuning the model's parameters, as extensively discussed in [12]. This crucial novel ingredient also generates enhanced dipole operators in the quark sector, which can lead to other desirable effects. In particular, we contemplate here the possibility that CP violation in charm decays, which has been recently established by LHCb [20], is also a NP effect and is accounted for by our simple model.
In the rest of the paper, after presenting the model in Section 2, we thoroughly study its phenomenological implications. In Section 3, we discuss the flavour effects we are interested in and the relevant constraints set by other flavour observables. In Section 4, we discuss in detail LHC and DM phenomenology of our model and we combine it with the flavour constraints. We conclude in Section 5, while we present some useful formulae in the Appendices.

Field content and interactions
We introduce the following set of new fields that are all odd under an unbroken Z 2 symmetry under which the SM fields are even: a singlet complex scalar, a complex scalar doublet, and two vectorlike pairs of Weyl fermions (that combine into two Dirac fermions) with the quantum numbers of the SM quark and lepton doublets. To summarise, the gauge quantum numbers of the extra fields are as follows: In terms of SU (2) L components the Dirac fermions can be written as: Given the unbroken Z 2 that we assumed, these fields do not mix with the SM fermions. For the same reason, the scalars do not mix with the SM Higgs, although they interact with it via trilinear and quartic 'Higgs portal' couplings. The scalar sector can be decomposed as follows: The physical states are thus two neutral and one charged complex scalar. The part of the Lagrangian involving the new fields is given by the following expression: where we omitted the quartic couplings of the scalar potential, we defined Φ D ≡ iσ 2 Φ * D , and we denoted the left-handed (LH) and right-handed (RH) SM fermions respectively as Q i , L i , and U i , D i , E i , with i = 1, 3 being a flavour index.
Upon electroweak-symmetry breaking, the scalar coupling a H (that has the dimension of a mass) induces mixing between the neutral components of Φ S and Φ D . The mass matrix and our definition of the mixing matrix U are the following: where v is the Higgs field vev 246 GeV. We denote the mass eigenstates as S 1 and S 2 and by convention we take M 2 S 1 ≤ M 2 S 2 . Physical masses and mixing are then given by Notice in particular that the entry U 1α (U 2α ) represents the singlet (doublet) component in the mass eigenstate S α , namely: If lighter than the vectorlike fermions S 1 is a good candidate for cold dark matter, as we will discuss in the Section 4. 1 Finally, the charged scalar mass is at the tree level simply given by the mass parameter of the scalar doublet: The Lagrangian written in terms of the mass eigenstates can be found in the Appendix A. In Figure 1, we sketch the spectrum of the new particles and their interactions, assuming for illustration purposes the hierarchy M Q > M L > M D > M S , and a moderate scalar mixing, so that M S 2 ≈ M S ± = M D and M S 1 ≈ M S .

Flavour observables and phenomenology
The purpose of this section is to illustrate how the new fields of the model contribute to the flavour observables we are interested in, and to discuss the relevant constraints. This discussion also allows us to identify the interactions (and quantify their strength) that lead to the desired effects. The resulting constraints and benchmark values of the couplings will be employed in the following section in order to study parameter space and spectrum compatible with the B-physics anomalies, CPV in charm decays, muon g − 2, and dark matter. 1 Quartic interactions in the scalar potential can introduce a mass splitting between the CP-odd and CPeven components of Sα, see e.g. [21]. We are going to assume that this is a small effect and ignore it in the discussion of the flavour phenomenology. Such a mass splitting does however play an important role for DM direct detection, cf. Section 4.2.

LFU violation in b → s transitions and B-physics constraints
The simplest way to address the anomalies observed in the semi-leptonic B decays LFU observables R K ( * ) and in branching ratios and angular distributions of several b → sµµ modes is adding non-standard contributions to the following operators (for the latest fits see [22][23][24][25][26][27]): where the Wilson coefficient are normalised by the SM contribution Adapting to our specific model the formulae of [19] (see also [10]), we get for the contribution to C bsµµ 9,10 from diagrams involving Q , L , and the scalars Φ S and Φ D (shown in Figure 2): where the loop function is Notice that the second term of both expressions come from the second diagram in Figure 2 (involving RH muons) and vanish in absence of scalar singlet-doublet mixing. In such a case the contribution of our model takes the form ∆C bsµµ 9 = −∆C bsµµ 10 , typical of new physics coupled to LH leptons only. 2 As it is apparent from the expressions given in the Appendix B.1, contributions to additional b → s operators involving RH quarks (and to dipole operators as those in Appendix B.2) depend on the couplings to RH down quarks, λ D i , and are thus suppressed if such couplings are small. In the following we are going to assume that this is the case (i.e. λ D i λ Q i ), although some degree of RH currents may help fitting the b → s data (see e.g. [22]). This choice is also motivated by the constraint from b → sγ transitions, B s → µµ, and B s -B s mixing in presence of RH currents (cf. the discussion below).
According to the latest fits to the data [25], a non-standard contribution in the following 2σ range is preferred with the best-fit value ∆C bsµµ 9 = −∆C bsµµ 10 = −0.52 improving the fit at the level of 6.5σ with respect to the SM. For similar global analyses see [22-24, 26, 27]. As discussed above, in presence of scalar mixing, our NP contribution is not exactly of the type ∆C bsµµ 9 = −∆C bsµµ 10 . Therefore, in the following we employ a parameterisation of the two-dimensional (∆C bsµµ 9 , ∆C bsµµ 10 ) fit result presented in [25]. The SU (2) L counterpart of the left diagram in Figure 2 contributes to processes such as B → K ( * ) νν, which can pose a substantial constraint to theories addressing the b → sµµ anomalies, as pointed out in [28,29]. However, as we will see below, these bounds are subdominant within our model. The relevant expressions can be found in the Appendix B.3.
The most relevant constraint on the product λ Q 3 λ Q * 2 , which enters the NP contribution to ∆C bsµµ 9 = −∆C bsµµ 10 , is given by B s -B s oscillations. Similarly, in presence of a sizeable λ Q 1 coupling, we will have a contribution to B -B mixing. Assuming as above small λ D i couplings to RH down quarks, our NP will contribute to the following ∆B = 2 operators: Using the results of [10,19], we find for the contribution of a Q − Φ S box diagram: where Real and imaginary parts of these operators are constrained by, respectively, B s -B s and B -B mass differences and CP violation observables. Given that a sizeable value of λ Q 1 would be subject to analogous (but more stringent) constraints from K -K mixing, 3 and more importantly from the neutron EDM (as we will discuss in the following subsection), here we 3 We checked that for the benchmark point of Figure 3 and a real λ Q 2 the bounds from ∆mK and K translate to, respectively, |λ Q All the other couplings were set to 0.
and focus on B s -B s mixing only. For simplicity, we also assume that λ Q 3 λ Q * 2 (hence our contribution to ∆C bs 1 ) is real and consider the bound from the B s -B s mass difference ∆m s . Using the formalism in [30][31][32], and taking the recent sum-rule and lattice based calculation giving ∆m SM s = (18.4 +0.7 −1.2 ) ps −1 [33], we obtain the following bound: where we take the matching scale of the Wilson coefficient at 1 TeV scale for definiteness. The above bound is consistent with that calculated in [34]. The B s -B s mixing constraint and the 2σ-favoured region for b → s are shown in Figure 3 for an illustrative choice of the parameters of the model. In this example, the vectorlike quark only couples to 2nd and 3rd generation LH quarks, and the vectorlike lepton only couples to muons. The values of the parameters adopted in the above example (especially the large coupling to LH and RH muons) will be better justified in the following subsections. As we can see, it is possible to find a setup of the parameters for which a good fit of b → s data is compatible with the B s -B s mixing bound, although such a constraint is particularly severe. An improvement of the theoretical determination of ∆m SM s would allow therefore to test an explanation of this kind of the B-physics anomalies. This is a common feature of models addressing b → s at one loop (cf. for instance the discussion in [18]). The figure also shows that in the (orange) region where b → s and B s -B s are compatible the rate of B → K ( * ) νν (cf. Appendix B.3 for the relevant expressions) deviates from the SM prediction by at most 5% and thus does not further constrain the model at present [35].
Finally, let us notice that non-vanishing λ D 2,3 would generate other operators contributing to B s -B s , including the LR and RR currents listed in the Appendix B.4. The coefficient LR operators are subject to a bound that is about a factor 3 stronger than the one given above for ∆C bs 1 [34]. Moreover, dipole operators and scalar operators would arise giving substantial contributions to respectively b → sγ and B s → µµ, cf. Appendix B.1 and B.2. 4 Considering that the improvement to the b → s fit in presence of RH currents is not dramatic [22][23][24][25][26][27], the compatibility between b → s and other b − s transitions prefers that the couplings λ D 2,3 are suitably suppressed. Here and in the following, we just set them to zero for simplicity.

CP violation in charm decays
The LHCb experiment has been recently established CP violation in the charm sector, by measuring the difference of the time-integrated CP asymmetries in the |∆C| = 1 decays where This observable is mostly sensitive to direct CP violation [36].
Interpreting the LHCb result is not straightforward, given the notorious difficulty of performing calculations at the charm mass scale. In the SM one gets ∆A SM CP ≈ −0.0013 × Im(∆R SM ) (see e.g. [37,38]) where ∆R SM encodes ratios of hadronic amplitudes naively expected to be of the order ∆R SM ≈ α s (m c )/π ≈ 0.1. This estimate is supported by the recent calculation in [39] (giving |∆A SM CP | ≤ 3 × 10 −4 ) and would imply a large discrepancy with the measured value. However, it is not possible to exclude that large non-perturbative effects in ∆R SM enhance the SM prediction up to the value observed by LHCb [40][41][42][43][44][45]. Here, we are going to speculate about the possible NP origin of ∆A CP . For the implications on other NP models of large CP violation in charm decays see also [38,39,46,47].
Possible NP effects in ∆A CP are encoded in the |∆C| = 1 effective Hamiltonian: The full list of operators can be found in [37]. Following [37,38], here we are interested in the NP contribution to the chromo-magnetic dipole operators that can give rise to sizeable 4 Using expressions and bounds reported in the Appendix (cf. Eqs. (B.14, B.21)) we checked that, despite the helicity-enahanced scalar contributions, λ D 2,3 ∼ O(1) are still compatible with the measured rate of Bs → µµ, whereas b → sγ sets a substantial constraint (|λ D 2 | 0.01 for our benchmark spectrum).
The resulting ∆A CP is [37,38] where ∆R SM and ∆R NP i are combinations of hadronic amplitudes, and ∆C cu i (m c ) are the NP contributions to the coefficients of Q cu 8 and Q cu 8 at the m c scale. In our model the chromo-magnetic operators are generated by the two diagrams shown in Figure 4. Adapting again the general formulae derived in [19], we obtain: where we only show the dominant chirally-enhanced LR contributions (the subdominant LL and RR terms can be found in [19]), Λ is the matching scale (≈ 1 TeV) and the loop function reads: The evolution of the coefficients down to m c can be computed with the standard formulae for the QCD running (see [48]) and numerically gives ∆C cu 8 (m c ) 0.41 × ∆C cu 8 (1 TeV). In the following we are assuming that the dominant contribution to ∆A CP is due to new physics, and we are employing the estimate for ∆R NP 8,8 given in [37,38]: The other parameters were set as in Figure 3.
which was obtained using naive factorisation and assuming maximal strong phases. Under the above assumptions (which are subject to O (1) uncertainties), the value of ∆A CP measured by LHCb is saturated for where we considered the 2σ range of Eq. (17). By inspecting Eqs. (22,23), we can see that a large effect in ∆A CP is more easily induced by ∆ C cu 8 . The reason is that a sizeable value of λ Q 2 is required by fitting the b → s data, so that λ Q 1 is tightly constrained by K -K mixing, as discussed in the previous subsection. On the other hand, a complex λ U 1 can easily account for the observed value of ∆A CP . This is illustrated in Figure 5, where we show that, for our benchmark point, an overlap between the regions favoured by ∆A CP (according to Eq. (26)) and b → s is obtained for |Im(λ U 1 )| 0.2. Notice that such large contribution to ∆A CP is achieved thanks to the chiral enhancement that follows from the singlet-doublet mixing (which is a peculiarity of this model), as shown in Figure 4. Let us finally discuss other possible constraints from the up-type quark sector. As pointed out in [37], the D -D mixing constraint is irrelevant when the above dipole contributions saturate the observed ∆A CP value. By employing the expressions reported in the Appendix B.4, we have explicitly checked that this is indeed the case: the limit on ∆C cu 2 reported in [37] translates into the mild bound |λ U 1 | 1/|λ Q 2 | for the benchmark point of Figure 5. More importantly, a sizebale (complex) value of λ U 1 can contribute to the up quark (and thus to the neutron) EDM, via the flavour-conserving counterpart of the diagrams in Figure 4. This sets a further constraint on λ Q 1 . Employing the formalism presented in the Appendix B.5, we find

Muon g − 2
Loop diagrams involving the extra scalars and the vectorlike lepton L contribute to the anomalous magnetic moment of the muon, a µ ≡ (g−2) µ . For a recent review see [2]. According to the classification of [12], this is a 'scalar LR' (SLR) model, namely one that yields a chirallyenhanced contribution 5 to a µ , as a consequence of the scalar singlet-doublet mixing. The relevant diagram is depicted in Figure 6. The leading chirally-enhanced contribution to a µ reads: where The expression of the subdominant terms, which are suppressed by a factor ∼ y µ relative to the above contribution, can be found in [12]. If confirmed, the present discrepancy between the measurement and the SM prediction of a µ would require at 1σ [49][50][51]: In Figure 7, we show the value of λ L 2 = −λ E 2 for which Eq. (27) provides a (positive) contribution ∆a µ of the size required by the above discrepancy (within 2σ). As we can see, for large enough couplings to the LH and RH muons, such region overlaps to that favoured by the b → sµµ anomalies (and still allowed by the B s -B s mixing constraint). The figure shows that a large value of the coupling λ L 2 (≈ 1.5 for this numerical example) is needed such that plane. The other parameters were set as in Figure 3.
the contribution ∆C bsµµ 9 = −∆C bsµµ 10 (that depends on λ E 2 only via the singlet-doublet mixing, hence mildly) accounts for the b → sµµ anomalies and the B s -B s bound is simultaneously evaded. Given the chirally-enhanced contribution of Eq. (27), accommodating the observed value of a µ and b → sµµ simultaneously thus requires either sizeable singlet-doublet mixing and small λ E 2 , or small mixing and large λ E 2 (as in the example shown in Figure 7). Sizeable couplings to electron or tau (λ L,E 1 , λ L,E 3 ) would induce lepton-flavour-violating (LFV) dipole operators through diagrams similar to Figure 6, thus being subject to the tight constraints from searches of LFV decays such as µ → eγ and τ → µγ (for a recent overview, cf. [52]). As discussed in [12], one indeed finds that the current limits on LFV processes [53,54] and the central value in Eq. (29) imply: The stringent limit on the electron couplings, in particular, does prevent any sizeable contribution to the g − 2 of the electron. This observable also exhibits a mild tension with the SM prediction: ∆a e = a EXP e − a SM e = −(0.88 ± 0.36) × 10 −12 [55]. In order to account for that, one would then need to extend our model, e.g. introducing multiple generations of vectorlike leptons (coupling either to electrons or to muons, not to both) along the lines of the models discussed in [56].

Summary: flavour structure of the couplings
We conclude this section by summarising the structure of the couplings of our new particles that we can infer from the above discussion. For a TeV-scale spectrum of the new fields Q , L , Φ S , Φ D and a moderate singlet-doublet mixing (as in the illustrative example adopted in Figs. 3, 5, 7), the model can successfully account for b → sµµ, ∆A CP , muon g − 2 if the following minimal set of ingredients is present: • Sizeable couplings of the vectorlike quark Q to LH bottom and strange quarks (with opposite signs): λ Q 2 λ Q 3 ≈ −(0.5) 2 , cf. Figs. 3, 5; • O(1) couplings of L to LH and RH muons, for the sake of the combined explanation of b → sµµ and ∆a µ : |λ L 2 | 1.5, |λ E 2 | 1, cf. Fig. 7; • A substantial (complex) coupling of Q to the RH up quark, in order to induce large CP violation in charm decays: |Im(λ U 1 )| 0.2, cf. Fig 5; • Suppressed coupling to LH up and down quarks (due to bound from the neutron EDM and K -K mixing): |λ Q 1 | 10 −3 ; • Small to mildly-suppressed couplings of Q to RH down-type quarks: Although the above pattern is not generic, it is certainly conceivable, especially if enforced by a flavour symmetry. In particular, notice that the couplings to quarks are in principle compatible with a SM-like hierarchical structure: (the couplings to RH charm and top being virtually unconstrained).
In the following section, we are going to assume the pattern summarised above, and discuss in better detail the new particle spectrum selected by the flavour anomalies and its consequences for LHC and dark matter.

LHC phenomenology
The new states of our model can only be produced in pairs at colliders, as a consequence of the Z 2 symmetry. For the same reason, they undergo decays ending in a SM particle plus the lightest Z 2 -odd particle, which we assume to be the lightest neutral scalar S 1 , in order to address the DM problem (see 4.2 for further details). All these features remind of supersymmetric models and, likewise, collider signatures will include energetic jets or leptons plus missing transverse momentum / E T . Searches for supersymmetry at the LHC can be thus used to set limits on the masses of our new particles too. A detailed study of the bounds on the different production modes and decay chains would be beyond the scope of this work.
Here we focus on a number of simplified topologies, in order to demonstrate that large regions of the parameter space that are relevant for the flavour processes discussed in the previous section are not excluded by current LHC searches (but are possibly in the reach of future LHC runs). In particular, we consider: pp → S + S − , pp → S ± S 2 , etc.
The decays of these particles can be visualised in the sketch of Figure 1. In the following, we are going to discuss them in turn, together with the resulting LHC signatures and searches.
1. Q production. Given the pattern of the couplings discussed in Section 3.4, the Q states will mostly decay through λ Q 3 to top and bottom (leading to U → t S 1 and D → b S 1 ), and through λ Q 2 to strange and charm (thus giving U → j S 1 and D → j S 1 ). Rates of decays into mostly doublet scalar states such as S 2 are suppressed as they require singlet-doublet mixing. Furthermore, the decays controlled by λ U 1 are typically subdominant since ∆A CP prefers a moderate value of this coupling. They would anyway lead to more complicated (and possibly phase-space suppressed) decay chains, such as U → j S 2 → j h S 1 , which are arguably less clean than the above signatures. A recent analysis performed by the CMS collaboration [57], which employs the full data set of the 13 TeV run, addresses the signatures relevant for this production mode and the direct decays into S 1 discussed above: tt + / E T , 2b-jets + / E T , and 2j + / E T . This search sets a limit on the production cross section of stops, sbottoms, and (a single generation of) squarks that is approximately σ 1.7 fb. Given that for states above ≈ 1 TeV decaying to much lighter particles the efficiency times acceptance of the search is virtually constant, we can directly translate this limit into a bound on the mass of the Q fermions (valid if M Q M S 1 ): M Q 1.5 TeV. 6 For simplicity, in the previous section, as well as in the following discussion, we set M Q = 1.5 TeV, a value that should be still borderline viable according to the above estimate. We have to keep in mind though that strong production of Q could be a way to test our scenario at future LHC runs. 6 In order to get this, we employed the production cross section as calculated at LO by means of MadGraph5 [58] and we rescaled it by a k-factor of 1.44 obtained by comparing LO and NLO-NLL squark production cross sections [59,60]. Furthermore, notice that, being the limit reported in [57] basically the same for stops, sbottom, and squarks, our estimate does not strongly depend on the branching fractions of U (D ) → t(b) S1 and U (D ) → j S1 (controlled by λ Q 3 /λ Q 2 ) .
2. L production. The charged states can decay directly into muons and S 1 through the coupling λ L 2 , L ± → µ ± S 1 , while the coupling to the scalar doublet λ E 2 would induce longer decay chains, e.g. L ± → µ ± S 2 → µ ± S 1 h/Z. Similarly, L 0 decays as L 0 → µ ± S ∓ → µ ± S 1 W ∓ due to λ E 2 , while the decay induced by λ L 2 is completely invisible: L 0 → ν S 1 . In the following we are focusing on the simplest topology pp → L + L − → µ + µ − + / E T . The latest available analysis of this signal has been presented by ATLAS in [61] (see also the results with a smaller data set in [62]). The resulting bound on the production cross section can be as strong as σ 0.2 fb (for M L M S 1 ), which corresponds to M L 900 GeV (according to the LO production cross section as calculated by MadGraph5 [58]). This limit is slightly above the benchmark value of M L employed in the last section, but it is also likely too tight, as the longer decay chain induced by λ E 2 would partially dilute the signal and lead to other signatures, which are possibly more challenging to constrain at the LHC (at least if the mass difference between the vectorlike lepton and the scalar doublet is not very large). A more quantitative discussion of the bound on L ± will be presented in Section 4.3.
3. Scalar doublet production. The production of the states of the scalar doublet, decaying to SM bosons and DM, leads to topologies similar to those sought for in the case of electroweak production of supersymmetric charginos and neutralinos: The most sensitive signature is thus again 2 [61] or 3 [63] leptons (from the leptonic decays of the gauge bosons) and / E T . This searches can constrain Higgsino-like charginos and neutralinos with masses up to about 600 GeV. However, the production cross section for our scalars is much smaller than for a fermion doublet of the same mass. As a consequence, we can estimate that searches as in [63] are at most sensitive to doublet masses up to 200-300 GeV for a very light singlet-like S 1 , M S 1 < 100 GeV. Therefore, as we will be clear from the plots presented in Section 4.3, these modes do not represent yet a relevant constraint of the region of the parameters space selected by the flavour observables.

Dark matter phenomenology
As discussed above, the extra fields we introduce are assumed to be odd under an unbroken Z 2 parity, which ensures that the lightest new state is stable. In the following, we are considering the case that such state is neutral so that it can provide a candidate of dark matter. In particular, we focus on the lightest scalar S 1 . Furthermore, we assume thermal dark matter production, i.e. that the standard frieze-out mechanism is at work. Despite the reduced field content of the model, a substantial number of annihilation and co-annihilation processes can control the DM relic density. Some of the most relevant modes are depicted in Figure 8. The relative importance of a single process depends on the size of the new couplings, as well as on the nature of the DM candidate S 1 that, we remind, is a mixture of a SM-singlet scalar and the neutral component of a scalar SU (2) L doublet: S 1 = U 11 S 0 s + U 21 S 0 d (cf. Section 2). In particular, we can identify the three following regimes with distinctive features. (ii) S 1 is mainly doublet. In this case gauge processes as those of the second column of Figure 8 are very efficient in depleting the DM density in the early universe. If S 1 is a pure doublet, the relic density matches the value observed today, Ω DM h 2 0.12 [64], if m S 1 ≈ 540 GeV (see e.g. [65]), while a lighter S 1 would be a subdominant DM component. DM direct detection experiments are sensitive to our scenario. Indeed, a Higgs-mediated DM-nucleon interaction (arising from the S 1 S 1 h coupling on one side, and the Higgs coupling to gluons through a top loop, on the other side) can induce a sizeable spin-independent (SI) cross section. The S 1 S 1 h interaction arises from the mixing of the singlet and the doublet and requires substantial components of both in S 1 , in order to be effective: as we can see from Eq. (A.2), the coupling is proportional to a H U 21 U 11 . Thus we expect direct detection experiments to best constrain the large mixing case (iii). Moreover, if S 1 is mainly doublet, as in case (ii), or through singlet-doublet mixing in the other cases, it can interact with the Z boson. Thus a tree-level Z exchange may induce a scattering cross section with nuclei several orders of magnitude larger than the present limits. However, notice that the term Z µ S 1 3) only couples the CP-even to the CP-odd component of S 1 , thus leading to an inelastic DM-nucleus scattering. A mass splitting of just O(100) keV between real and imaginary part of S 1 (naturally achieved via the quartic couplings in the scalar potential) then is sufficient to kinematically forbid Z-mediated scattering with nuclei [66]. In the following, we are assuming that this is the case and only focus on Higgs-mediated elastic DM-nucleon interactions.
Finally, we comment about another possible DM candidate in our model: the neutral component of L . This would constitute a pure fermion doublet DM candidate, akin to a supersymmetric Higgsino. There are two difficulties related to this possibility. First of all, as in the case of Higgsino DM, the observed relic abundance would require M L ≈ 1.1 TeV and all the other particles of course heavier than this. The spectrum would be thus too heavy to account for all the flavour effects we are interested in (in particular b → sµµ), as it will appear clear from the quantitative discussion in the rest of this section. The second problem is that L 0 interacts with the Z boson, cf. the second line of Eq. (A.3). As discussed above, an unacceptably large scattering cross section with nuclei can be avoided if a small Majorana mass term splits the Dirac fermion into two Majorana states, e.g. through mixing with another Majorana fermion (like in the Higgsino-Bino system) but an extension of the model would be required. For these reasons we are not going to consider this possibility further.
In the following, we will numerically calculate the S 1 relic density and SI cross section with nuclei by means of the routine micrOMEGAs [67,68] and show on our parameter space where Ω DM h 2 0.12 [64] is fulfilled and what are the regions excluded by the latest limit of the XENON1T experiment [69].

Combined results
We end this section discussing the combined impact of the flavour observables presented in Section 3 and the DM/LHC constraints on the parameters of our model. The outcome is summarised in Figures 9 and 10 for several representative slices of the parameter space.
In Figure 9 we show the singlet-doublet mass plane (M S , M D ) while setting the mass of the vectorlike quark to a value close to the LHC bound discussed above, M Q = 1.5 TeV, and the vectorlike lepton to M L = 800 GeV (left panel) and M L = 1 TeV (right panel), cf. the discussion below on the implication of these choices for µ + µ − + / E T searches at the LHC.
. All other couplings are set to zero. Cf. the main text for further details.
The couplings are set to values consistent with the findings of Section 3, as indicated in the caption of Figure 9. The coloured areas highlight the portions of the parameter space that are preferred by our flavour observables: in the orange region b → sµµ data can be fitted within 2σ simultaneously evading the B s -B s mixing bound, the green region shows where the observed ∆A CP is completely accounted for by our NP contribution (at 2σ), while in the blue area the muon g − 2 discrepancy is solved at the 1σ level. The hatched areas are excluded by LEP searches for new charged states with M S ± 100 GeV [70,71] (yellow) and the DM direct detection experiment XENON1T (purple). Besides the value of M L , the main difference between the two panels is the singlet-doublet mixing parameter, set to a H = 20 GeV (left) and 40 GeV (right). As we can see, by comparing the two plots, a larger value of a H implies a boost to the effects that depend on the singlet-doublet mixing, such as the chirality-enhanced contributions to the muon g − 2 and the ∆C = 1 chromomagnetic operator 7 , as well as the nucleon-DM interaction.
The line where the S 1 relic abundance approximately saturates the observed DM relic density Ω DM h 2 = 0.12 is indicated in red. Given the fact that in both examples the chosen values of the mixing parameter a H are quite moderate, in the M S < M D region of Figure 9, S 1 is typically singlet-dominated and thus in general overabundant: we are in the regime (i) discussed in Section 4.2. The correct relic density is obtained either due to the Higgs resonance, for M S ≈ m h /2, or when the DM mass approaches the vectorlike lepton mass, in which case the t-channel annihilation to muons and the co-annihilations modes become effective (this In the left plot, not only the DM annihilation to W + W − is very efficient but also, given the moderate value of M L and the large coupling λ E 2 , DM annihilation and co-annihilation rates mediated by L are very large. As a result S 1 is everywhere underabundant. In the right panel instead, due to the choice of a heavier L , the observed relic density Ω DM h 2 = 0.12 can be saturated when the rates of either the gauge or the Yukawa modes decrease to a sufficient extent.
As we can see, in both examples of Figure 9, the red line does overlap with the coloured regions, hence one can find suitable spots where the correct relic density is obtained and all our flavour observables are accounted for.
In Figure 10, we show the effect of varying the vectorlike lepton mass, M L : we plot the same observables as above on the (M L , M S ) plane, while keeping a constant ratio between singlet and doublet masses, M D = 1.5 × M S . All other parameters are as in Figure 9. In addition, we show the constraint from µ + µ − + / E T searches at the LHC under the simplifying assumption BR(L ± → µ ± S 1 ) = 1. The hatched cyan area corresponds to the region excluded by the ATLAS search [62] as recast in [12]. The dashed cyan line shows how the bound increases due to the updated analysis in [61]: this limit is an estimate based on the excluded production cross section as reported in the auxiliary material of the ATLAS article. As we mentioned in Section 4.1, we expect that this search can exclude our vectorlike lepton up to This is a stringent constraint, but the plots of Figure 10 shows that it does not prevent a simultaneous explanation of DM and our flavour observables. Indeed, this seems to be possible either for a rather heavy L (up to M L ≈ 1.2 − 1.3 TeV) or for the 'compressed spectrum' region, where the L − S 1 mass difference is reduced and the LHC searches quickly lose efficiency (because the muons are less energetic) while the correct relic density can be achieved through L − S 1 co-annihilations. If the latter option may be challenging to test at the LHC (barring perhaps searches for soft leptons as in [72]), the former one will be surely within the sensitivity of future LHC runs.

Conclusions
We have presented and thoroughly discussed the phenomenological consequences of an extension of the SM, featuring a heavy vectorlike quark, a heavy vectorlike lepton, and two scalar fields (a singlet and a doublet) that couple to the Higgs field and hence mix through EW symmetry breaking. We have shown that this rather simple setup can provide a simultaneous explanation of the B-physics anomalies and the muon g−2, give a large contribution to the CP violation in charm decays ∆A CP (to the extent of easily saturating the value recently measured by LHCb), and account for the observed DM abundance, while evading all constraints set by other flavour observables, LHC searches, DM searches. We found that the novel ingredient of our model (compared to e.g. [10,15]), namely the singlet-doublet mixing, is crucial in order to achieve that. This is because the mixing can give rise to chirally-enhanced dipole transitions that allow to account for the muon g − 2 and ∆A CP for TeV-scale masses of the vectorlike quark and lepton. In Section 3.4, we have shown the pattern of the new fields' couplings that can address our flavour observables and be compatible with the bounds from other flavour processes. In the spirit of simplified models, we have not discussed how plausible such flavour structure is. It is however encouraging that, at least in the quark sector, the couplings are compatible with a SM-like hierarchical pattern. Our model could be regarded as a building block of a more complete theory addressing other shortcomings of the Standard Model. Still, it is remarkable that, despite its simplicity, it can consistently account for so many phenomena. The model can be tested (at least in part) via flavour observables, in particular by the upcoming results of the Fermilab Muon g-2 experiment, and future determinations of the SM prediction of the B s -B s mass splitting with increased accuracy, and also at the future runs of the LHC, given the necessary presence of charged scalars below 1 TeV and vectorlike fermions in the 1-2 TeV range.

A Lagrangian
After electro-weak symmetry breaking, the Lagrangian in terms of the mass eigenstates can be written as L ⊃ L mass + L mix + L gauge + L yuk , where , τ ), and for the LH quarks we chose the basis Q T i = (V * ji u L j , d L i ) with V ij being elements of the CKM matrix. As usual, we The expressions for the masses of the neutral scalar eigenstates S α (α = 1, 2) and the mixing matrix U are given in Section 2.
For simplicity, we did not display the terms in the scalar potential (whose coefficients can be assumed to be small enough to have vanishing phenomenological impact apart from providing a mass splitting between CP-even and CP-odd components of S α ), but the Higgsscalar coupling in L mix arising from the scalar singlet-doublet mixing term.

B Wilson coefficients and further observables
We use the following definition of the effective dimension-6 Hamiltonian controlling b → s transitions (cf. [25] and references therein): where the normalisation is and x run over the semi-leptonic operators defined as and over the O bs x operators obtained by exchanging P L ↔ P R in the above expressions. Within our model, the Wilson coefficients, as obtained from the general results presented in [19], read: ∆C bsµµ Sα , (B.10) where the loop functions are defined as , (B.12) . (B.13) Besides b → s transitions, the above operators also contribute to B s → decays, such as B s → µµ. In particular, (pseudo) scalar operators provide an helicity-enhanced contribution compared to the SM one, controlled by O bs 10 . From the measured value of B s → µµ, that agrees with the SM prediction within 1σ [75], one thus obtains the following bound on the scalar coefficients (calculated at the matching scale of 1 TeV) [19]: |∆C bsµµ S,P |, |∆ C bsµµ S,P | 0.03 (2σ). (B.14)

B.2 b → sγ
This kind of transitions can be accounted for by adding the following electro-and chromomagnetic dipole operators to the above Lagrangian: plus the corresponding O bs x operators obtained by exchanging P L ↔ P R . The leading (chirally-enhanced) contributions of our new fields to the coefficients of the dipole operators are [19]: Following [35], we employ the effective Hamiltonian where N is as in Eq. (B.2). Our model's fields contribute to the above operators as follows: Given the constraints on RH currents form B s -B s mixing and the fit to b → s data, we work in the limit of vanishing λ D i couplings, resulting in ∆C bsνν R ≈ 0. In this limit, one simply finds that [19] BR

B.4 Meson mixing
We work with the following ∆B = 2 effective Hamiltonian: The operators are defined as where a, b are (summed-over) colour indices.