# B anomalies and dark matter: a complex connection

## Abstract

We study an extension of the Standard Model that addresses the hints of lepton flavour universality violation observed in \(B\rightarrow K^{(*)} l^+l^-\) decays at LHCb, while providing a viable candidate for dark matter. The model incorporates two new scalar fields and a Majorana fermion that induce one-loop contributions to *B* meson decays. We show that agreement with observational data requires the new couplings to be complex and that the Majorana fermion can reproduce the observed dark matter relic density. This combination of cosmological and flavour constraints sets an upper limit on the dark matter and mediator masses. We have studied LHC dijet and dilepton searches, finding that they rule out large regions of parameter space by setting lower bounds on the dark matter and mediator masses. In particular, dilepton bounds are much more constraining in a future high-luminosity phase. Finally, we have computed the scattering cross section of dark matter off nuclei and compared it to the sensitivity of current and future direct detection experiments, showing that parts of the parameter space could be accessible in the future to multi-ton experiments. Future collider and direct DM searches complement each other to probe large areas of the parameter space of this model.

## 1 Introduction

LHCb has reported anomalies in the measured decay rates of the *B* meson, which have been interpreted as hints of lepton flavour universality violation [1, 2]. The SM predicts equal rates for the processes \(B\rightarrow K^{(*)} \mu ^+\mu ^-\) and \(B\rightarrow K^{(*)} e^+e^-\), and it is customary to study the ratios of these branching ratios, defined as *R*(*K*) and \(R(K^*)\), since the dependencies on hadronic matrix elements (and associated uncertainties) cancel out [3]. The measurements of these hadronically clean observables deviate consistently (although perhaps with not enough statistical significance) from the SM prediction \(R(K^{(*)})=1\) [4]. These hints are complemented by measurements of other observables that are more sensitive to hadronic physics. In particular, the differential branching fractions [1, 2, 5] and angular observables [6, 7, 8, 9, 10, 11, 12, 13] associated to the processes \(B\rightarrow \phi \mu ^+\mu ^-\) and \(B\rightarrow K^{(*)} \mu ^+\mu ^-\) also deviate from the SM predictions. Interestingly, all the apparent anomalies involve the transition \(b \rightarrow s\mu ^{+}\mu ^{-}\).

In order to account for these experimental results, one can modify the SM effective Hamiltonian, which involves penguin and box diagrams, by including one-loop contributions from new exotic particles. A full classification of the various particle combinations, considering different gauge representations, was presented in Refs. [14, 15]. Among the different models, some featured neutral scalar or fermions that, if stable, could play the role of dark matter (DM).^{1} The first possibility was investigated in Ref. [41], where it was found that the large new couplings required to reproduce the correct DM relic abundance induce sizeable 1-loop contributions to DM-nucleon scattering, leading to very strong limits from direct detection experiments. In addition, as reported by [42], the Higgs portal coupling typically dominates over other new physics effects. The second possibility was addressed in Ref. [43], where the fermionic dark matter field was accompanied by one additional scalar and one additional coloured fermion.

In this work, we consider a modification of the model of Ref. [43]. Namely, we will also assume a fermionic dark matter particle, but with two extra scalar fields, one of which has a colour charge. On top of this, we include the latest SM theoretical prediction for the mass difference in \(B_{s}-\)mixing [44], which differs from the experimental observation by \(1.8\,\sigma \). In order to reduce this tension and provide an explanation for the *B* anomalies, complex couplings are needed, leading to new CP-violation sources, a scenario that has not been studied in the context of one-loop models so far. We explore the parameter space of this model, taking into account all the flavour observables, DM constraints, and LHC collider signatures.

This paper is organised as follows. In Sect. 2, we introduce the details of the particle physics model, address the constraints from the observed DM relic abundance and \(B_{s}-\)mixing and discuss the implications on the model’s parameter space. In Sect. 3, we investigate the possibility of observing this scenario at the LHC, for which we take into account dijet and dimuon searches. We also include a projection of the potential reach of the High Luminosity phase of the LHC. Finally, in Sect. 4, we compute the DM-nucleus scattering cross section and study current constraints and the future reach of direct DM detection experiments. The conclusions are presented in Sect. 5.

## 2 The model

^{2}The interactions between the new particles and the SM are described by the Lagrangian,

Quantum numbers of the new fields. We also indicate the charges under \({\mathcal {Z}}_2\)

| \(SU(2)_{L}\) | \(U(1)_{Y}\) | \({\mathcal {Z}}_2\) | |
---|---|---|---|---|

\(\phi _{q}\) | 3 | 2 | 1 / 6 | \(-\,1\) |

\(\phi _{l}\) | 1 | 2 | \(-\,1/2\) | \(-\,1\) |

\(\chi \) | 1 | 1 | 0 | \(-\,1\) |

Global fits [20, 45, 46, 47, 48, 49, 50, 51, 52, 53] have been used to determine the new physics contribution to the Wilson coefficients in order to reproduce the observed experimental results. These fits favour \(C_{9}^{\text {NP}} = -C_{10}^{\text {NP}}\), and suggest that no new physics is required for operators involving electrons or tau leptons. Motivated by these results, we assume negligible couplings to the first quark generation (i.e., \(\lambda _{Q_1} = 0\)) and to the first and third lepton generations (i.e., \(\lambda _{e}=\lambda _{\tau }=0\)). This provides an explanation for the \(R_{K^{(*)}}\) anomalies, while relaxing the bounds from other searches.

Therefore, in total, we are left with six free parameters in this model, namely the masses of the three new particles (\(m_{\chi }\), \(m_{\phi _{l}}\), \(m_{\phi _{q}}\)), and the couplings to \(b-\)type quarks, \(s-\)type quarks, and leptons (\(\lambda _b\), \(\lambda _s\), \(\lambda _{\mu }\)).

It should be noted that the couplings \(\lambda _{1}|\phi _{l}|^{2}|H|^{2}\) and \(\lambda _{2}|\phi _{q}|^{2}|H|^{2}\) are allowed by gauge symmetry in the Lagrangian of Eq. (2.1). However, they only lead to an overall shift to the masses of \(\phi _{l}\) and \(\phi _{q}\) after electroweak symmetry breaking since the couplings to the Higgs play no phenomenological role in the relevant range of \(\phi _{l,q}\) masses. Likewise, the terms \(\lambda _{3}|\phi _{l}H|^{2}\) and \(\lambda _{4}|\phi _{q}H|^{2}\) are also allowed by gauge symmetry. They typically induce a small split in the masses of the neutral and charged components of the doublets \(\phi _{l}\) and \(\phi _{q}\) in the range of \(\phi _{l,q}\) masses that survive the collider constraints. Finally, a term of the form \((\phi _{l}H)^{2}\) can lead to large contributions to neutrino masses at one loop, which forces the corresponding coupling to be extremely small [43]. We will neglect these couplings in the following.

As mentioned in the Introduction, similar models have been discussed in the literature, featuring either scalar DM [41, 54, 55, 56] or fermionic DM [43]. Our model differs from that of Ref. [43] in that we have two extra scalar fields which couple to the lepton or quark sectors.

### 2.1 Dark matter relic abundance

*x*[58],

We will use this relation to fix \(m_{\phi _{l}}\) as a function of the other parameters, thus effectively reducing by one the number of free parameters. Furthermore, due to the suppression of the velocity-independent term for \(\langle \sigma v\rangle \), indirect detection bounds are not expected to constrain our model.

### 2.2 \(B_{s}-\)mixing and other flavour constraints

This model introduces new couplings to the *s* and *b* quarks (and to the rest of the quarks by rotation of the CKM matrix). We must therefore incorporate constraints from *B* meson physics.

^{3}

In order to constrain the Wilson coefficients we use the first global fit that takes into account the possibility that \(C_{9}\) and \(C_{10}\) are complex [59]. This is a natural scenario that arises when new CP-violation sources are introduced, and has not been studied in detail in the literature so far.

*F*and

*G*were already defined in Eq. (2.11).

In Fig. 3, the effect of all of these constraints on the real and imaginary parts of the couplings \(\lambda _{s}\lambda _{b}^{*}\) for several benchmark points is shown. Regions that are allowed by \(b \rightarrow s\mu ^{+}\mu ^{-}\) observables and \(B_{s}-\)mixing (given by Eqs. (2.17) and (2.18)) are shaded in green and blue, respectively. For illustrative purposes, the figure shows the constraints for multiple values of DM and mediator masses, while keeping \(\lambda _\mu =\sqrt{4\pi }\) fixed. We remind the reader that the mass of \(m_{\phi _{l}}\) is fixed so as to reproduce the correct relic density using Eq. (2.9).

As we can observe, in order to simultaneously satisfy both types of constraints, complex couplings are needed (\(\text {Im}(\lambda _{s}\lambda _{b}^{*})\ne 0\)). Also, as the mass of the dark matter particle and the mediators increase, both areas are more difficult to reconcile. In practise, this leads to an upper bound on the masses of the exotic new particles. The precise limit depends on the choice of couplings, which we will discuss in Sect. 3.

### 2.3 Benchmark scenarios

- (A1)
\(|\lambda _{\mu }| = 2\), with \(\lambda _{b} = \lambda _{s}^*\);

- (A2)
\(|\lambda _{\mu }| = 2\), with \(\lambda _{b} = 4\lambda _{s}^*\);

- (B1)
\(|\lambda _{\mu }| = \sqrt{4\pi }\), with \(\lambda _{b} = \lambda _{s}^*\);

- (B2)
\(|\lambda _{\mu }| = \sqrt{4\pi }\), with \(\lambda _{b} = 4\lambda _{s}^*\),

## 3 LHC constraints and prospects for high-luminosity

In this section, we study the experimental signatures that this model would produce at the LHC. DM search strategies in both ATLAS and CMS involve analysing final states containing jets and leptons produced in association with a DM particle, identified from missing transverse energy. In this model, direct production of the coloured and leptonic scalar doublets \(\phi _q\) and \(\phi _l\), respectively, typically leads to such final states.

*W*or

*Z*bosons and involves the electroweak coupling, as shown in Fig. 5. The decays of \(\phi _l\) lead to clean final states with one or two leptons and missing energy. Although flavour constraints require \(\lambda _\mu \gg \lambda _q\), the cross section of this process is smaller than the production of the coloured mediator for similar mediator masses. However, since \(m_{\phi _l}\) is fixed for every value of \(m_{\chi }\) to reproduce the correct relic abundance, there are regions of the parameter space where both searches are complementary. We will here consider the process

### 3.1 Simulation details

We have implemented this model in Feynrules 2.3[67]. The calculation of the matrix elements and the event generation is done using MadGraph5_aMC@NLO 2.6.3[68]. Production and decay of the new particles are considered independently using the narrow width approximation, as implemented in MadSpin[69], which further accounts for spin correlations in decay chains.^{4} We then use Pythia 8.235[70] to shower the parton-level events and we pass the output to CheckMATE 2.0.26 [71], which compares the expected signal with supersymmetric searches at the LHC and derives an exclusion limit. As we have explained above, we can apply squark and slepton searches to constrain the coloured and leptonic mediator, respectively.

In order to describe initial and final state radiation and reproduce the correct jet structure precisely, we consider leading order (LO) production with parton shower matching and multijet merging when needed. The LO multijet merging techniques describe how parton shower emissions can be combined with full matrix element calculations to achieve a better accuracy in the description of the radiation spectrum. Using this technique, every jet is classified according to its \(p_T\) and then compared to a hardness scale \(Q_{\text {cut}}\). In this way, emissions above the hardness scale \(Q_{\text {cut}}\) are described at LO accuracy using the corresponding matrix element calculation for an extra hard, wide-angle QCD emission in the final state, while emissions below this scale are defined as soft or collinear jets and the all-orders resummation description from the parton shower is preserved. Note that even though \(\mathcal {O}(\alpha _s)\) corrections are included using this procedure, the calculation remains formally \(\text {LO}+\text {LL}\) accurate after parton shower due to missing virtual corrections.

After hadronization, the showered events and the production cross sections are passed to CheckMATE. Each model point is tested against all the implemented experimental analyses to determine the optimal signal region. For this signal region, CheckMATE compares the simulated signal with the actual experimental observation and determines whether the model point is excluded at the \(90\%\) confidence level.

### 3.2 Results

List of experimental searches sensitive to our model, where *l* denotes electron and muon. The third column describes the final state targeted by the analysis and the last column displays the total integrated luminosity

Regarding the Open image in new window search, the limits in every scenario show that for the lightest DM mass, coloured mediators with masses below \(\sim \)1 TeV are excluded. Even though heavier DM produces larger amounts of missing energy in final states, the cross section decreases rapidly with the \(m_{\chi }\), leading to similar exclusion limits. It is interesting to note that exclusion limits are slightly stronger for the scenarios with \(|\lambda _{b,t}| > |\lambda _{s,c}|\), where mediators with masses below \(\sim \)1.1 TeV are excluded. The reason for this is that final states with either top or bottom quarks are more sensitive than final states with light jets to some experimental searches, which are specifically designed to target topologies with top and bottom quarks signatures. The most stringent experimental search involves final states with at least two (\(b \bar{b}\) production) or four (\(t \bar{t}\) production) jets or exactly two leptons and missing energy [72]. In particular, the most sensitive signal region is optimised to detect events featuring a DM particle produced in association with a \(t \bar{t}\) pair, which decays fully hadronically.

Regarding the Open image in new window search, the limits show that models with dark matter masses below approximately \( 30 \; {\mathrm {GeV}}\) are ruled out for \(|\lambda _\mu | = 2\), with the exclusion limit going down to \(\sim 13 \; {\mathrm {GeV}}\) for \(|\lambda _\mu | = \sqrt{4\pi }\). This corresponds to mediator masses below \(360 \; {\mathrm {GeV}}\) for \(|\lambda _\mu | = 2\) and \(410 \; {\mathrm {GeV}}\) for \(|\lambda _\mu | = \sqrt{4\pi }\). The Open image in new window cross section mainly depends on \(m_{\phi _l}\), so the limits on \(m_\chi \) can be understood through its relation with \(m_{\phi _l}\) given by the DM relic condition (2.9) for a particular value of \(\lambda _\mu \). The most stringent search involves final states with \(2l + 0 j\), 2*l* and at least 2 jets, or 3*l* and missing energy [73]. In particular, the most sensitive signal region is characterised by \(2l + 0j\) and a dilepton invariant mass \(m_{ll} > 300 \; {\mathrm {GeV}}\), and it is optimised to target slepton pair production.

The most remarkable result is that LHC limits completely exclude the scenario with \(|\lambda _\mu | = 2\) and \(\lambda _b = \lambda _s^*\), as well as a sizeable region of the scenario with \(\lambda _b = 4\lambda _s^*\) for the same \(|\lambda _\mu |\). These constraints become weaker for larger values of \(|\lambda _\mu |\) and, for the scenarios with \(|\lambda _\mu | = \sqrt{4\pi }\), most of the parameter space is allowed. It is crucial to note that the limits coming from final states with jets and leptons are complementary to each other. While the former exclude regions of the parameter space with large \(m_\chi \) and small \(m_{\phi _q}\), the latter rule out models with very heavy mediator masses \(m_{\phi _q}\) and light dark matter. Importantly, these limits are also complementary to the ones coming from direct detection, where dark matter masses below \(12 \; {\mathrm {GeV}}\) lie below the neutrino floor. Therefore, it is fundamental to consider both approaches to explore the model.

It is worth mentioning that the small couplings required by flavour constraints lead to decay widths slightly below the QCD scale for \(m_{\phi _{q}}\lesssim 370 \; {\mathrm {GeV}}\). Strictly speaking, this means that the computation of the decay width cannot be handled perturbatively and that the new particle \(\phi _q\) may hadronize into bound states with SM quarks, analogous to R-hadrons [76], before decaying. However, the typical width involved is \(\Gamma _{\phi _q} \sim \mathcal {O}(10^{-2})-\mathcal {O}(10^{-3}) \; {\mathrm {GeV}}\), which means lifetimes of the order \(\tau \sim 10^{-22} \; \text {s}\), so any potential bound state would decay promptly in the detector. This region of the parameter space is excluded by ATLAS and CMS R-hadron searches [77, 78].

We have also studied the limits that could be obtained with \(3000 \; \text {fb}^{-1}\) of 14 TeV data once the LHC High Luminosity phase [79] is completed. As we can observe in the plots, the main gain would come from the leptonic channels, which would allow to test a considerable amount of the model’s parameter space. In particular, scenarios with \(|\lambda _\mu | < 2\) would be completely excluded. The experimental searches giving the strongest exclusion limits target the same final states and are shown in the low panel of Table 2.

## 4 Direct DM detection prospects

*N*is the corresponding nucleon, and \(\mathcal {O}_i\) is the set of non-relativistic operators [80, 81]. The values for the coefficients \(c_i^N\) can be derived as the non-relativistic limit of the original interaction Lagrangian, and the differential rate can be computed using the corresponding nuclear form factors from Refs. [81, 82], and for a given choice of the DM halo properties. We have adopted the so-called standard halo model [83] with local DM density \(\rho _{\chi }=0.4\ \text {GeV/cm}^3\), a central velocity of \(v_0=220\ \text {km s}^{-1}\), and a escape speed of \(v_{\text {esc}}=544\ \text {km s}^{-1}\) to calculate the number of expected recoils in a specific experiment.

*e*is the electron charge, \(Q_N\) is the nucleon charge, and \(g_N\) are the nucleon g-factors (\(g_p=5.59\) and \(g_n=3.83\)). The effective coupling to the anapole interaction term, \(\mathcal {A}\), reads [88]

Given the range of DM masses that we consider in this study, the main constraint is due to Xenon1T results [92], which we simulate using the prescription outlined in appendix A of Ref. [93], achieving good agreement. As we can see in Fig. 7, the theoretical predictions for this model are beyond the reach of current experimental searches. We also show the reach of future direct detection experiments. The LZ detector, will employ 5.6 tons of liquid xenon with 1000 days exposure as outlined in [94, 95]. The DarkSide-20k experiment [96], is an argon detector which will employ 20 tons of fiducial mass for a duration of 10 years. We have assumed that the DarkSide collaboration will be able to achieve a threshold energy of 5 keV, a reasonable assumption considering the results from DarkSide-50 [97]. For reference we have also calculated the neutrino floor for anapole interactions in the \((\mathcal {A},\, m_{\chi })\) plane. We have used the prescription described in Ref. [98] and the expected neutrino fluxes from Refs. [99, 100, 101, 102, 103, 104]. It is clear that our model favourably lays in a region of parameter space that would be probed by a generation of experiments with multi-ton targets, that can probe near or even slightly beyond the neutrino floor. Spectral analysis with the neutrino background compounded with annual modulation data, could provide complete discrimination between model and the anapole moment which is both velocity and momentum dependent.

For completeness, we have also calculated the effect on the total scattering cross section from aforementioned twist-2 operator and spin-dependent interaction. The former contribute to the spin-independent scattering cross section (operator \(\mathcal{{O}}_1\)) and can be sizeable if the new coupling to quarks is large or the colour mediator is very light. We have explicitly checked that once LHC constraints are included in the parameter space of the model, these terms are always subdominant to the anapole term discussed above.^{5} We represent in Fig. 8 the theoretical predictions for \(c_1\) as a function of the DM mass from this contribution. For the spin-dependent interaction, we found that the predicted rate for our sampled parameter space is always sub-dominant.

Had we chosen to work with a Dirac fermion, the dipole and charge radius contributions should have been added. As it has been pointed out in Ref. [84], the fairly large coupling to muons that is required to explain the flavour anomalies leads to effective DM couplings that are orders of magnitude higher than those coming from the tree level contribution, the most important being the charge-radius interaction. This we have checked, and in fact above \(m_{\chi }\sim 10\) GeV, all our parameter points are excluded by Xenon1T. Below \(m_{\chi }\sim 10\), the model is excluded by both LHC constraints and indirect detection bounds. Unlike in the Majorana case, the s-wave contribution to the thermal cross section \(\langle \sigma v\rangle \) is no longer helicity suppressed and hence excluded [105].

Our results suggest that future multi-ton direct detection experiments, such as DarkSide [96], would be able to probe this model in the mass range \(m_{\chi }\sim \) 10–60 GeV. It is very interesting to point out that many of the points in this DM mass range feature very heavy \(\phi _q\) and therefore would be beyond the reach of collider searches. In a sense, future direct DM detection and the LHC complement each other to probe a large part of the model’s parameter space.

## 5 Conclusions

In this article, we have studied a particle physics model that addresses the hints of lepton flavour universality violation observed by LHCb in \(b \rightarrow s\mu ^{+}\mu ^{-}\) transitions, and that provides a solution to the dark matter problem. The scenario that we have analysed incorporates two new scalar fields and a Majorana fermion that provide one-loop contributions to *B* meson decays.

The Majorana fermion is stable and can reproduce the observed DM relic abundance. We have studied the effect of new physics in flavour observables, for which \(B_{s}-\)mixing and \(b \rightarrow s\mu ^{+}\mu ^{-}\) processes provide the most important constraints. In order to find an explanation for the *B* anomalies and to reduce the \(1.8\,\sigma \) tension between the predicted and measured mass difference in \(B_{s}-\)mixing, complex couplings are needed. We have used results from the first global fit that takes into account this possibility. The combination of flavour bounds and constraints on the DM relic abundance leads to upper limits on the masses of the exotic states, and in general points towards a rather light DM candidate (with a mass \(m_{\chi }\lesssim 200\) GeV).

We have studied the signatures that this model would produce at the LHC. The dominant processes are the pair production of the coloured and leptonic scalars. For the former, the strongest exclusion limits are given by Open image in new window searches. For the latter, the final states are very clean, containing 1 or 2 leptons and missing energy. Both searches are complementary and exclude different regions of the parameter space, setting lower bounds on DM and mediator masses. The high-luminosity phase improves bounds coming from both searches, with dilepton being the most pronounced. The collider constraints are weakened when the \(\lambda _{\mu }\) parameter is pushed towards the perturbative limit.

Finally, we have investigated how DM direct detection experiments constrain this model. Given the range of DM masses that we consider in this study, the main constraint is due to Xenon1T results. The small new couplings required by flavour constraints means that one-loop contributions to the DM-nucleon scattering cross section are generally larger than the tree level process. In particular, the dominant loop induced interaction is the anapole moment. We have shown that this model is not excluded by current data and could be probed by the next generation of experiments with multi-ton targets in the mass range \(m_{\chi }\sim \) 10–60 GeV.

## Footnotes

- 1.
An alternative to this one-loop solution is to consider \( Z^\prime \) [16, 17] or leptoquark [18, 19] tree-level contributions, see e.g., Ref. [20] and references therein. The DM problem has been addressed in the framework of these constructions [21], see e.g., Refs. [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] for the \(Z^\prime \), and Refs. [37, 38, 39, 40] for the leptoquark models.

- 2.
As we will comment in Sect. 4, the alternative construction with Dirac DM is ruled out mainly by experimental results from direct DM detection.

- 3.
We have neglected the

*Z*-penguin contribution to \(C_{9}^{\text {NP}}\), since it is suppressed by \((m_{b}/m_{Z})^{2}\) and is subdominant compared to the photon exchange. - 4.
The narrow width approximation is not valid in benchmark points B1 and B2, for which we have taken interference effects into account.

- 5.
Note that box diagrams containing a

*W*boson give an additional contribution, which is further suppressed by CKM factors and electroweak couplings and can thus be safely neglected.

## Notes

### Acknowledgements

We would like to thank Alejandro Ibarra for useful discussions. DC and AC are grateful for the support from the Science and Technology Facilities Council (STFC). DC acknowledges the partial support of the Centro de Excelencia Severo Ochoa Program through the IFT-UAM/CSIC Associate programme. PMR and JMM acknowledge support from the Spanish Research Agency (Agencia Estatal de Investigación) through the contract FPA2016-78022-P and IFT Centro de Excelencia Severo Ochoa under Grant SEV-2016-0597.

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