R({K^{(*)}}) from dark matter exchange

# $R({K^{(*)}})$ from dark matter exchange

## Abstract

Hints of lepton flavor violation have been observed by LHCb in the rate of the decay relative to that of . This can be explained by new scalars and fermions which couple to standard model particles and contribute to these processes at loop level. We explore a simple model of this kind, in which one of the new fermions is a dark matter candidate, while the other is a heavy vector-like quark and the scalar is an inert Higgs doublet. We explore the constraints on this model from flavor observables, dark matter direct detection, and LHC run II searches, and find that, while currently viable, this scenario will be directly tested by future experiments.

Introduction. The LHCb experiment has observed intriguing deficits in and , defined as the ratio of branching ratios Aaij et al. (2014, 2017). These “hadronically clean” ratios are free from theoretical uncertainties in hadronic matrix elements, which cancel out Hiller and Kruger (2004). In the standard model (SM) it is expected that Bordone et al. (2016), while experimentally deficits of approximately 20% are observed. Although the significance in either observation or is not high, model-independent fits to both data, and possibly including quantities more sensitive to hadronic physics, including , and the angular observable , indicate a higher significance of Capdevila et al. (2017); Geng et al. (2017); Ciuchini et al. (2017); Hiller and Nisandzic (2017); Altmannshofer et al. (2017) Ref. D’Amico et al. (2017) shows that the best fits and significance do not change appreciably whether one includes the hadronically sensitive observables or not, and that it is possible to find a good fit to the data by including a single dimension-6 operator in the effective Hamiltonian,

 Heff∋ObLμL=1Λ2(¯sLγαbL)(¯μLγαμL) (1)

with , which is approximately times the SM contribution at one loop.

The new physics contribution (1) can be obtained from tree-level exchange of a heavy vector boson Di Chiara et al. (2017); Chiang et al. (2017a); Dalchenko et al. (2017); Agrawal et al. (2018); Romao et al. (2017); Faisel and Tandean (2018); Falkowski et al. (2018); Kohda et al. (2018) or leptoquark Chen et al. (2017); Crivellin et al. (2017); Cai et al. (2017); Chauhan et al. (2017); Diaz et al. (2017); Doršner et al. (2017); Crivellin et al. (2018); Aloni et al. (2017); Assad et al. (2018); Di Luzio et al. (2017); Calibbi et al. (2017); Dey et al. (2017); Chauhan and Kindra (2017); Müller (2018); Doršner and Greljo (2018); Hiller et al. (2018); Fajfer et al. (2018); Monteux and Rajaraman (2018), or through loop effects of new particles. In ref. Arnan et al. (2017), an exhaustive classification and study of the simplest loop models was carried out, where it was shown that one needs either two new scalars and one new fermion, or two new fermions and one new scalar, to explain the decay anomalies. Many possible quantum numbers of the new particles are possible. Here we note that these include cases where one of them can be neutral under the SM gauge interactions, opening the possibility that it could be dark matter (DM), and thus allowing the model to explain two observed phenomena requiring new physics.

We prefer to minimize the number of new scalars so there is just one, thereby allowing the DM candidate to be one of the new fermions.1 Fermionic dark matter is free from relevant Higgs portal couplings, making for a more predictive theory in which the dark matter properties are determined by the same couplings that explain the flavor anomaly. It will be shown that considerations of the dark matter relic density and direct detection give interesting additional restrictions on the model, and that it is also constrained by existing LHC searches as well as flavor-changing neutral current processes. The model therefore has high potential for discovery by a variety of complementary experimental searches.

Model and low-energy effective theory. We introduce a Majorana fermionic DM particle , a vectorlike heavy quark that carries SM color and hypercharge, and a scalar that is an inert SU(2) doublet. The quantum numbers are shown in table 1. The only couplings of the new fields to SM particles allowed by gauge and global symmetries (see table 1) are

 −L ∋ ~λi¯Qi,aϕaΨ+λi¯Sϕ∗aLai+H.c. (2) + λH,1|H|2|ϕ|2+λH,2|H†ϕ|2

where are the SM quark and lepton doublets, is the SU(2) index and is the flavor index. The relevant interactions at low energy are generated at one loop and thus require sizable couplings. Since there is no flavor symmetry, we will see that this model lives in a corner of parameter space where meson mixing constraints are nearly saturated. In a more complete model, the global symmetries could be an accidental consequence of a spontaneously broken gauge symmetry under which the new physics particles are charged.

The Higgs portal couplings play no important role in the following; gives an overall shift to after electroweak symmetry breaking, while splits the charged and neutral components of by a small amount (relative to as constrained by LHC searches). A coupling of the form

 λH,3(H†ϕ)2+H.c. (3)

violates lepton number conservation, as can be seen from the charge assignments in table 1. (Notice that cannot be assigned lepton number since it is Majorana.) Of course one expects that is only an approximate symmetry, if neutrinos have Majorana masses, which constrains the size of . In fact this operator could be the origin of one of the neutrino masses through the loop diagram shown in fig. 2, with mass matrix (where GeV), which has a single nonvanishing eigenvalue given by the trace.2 If eV for example, .

To make definite predictions from (2), we must specify which field bases are referred to. We will assume that for the leptons and down-type quarks, it is the mass eigenbasis. This implies that up-type quarks have couplings that are rotated by the CKM matrix:

 ~λi¯Qi→~λj(¯uL,iVij,¯dL,j)≡(~λ′i¯ui,~λi¯di) (4)

The box diagrams relevant for , , neutral meson mixing and DM scattering on nucleons are shown in fig. 1.3 Evaluating them we find the effective dimension-6 operators of the same form as (1) but different external states. The operator coefficients are shown in table 2, where for simplicity we take . Below we will see that  TeV to meet LHC constraints, but can be light since it is dark matter. The loop functions are given by and , normalized such that and and .

Flavor constraints. To match the observed anomalies, we require that D’Amico et al. (2017). Therefore the couplings must be of order unity, since LHC searches discussed below require TeV. On the other hand, strong mixing constraints, as determined by the mass splitting between and , limit the coefficient of in table 2 to be less than at 95% confidence level (c.l.) Arnan et al. (2017), giving the bound . Combined with the previous determination, this demands large ,

 |λ2|>2.9(M/TeV)1/2. (5)

Analogous bounds arise from , and Bona et al. (2008); Bona (2016) mixing: , . .

As an example, suppose that TeV and the bound on mixing is saturated. We can satisfy all other constraints with hierarchical quark couplings

 |~λ1|=0.014,|~λ2|=0.14,|~λ3|=1.1,|λ2|=2.9 (6)

If all of the couplings are positive and real, , right at the mixing 95% c.l. limit. If has the opposite sign to , is smaller, .

The hierarchical nature of the quark couplings is preserved under renormalization group running, since they are multiplicatively renormalized. The one-loop beta functions take the form Luo and Xiao (2003); Machacek and Vaughn (1983)

 β(~λi)≡μddμ~λi=316π2~λi(12|~λi|2+∑k|~λ2k|) (7)

For the choice of couplings in (6), this leads to a Landau pole in at a scale of around , indicating the need for further new physics at such scales. For example a spontaneously broken nonabelian gauge symmetry, such as we already suggested for explaining the global symmetries of the model, could avert the Landau pole.

It is technically natural to assume the other leptonic couplings are negligible, since they are generated radiatively only through neutrino mass insertions. However aesthetically it may seem peculiar to have . If , the box diagrams leads to lepton flavor-violating decays such as and . However because of the Majorana nature of , there are crossed box diagrams, shown in fig. 1, that exactly cancel the uncrossed ones in the limit where external momenta are neglected in the loop. Their amplitudes then scale as and respectively. After comparing them to those of leptonic decays in the SM, , and imposing the experimental limits on the forbidden decay modes Patrignani et al. (2016) we find no significant constraints on or .

Radiative transitions are another flavor-sensitive observable, as shown in fig. 3. For , fig. 3(a) generates the dipole operator

 ~λ∗3~λ2emb32π2(qψf(R)m2ϕ−qϕf(R−1)m2ψ)(¯sL⧸qγμbR) (8)

where , , is the photon momentum and . The electric charges of and are as in table 1. Due to operator mixing, the chromomagnetic moment also contributes. Using the results of ref. Arnan et al. (2017), the Wilson coefficients for our benchmark model with TeV give , a factor of 10 below the current limit on this combination from measurements of the branching ratio of .

Fig. 3(b) gives a contribution to the anomalous magnetic moment of the muon of , by saturating (5) and taking TeV. Ultimately this model increases the tension between the measured and predicted values of , but the effect is minimal, 20 times smaller than the SM discrepancy Patrignani et al. (2016). A similar diagram with the photon replaced by the leads to a correction of the coupling of the to left-handed muons of the form Arnan et al. (2017). This is significantly smaller than the uncertainty on the most accurate measurements of this coupling by LEP, Schael et al. (2006), which has a 0.4% error at the level.

If the couplings , are nonzero, there are contributions to , , and , with partial width Giunti and Studenikin (2015) where . Using and TeV, the requirement that the partial width of induced by the new physics contributions not exceed the measured value requires , while leads to the strong limit .

Dark matter constraints. The dark matter candidate in our model has tree-level annihilation to and . The -wave contribution to the cross section is helicity suppressed, so the term dominates Chang et al. (2014). The total thermally averaged annihilation cross section, counting both final states, either muons or neutrinos, is

 ⟨σvrel⟩(x)=|λ2|4m2S(m4ϕ+m4S)4π(m2ϕ+m2S)4x (9)

where . To get the observed relic density Ade et al. (2016), at the freeze-out temperature this should be roughly equal to the standard value /s Steigman et al. (2012) appropriate for -wave annihilating Majorana dark matter in the mass range GeV, that we will see is required by collider constraints. By assuming that saturates the inequality (5) so that it is no larger than needed to satisfy the flavor constraints, the relation requires

 mS=0.026√xfmϕ. (10)

This is valid if ; one can show that (10) is further reduced by the factor if .

We verified the previous estimate by numerically solving the Boltzmann equation with micrOMEGAs 4.3.5 Belanger et al. (2007); contours corresponding to the cosmologically preferred value Ade et al. (2016) are displayed in fig. 4. annihilations can lead to indirect signals in gamma rays and charged cosmic rays, but the -wave suppression of the cross section makes the limits from such searches very weak. Collider limits are far more constraining, notably ATLAS searches for 2 leptons and missing transverse energy collaboration (2017), which exclude the green region in fig. 4.

Because is a Majorana particle, the box diagram for scattering of off quarks leads only to spin-dependent or velocity-suppressed scattering off nucleons. The spin-dependent cross section for DM scattering off a single nucleon is given by where for low-energy scattering (e.g. Belanger et al. (2009)). The determination of the parameters is reviewed in Bringmann et al. (2017). For our benchmark model with TeV this leads to cm for scattering off neutrons, far below current experimental limits on spin-dependent scattering from the PICO-60 direct detection experiment Amole et al. (2017).

Had the dark matter been Dirac, diagram (c) of fig. 3 would give both a magnetic moment for the dark matter , [approximating consistently with eq. (10)], and a charge-radius interaction that lead to scattering on protons. Although the former is below current direct detection limits, the latter is far too large, which obliges us to take to be Majorana.4  Then there is only an anapole moment , which has been computed and constrained (using 2013 LUX results) for our class of models in ref. Kopp et al. (2014). We rescale their limit on to reflect more recent results from LUX Akerib et al. (2017), as well as the projected eventual sensitivity of DARWIN Aalbers et al. (2016), in fig. 5. The predicted value is also shown, using (5) and (10) with to eliminate and in favor of . For the lowest allowed value of GeV (considering that GeV from LHC constraints), the limit is a factor of weaker than the prediction, corresponding to a factor of 500 in the cross section. This is below the reach of the LZ experiment Szydagis (2016), but slightly above the expected sensitivity of DARWIN, leaving open the possibility of direct detection.

Collider constraints. The new states and carry SM quantum numbers, and can therefore be pair-produced in particle collisions. Fig. 6 shows the main production modes at a hadron collider and their decays. The final states necessarily include hard lepton pairs, since the splitting between and must be large, eq. (10). This also produces missing energy as the decay products inevitably include dark matter pairs. Moreover hadronic jets appear if is produced, since decays into plus quarks.

For Drell-Yan production of - pairs, the signal is lepton pairs and missing energy, with no jets. (One of the leptons is a neutrino if occurs). This is the same final state as in production of slepton pairs, so SUSY searches collaboration (2017) may be applied.5 The excluded region is shown in fig. 4, constraining GeV for all for which the relic density can be accommodated.

In diagrams 6(b,c,d), is produced, which subsequently decays to or . Such final states have been searched for by ATLAS in 13.3 fb of TeV data, including events with one or two leptons, jets and missing transverse momentum collaboration (2016a, b). These analyses has been implemented in CheckMATE 2.0.14 Drees et al. (2015), which we used to constrain our model, in conjunction with FeynRules 2.3 Alloul et al. (2014) and MadGraph 2.6.0 Alwall et al. (2014). 20,000 events per model point were generated for the process ( is suppressed by the small couplings of and to first generation quarks, or the parton distribution function of or ). The subsequent showering and hadronisation of the final state partons was modelled with Pythia 8.230 Sjöstrand et al. (2015) and detector simulation was done with Delphes 3.4.1 de Favereau et al. (2014).

Fig. 7 shows the resulting 95% c.l. limits on versus for models which both explain the flavor anomalies and give the correct DM relic density. Here is set by eq. 10 with and the couplings are scaled relative to (6) by the factor , where ; this choice keeps all the box diagrams approximately constant. At values of GeV, the lowest values that allow for the correct relic density while avoiding slepton search constraints, the one-lepton search limits GeV, except for a narrow window with just below . The two-lepton search does not constrain as strongly but is more sensitive to larger DM masses.

Conclusions. The indications from LHCb of lepton flavor universality breaking down are currently our best hint of physics beyond the standard model from colliders. These anomalies should be verified within a few years by further data from LHCb and Belle II Albrecht et al. (2017). If confirmed, it is not unreasonable to expect that the relevant new physics could also shed light on other shortcomings of the standard model. We have shown how a very economical model, in which dark matter plays an essential role, could be the source of anomalies, while predicting imminent tensions in other flavor observables, notably mixing. The model may be tested by the next generation of direct detection searches and can be discovered at the LHC via searches for leptons, jets and missing energy.

Acknowledgment. We thank J. Martin Camalich and D. London for helpful discussions, and S. Okawa for alerting us to a problem with the first version of this paper. Our work is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

### Footnotes

1. Ref. Gripaios et al. (2016) focuses on the opposite choice, and observes that the possible scalar dark matter candidate cannot satisfy direct detection constraints because of its coupling to . Previous attempts to connect to dark matter can be found in refs. Aristizabal Sierra et al. (2015); Bélanger et al. (2015); Celis et al. (2017); Altmannshofer et al. (2016); Cline et al. (2017); Baek (2017); Cline (2017); Sala and Straub (2017). In addition, refs. Kawamura et al. (2017); Chiang et al. (2017b) recently studied models similar to ours, but in which the DM is chosen to be a new scalar. These studies do not fully consider the impact of the Higgs portal coupling on the DM relic density and direct detection. In ref. Bhattacharya et al. (2015) it was shown that tends to dominate over any other new physics effects. Even if it vanishes at tree level, the one-loop correction tends to be too large to ignore without fine tuning.
2. A more complicated model with two or more flavors of dark matter would allow for nonsingular mass matrices.
3. The charges of the fields in this theory do not allow it to contribute to at one loop.
4. We thank S. Okawa for pointing out the importance of the charge radius contribution.
5. These limits assume annihilation to all flavors of both right and left handed sleptons, taken to be degenerate. Comparing production cross sections of all sleptons to that of a pair using MadGraph Alwall et al. (2014) indicates that they may be overly stringent for our model; at 13 TeV, the slepton production cross section is fb for  GeV, whereas  fb for production with  GeV.

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