$B \to K \nu\bar\nu$, MiniBooNE and muon $g-2$ anomalies from a dark sector

Belle II has reported the first evidence for $B^+ \to K^+\nu\bar\nu$ with a branching ratio $2.7 \sigma$ higher than the standard model expectation. We explain this, and the MiniBooNE and muon anomalous magnetic moment anomalies in a model with a dark scalar that couples to a slightly heavier sterile Dirac neutrino and that communicates with the visible sector via a Higgs portal. We make predictions for rare kaon and other $B$ meson decays.

Decays of B mesons that involve invisible final states are excellent probes of new invisible or long lived states like a massive sterile neutrino, ν D .These states may be produced via mixing with the active neutrinos or through new mediators such as new vector bosons or leptoquarks that couple them to SM particles.We consider a mechanism in which ν D communicates with the SM sector through a light scalar mediator S [18] which couples to the SM sector through an extended Higgs portal [19][20][21].The scalar couples to active neutrinos via four-neutrino mixing and a contribution to B + → K + ν ν is generated by the two body decays, B + → K + S and S → ν ν.
Our framework also permits an understanding of the excess in electron like events in the MiniBoone experi-ment [22] in terms of a light neutrino upscattering into ν D which subsequently decays to νS(→ e + e − , γγ).Note that similar upscattering into ν D via a dark vector boson Z ′ [23,24] is excluded [25] by data from the CHARM-II [26] and MINERvA [27] experiments if the Z ′ is lighter than the sterile neutrino as in Ref. [23].As shown in Ref. [18], with a light scalar mediator, the solution to the MiniBooNE anomaly is consistent with the CHARM-II and MINERvA data even with S lighter than the sterile neutrino.It is noteworthy that this explanation is untested by the template analysis of MicroBooNE [28].
Another long-standing anomaly is that of the anomalous magnetic moment of the muon, a µ ≡ (g − 2) µ /2.The SM prediction [29] is more than 5σ smaller than the updated world average following the latest experimental measurement [30]: The anomaly can be resolved by including a higher dimensional coupling to two photons [18,20].This coupling can also help explain the MiniBooNE anomaly because it enables the scalar to decay to photon pairs which can be misidentified as electron events at MiniBooNE.Model.The scalar Lagrangian is given by where v ≃ 246 GeV, is the Higgs vacuum expectation value, d and ℓ correspond to down-type quarks and leptons with a universal coupling η d scaled by the respective SM Yukawa.The structure of the Lagrangian can arise from the mixing of singlet scalar with the neutral components of a two Higgs doublet model [19][20][21].We will however adopt an effective interaction in the spirit of Ref. [18] and Observable SM expectation Measurement or constraint  take the couplings η f of the scalar to the up-type quarks to not be flavor universal.The parameter κ, of inverse mass dimension, induces an Sγγ coupling which contributes to (g − 2) µ via the one-loop Barr-Zee diagram [31].(Without this higher dimensional operator, the contributions to (g − 2) µ are via the vertex correction of the γµ + µ − vertex and self-energy diagrams which are insufficient to address the anomaly [20].) The light active neutrinos mix with the heavy sterile neutrino and induce a coupling of the dark scalar to the light neutrinos.The four flavor eigenstates ν α are related to the mass eigenstates ν i by where L, R indicate the handedness of the neutrino, and U is a 4 × 4 orthogonal matrix common to ν L and ν R .We take by unitarity.We require ν 4 to be a Dirac neutrino so that its nonrelativistic decays, ν 4 → νS, are not isotropic [32], as required by MiniBooNE data.Note that the sterile neutrino will have a much shorter lifetime The coupling of the light scalar to up-type quarks and light neutrinos yields several flavor changing neutral current transitions via the penguin loop.The rare hadronic decays in Table I provide constraints on the model.
Signals and constraints.We consider a dark scalar with mass in the range 10 < ∼ m S /MeV < ∼ 150.Since m S < 2m µ , S can only decay to photons, electrons and neutrinos, for which the decay widths are provided in the Supplemental Material.Below, we discuss the phenomenological implications for several observables of interest and the constraints imposed on the model.S decay length.We require the decay length of the dark scalar to be shorter than 0.1 mm to evade bounds from beam dump and other experiments that probe long-lived particles.
B and K meson decays.The Lagrangian in Eq. ( 2) induces two-body meson decays such as B → K ( * ) S and K → πS at one loop.The flavor changing neutral current (FCNC) Lagrangian for the b → s and s → d transitions is given by where the effective couplings g bs , g sd take the form, and The rates for the two body FCNC processes can be found in the Supplemental Material.The relevant B meson form factors are taken from Refs.[50,51] while the kaon form factor |f K 0 (m 2 S )| 2 is approximately unity [52].It is important to note that g bs is essentially determined by η t since the contribution proportional to η c is both helicity and charm-mass suppressed.On the other hand, for g sd , the contribution from η c is only helicity suppressed and can be sizable for η c ≫ η t .L FCNC also induces B s decays to γγ, µ + µ − and ν ν.The most important constraints on the couplings from the flavor changing B and K transitions are as follows: 1. B decay width: We require B(B → K ( * ) S) < 10% so that it does not exceed the uncertainty in the SM prediction of the B meson width [53].
2. B → Kν ν: We require B(B + → K + ν ν) to lie within 1σ of the Belle II measurement in the first row of Table I.
3. B → K * ν ν : We ensure that the upper limits on the branching fractions of the B → K * modes in Table I are satisfied.
4. B 0 → K * 0 e + e − : This decay has been measured by LHCb in the low dilepton mass region of 30-1000 MeV/c 2 [37] which overlaps the mass range of the dark scalar.We therefore require the branching ratio to lie within 1σ of the measured value.

B s decays:
We require the scalar contribution to B s → γγ to remain below the upper limit placed by Belle [39] and the contribution to B s → µ + µ − to remain within the 1σ uncertainty of the measurement in Table I.We take the decay constant f Bs = 230.3(1.3)MeVfrom Ref. [54].An interesting signature is B s → invisible which is currently not constrained by experiment but can be probed at Belle II.We make predictions for its branching fraction for some benchmark points.
6. K + → π + ν ν: The NA62 experiment at CERN has set stringent limits on B(K + → π + X) as a function of the X mass and lifetime for invisible X decays except in the range 110 < m X /MeV < 160 [55,56].However, since S has a very short lifetime, O(0.1) ps, these limits do not apply.
7. K L → π 0 ν ν: We require that the most recent upper limit on the branching fraction from the KOTO experiment at J-PARK be satisfied; see Table I B and K meson mixing.The Lagrangian in Eq. ( 4) also induces meson mixing.The measurement of the mass difference ∆M Bs is consistent with the SM prediction.Hence we require the additional contribution to not exceed the uncertainty in the SM expectation.For kaon mixing, the SM prediction suffers from large uncertainties.The long distance contribution is poorly estimated, so we only include the short distance contribution to ∆M K in Table I.The measured value, however, is quite precise and we do not allow the new contribution to ∆M K to exceed the 1σ uncertainty in the measurement.
(g − 2) ℓ .The dominant contribution to the anomalous magnetic moments of the muon and electron comes from the log enhanced term of the Barr-Zee diagram (see Supplemental Material) which depends on the cutoff scale Λ.We fix Λ = 2 TeV and require consistency with Eq. (1) within 1σ.Because ∆a e /∆a µ = m 2 e /m 2 µ , we find ∆a e ∼ few × 10 −14 which is much smaller than the magnitude of the inferred value, O(10 −13 −10 −12 ) [57][58][59].
MiniBooNE.The scattering of a muon neutrino off a nucleus may take place via the dark scalar exchange as shown in Fig. 1.Being heavy, the sterile neutrino produced promptly decays to a light neutrino and the dark scalar, whose subsequent decay to e + e − and γγ may mimic the signal observed in the MiniBooNE experiment.Since MiniBooNE is a Cherenkov detector that cannot distinguish between electrons and photons, two photons or two electrons with a small opening angle can be misidentified as a single electron.The details of the coherent scattering cross sections mediated by the dark scalar can be found in the Supplemental material.
It has been previously shown in Ref. [18] that the dark scalar model is able to explain the excess observed in Mini-booNE data while being consistent with the observations of the CHARM II experiment for m ν4 between 400 MeV and 500 MeV.The sterile neutrino has to be heavier than 400 MeV to ensure that less than 70% of the excess events are in the forward most bin (0.8 < cos θ < 1) of the angular distribution of electron-like events at MiniBooNE [25].
We compute the coherent and incoherent scattering cross sections at MiniBooNE and CHARM-II and recast the results of Refs.[23,25] for a dark Z ′ kinetically mixed with the electromagnetic field, to our case.The following constraints are imposed to implement the mapping between the dark Z ′ model and our model: 1.In terms of the total scattering cross sections σ S and σ Z ′ for the scalar and Z ′ mediators, respectively, we define, with the denominator evaluated at the benchmark point m Z ′ = 30 MeV, α Z ′ = 0.25, αϵ 2 = 2 × 10 −10 of Ref. [23] to explain the MiniBooNE anomaly.The ν µ flux at the Booster Neutrino Beam in the neutrino run [22] is denoted by Φ.To reproduce the MiniBooNE signal we require 0.95 ≤ R ≤ 1.05.
2. The Z ′ model of Ref. [23], however, is excluded by the CHARM-II constraint in Ref. [25].To satisfy this constraint we require for E νµ = 20 GeV, where the denominator is evaluated for the parameter values in Fig. 3 of Ref. [25] with Heavy neutral lepton searches.Several experiments including PS191 [60], NuTeV [61], BEBC [62], FMMF [63], CHARM II [64], T2K [65], NA62 [66] and Micro-BooNE [67] have placed limits on heavy neutral leptons with sufficiently long lifetimes that can reach the detector before decaying into SM particles.However, the limits on U µ4 from the nonobservation of the decay signal do not apply to our model because ν 4 has a lifetime of O(0.1) ps and decays promptly.The only relevant bound for m ν4 between 400 MeV and 700 MeV arises from lepton universality: |U µ4 | < ∼ 0.07 at the 99% C.L. [68].
Analysis and results.We analyze two cases: κ ̸ = 0 and κ = 0. We find that η t ≳ 0.005 is needed to explain the measured branching fraction of B + → K + ν ν.However, this leads to a correction to the K L → π 0 ν ν branching fraction that violates the upper limit set by the KOTO experiment.To lower the new contribution to K L → π 0 ν ν, a cancellation between the η t and η c dependent terms in g sd is required.This is achieved for η c ≈ −27η t .The large η c does not, however, affect g bs significantly due to the charm mass suppression.If nonzero, the coupling κ is primarily constrained by MiniBooNE and g − 2 data.Even though the FCNC transitions to invisible final states depend upon the sterile neutrino parameters g D and U µ4 , these are mainly constrained by the ν µ -nucleus scattering cross section.For κ ̸ = 0 and κ = 0, we set η u = 0 for simplicity and scan the other parameters in the following ranges: m S ∈ [10,150] MeV , m ν4 ∈ [400, 700] MeV .
For both cases, we choose some benchmark points (BPs) that have interesting consequences for rare K and B decays.
a. κ ̸ = 0: In this case, the dark scalar has a nonzero effective coupling to photons which permits an explanation of the discrepancy in the muon anomalous magnetic moment.The MiniBooNE signal is also enhanced due to the nonzero branching fraction to γγ.In Fig. 2, we show the allowed regions by the red points.As explained above, there is a strong correlation between η t and η c .A significant correlation is also observed between m S and m ν4 .For certain values of η d , the effective scalar-nucleus coupling responsible for neutrino-nucleus scattering becomes too small to produce a significant scattering cross   section due to the fine-tuning between η t and η c .Hence we observe gaps in the allowed regions of η d .We select five benchmark points, as listed in Table II, that simultaneously explain the B + → K + ν ν, Mini-BooNE and g − 2 anomalies.For each BP we also provide predictions for some important decays in Table III.
b. κ = 0: The allowed regions are shown by the cyan points in Fig. 2. Since the branching of S → ν ν is enhanced in the absence of the γγ mode, η t is restricted Correlations between the branching fractions for kaon and B meson decays are shown in Fig. 3.The branching fraction for K + → π + ν ν is predicted to lie in a quite narrow range.Note that K + → π + ν ν and B s → ν ν are currently unconstrained for a short-lived dark scalar and sterile neutrino.
Summary.Motivated by the recent Belle II evidence for B + → K + ν ν which shows a ∼ 3σ excess relative to the SM prediction, we explored a new physics explanation of the result in terms of new light states.The new physics contribution to this decay is interpreted as B + → K + S, where S is a light scalar in the mass range 10 < ∼ m S /MeV < ∼ 140 which then decays to light neutrinos.The interaction of S with light neutrinos occurs via its coupling to a heavy neutral lepton ν D which mixes with the light neutrinos.We demonstrated that our model is consistent with other measurements and bounds and has interesting predictions for several B and K decays.The excess in electron-like events observed by the Mini-BooNE experiment is explained by the upscattering of ν µ to ν D which subsequently decays to e + e − and γγ states.Finally, our model can accommodate the recent muon g − 2 measurement if S directly couples to photons.

TABLE I .
Experimental measurements and constraints used in the analysis.The upper limits are at 90% C.L.

TABLE III .
Predictions for the benchmark points in TableII.