A Dark Seesaw Solution to Low Energy Anomalies: MiniBooNE, the muon (g−2) , and BaBar
FFTPI-MINN-20-25, IPPP/20/32
A Dark Seesaw Solution to Low Energy Anomalies:MiniBooNE, the muon ( g − , and BaBar Asli Abdullahi, ∗ Matheus Hostert,
2, 3, 4, † and Silvia Pascoli ‡ Institute for Particle Physics Phenomenology, Department of Physics,Durham University, South Road, Durham DH1 3LE, United Kingdom School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,University of Minnesota, Minneapolis, MN 55455, USA Perimeter Institute for Theoretical Physics, Waterloo, ON N2J 2W9, Canada (Dated: August 19, 2020)A recent update from MiniBooNE has strengthened the observed . σ excess of e -like events.Motivated by this and other notable deviations from standard model predictions, such as the muon ( g − , we propose a solution to low energy anomalies through a dark neutrino sector. The modelis renormalizable and can also explain light neutrino masses with an anomaly-free and dark U (1) (cid:48) gauge symmetry broken at the GeV scale. Large kinetic mixing leads to s-channel production ofheavy neutral leptons at e + e − colliders, where we point out and explain a (cid:38) σ excess observed inthe BaBar monophoton data. Our model is also compatible with anomalous e -like events seen at oldaccelerator experiments, as well as with an excess of double vertex signatures observed at CCFR. I. INTRODUCTION
The discovery of neutrino oscillations [1–3], and con-sequently of neutrino masses and mixing, implies thatthe Standard Model (SM) of particle physics is incom-plete. Many extensions have been proposed to explainthe origin of neutrino masses, with the Type-I seesawmechanism [4–12] and its variants being the most wellstudied. Heavy neutral leptons (HNL) are the hall-mark of such models and carry a lepton number violating(LNV) Majorana mass, which, barring theoretical preju-dice, could take any value from sub-eV to GeV. Inrecent years, renewed attention has been devoted to theMeV - GeV mass scale, as such states can be searched forin an expanding program of fixed-target, meson decay,and collider experiments [13–19], having consequencesfor cosmology and the baryon asymmetry of the Uni-verse [20, 21]. Two approaches are typically adopted: oneof minimality, in which only new neutral fermions are in-troduced, e.g. [22], and, more recently, one in which theHNLs are considered part of a richer low energy dark sec-tor [23–36], all the more compelling in view of the largeabundance of dark matter in our Universe [37–40]. Ithas been pointed out that in the second approach thephenomenology can have unique features, requiring thereevaluation of existing bounds and offering new signa-tures, especially in the presence of multiple portals tothe SM [41]. Such an extension of the SM would leaveimprints, not just in neutrino experiments, but also ine.g. dark photon and dark scalar searches. Interestingly,some anomalies are present in these areas.In this letter , we propose a coherent explanation of sev-eral experimental anomalies, generating the correct scale ∗ [email protected]; 0000-0002-6122-4986 † [email protected]; 0000-0002-9584-8877 ‡ [email protected]; 0000-0002-2958-456X for the light neutrino masses. We simultaneously explainthe excess of e -like events observed at MiniBooNE [42]and the muon ∆ a µ = ( g − µ anomaly [43]. We alsopoint out some less-often discussed anomalies in exist-ing data which are compatible with the predictions ofour model. These include a mild excess of monopho-ton events at BaBar [44], the anomalous ν e -appearanceobserved by past accelerator experiments, such as PS-191 [45] and E-816 [46], and the double neutral vertexevents in CCFR [47, 48]. We show how these resultsemerge within a coherent picture and that they are, infact, highly correlated when interpreted under our hy-pothesis. This is achieved within an anomaly-free modelof a spontaneously-broken and secluded U (1) (cid:48) gauge sym-metry, providing a concrete model for the phenomenolog-ical idea put forward in Ref. [41]. The presence of ster-ile and dark vector-like neutrinos leads to light neutrinomasses via a generalized inverse seesaw [49–51], modifiedby the interactions in the dark sector. II. MODEL
We extend the SM gauge symmetry with a secluded U (1) (cid:48) , accompanied by a dark complex scalar Φ withcharge Q X that breaks the symmetry at sub-GeV scales.Generically, our fermionic sector comprises of d vector-like dark neutrinos, ˆ ν D =ˆ ν D L +ˆ ν D R , also charged underthe U (1) (cid:48) with charge Q X , guaranteeing anomaly cancel-lation in each dark neutrino family. A neutrino portal tothe SM is then achieved by n completely sterile states, ˆ N . In the following, we refer to particles charged under the U (1) (cid:48) gauge symmetry as “dark". a r X i v : . [ h e p - ph ] A ug The full Lagrangian is given by
L ⊃ L SM − X µν X µν − sin χ X µν B µν (1) + ( D µ Φ) † ( D µ Φ) − V (Φ) − λ Φ H | H | | Φ | +ˆ ν N i /∂ ˆ ν N +ˆ ν D i /D X ˆ ν D − (cid:20) ( L (cid:101) H ) Y ˆ ν cN + 12ˆ ν N M N ˆ ν cN +ˆ ν N (cid:0) Y L ˆ ν cD L Φ + Y R ˆ ν D R Φ ∗ (cid:1) +ˆ ν D M X ˆ ν D + h.c. (cid:21) , where flavor indices are implicit, and we write the ki-netic mixing between hypercharge and the U (1) (cid:48) me-diator X µ , as well as scalar mixing between the Higgsand Φ explicitly. Here, X µν ≡ ∂ µ X ν − ∂ ν X µ , /D X ≡ /∂ − iQ X g X /X , and Q X [ ν D L ] = Q X [ ν D R ] = 1 . Thescalars Φ and H acquire VEVs, v Φ (cid:39) O (500) MeVand v H (cid:39) GeV, respectively. After the electroweakand dark symmetries are spontaneously broken, taking ˆ ν f ≡ (cid:0) ˆ ν cα ˆ ν cN ˆ ν cD L ˆ ν D R (cid:1) T , the neutrino mass matrixreads L ν − mass = 12ˆ ν cf M D M TD M N Λ L Λ R TL M X TR M TX ˆ ν f + h.c. , (2)where M D ≡ Y v H / √ and Λ L,R ≡ Y L,R v Φ / √ . We di-agonalize the mass matrix with a unitary matrix U , de-fined in terms of sub-blocks U ≡ (cid:0) U α U N U D L U D R (cid:1) T ,such that ˆ ν m = U ˆ ν f ≡ (cid:0) ν N (cid:1) T contains the lightneutrinos ν and the ( n + 2 d ) HNLs N . At tree-level,the mostly-active neutrinos get a mass as in the in-verse [52, 53] and extended seesaw [54, 55] models. Atthe one-loop-level, however, we find an independent fi-nite contribution proportional to M N [56]. This is thesame contribution found in Ref. [31], and is analogousto the minimal radiative inverse seesaw [49–51]. Theseindependent tree- and loop-level contributions can haveopposite signs, leading to cancellations if M X (cid:46) M N .We exploit this fact to achieve neutrino masses compat-ible with current data. We neglect loop corrections toother mass parameters in the matrix.The massive dark photon, scalar, and HNLs only cou-ple to the SM via portal operators. After symmetrybreaking, the model has two CP-even scalars, the SMHiggs h (cid:48) , which contains a small Φ component with scalarmixing θ (cid:39) ( λ Φ H / λ H ) × ( v Φ /v H ) , where λ H is the quar-tic coupling of the Higgs, and a light mostly-dark ϕ (cid:48) . Inthe neutral gauge boson mass basis, we have a light Z (cid:48) vector boson that couples predominantly to the dark sec-tor current ( J µD ), as well as to the SM electromagnetic(EM), and neutral current (NC), L ⊃ Z (cid:48) µ (cid:18) eε J µ EM + g c W m Z (cid:48) m Z χ J µ NC + g X J µD (cid:19) , (3)where we assume m Z (cid:48) (cid:39) g X v ϕ (cid:28) m Z , and define ε ≡ c W χ . III. LOW ENERGY ANOMALIES
Our aim is to show that the model can explain severallow energy anomalies, while simultaneously generatingthe correct scale for light neutrino masses. Since mixingin the light neutrino sector can be generated by appropri-ate choices of the M D matrix, we work under the simpli-fying assumption of a single active neutrino generation,in our case ν µ , denoting the lightest non-zero mass eigen-state by ν . We require that m (cid:39) . eV, compatiblewith the scale suggested by neutrino oscillation experi-ments [57]. For concreteness, we pick n = 3 sterile and d = 1 vector-like dark neutrinos, although only the threelightest heavy neutrino mass eigenstates N j , j = 4 , , ,will be important for the phenomenology we discuss. Theheaviest states N and N have masses of several GeVs,and are mostly-sterile states.Our proposal is illustrated by two benchmark points(BPs), one exhibiting a left-right symmetry and one with-out. Their properties are shown in Table I but a detaileddefinition is left to Appendix A. The left-right symmetryin the dark sector of BP-A ( ν cD L ↔ ν D R ) is achieved bysetting Y L = Y R , and explains the vanishing entries inTable I. This can be shown to be related to CP conser-vation.Let us comment on the generic features of our twoBPs. We fix m Z (cid:48) = 1 . GeV and ε = 4 . × − for the dark photon. The three lightest HNLs all have O (100) MeV masses, and decay via neutrino and kineticmixing as N i → N i − e + e − . Specifically, N will typicallydecay with cτ (cid:46) cm, leading to displaced e + e − ver-tices, while N will decay more promptly, cτ (cid:46) mm.In the case of N , it can only decay into SM particles, N → νe + e − , making it much longer-lived, cτ (cid:46) km. In addition, N is mostly sterile, which naturallyleads to B ( Z (cid:48) → N N ) (cid:28) B ( Z (cid:48) → N { , , } N { , } ) ,and explains why cτ < cτ . For concreteness, we fix m ϕ (cid:48) = 1 GeV, forbidding fast N → N j ϕ (cid:48) decays and re-specting perturbativity limits on the dark scalar quarticcoupling λ Φ . ∆ a µ and BaBar – A discrepancy between the most pre-cise ∆ a µ measurement performed by the Muon ( g − col-laboration [43] and theoretical calculations [58–62] standsat more than . σ . In view of the efforts by the Muon ( g − collaboration to measure this quantity four timesmore precisely at FNAL [67], it is timely to reconsiderthe dark photon solution to the ∆ a µ puzzle [40]. Min-imal dark photon models are excluded by collider andbeam dump searches for Z (cid:48) → (cid:96) + (cid:96) − [68–71]. If a GeVdark photon decays invisibly, then it is subject to stronglimits from monophoton searches at BaBar [44]. Thisconstrains ε (cid:46) − for m Z (cid:48) < GeV by searching for a Recent lattice calculations [63] predict values closer to the ex-periment. However, this has been pointed out to lead to incon-sistencies with e + e − → hadrons data [64, 65]. For the latestconsensus in this field, see Ref. [66] BP MB ∆ a µ BB Acc α D m m m m | V | | V | | V | B ( Z (cid:48) → N j N k ) / % cτ /cm/eV /MeV / −
44 45 46 55 56 66 N N N A (cid:88) (cid:88) (cid:88) ( (cid:88) ) .
39 0 .
05 35 120 185 0 22 . . . × . . B (cid:88) (cid:88) (cid:88) (cid:88) .
32 0 .
05 74 146 220 13 . . .
15 11 0 .
48 1 . .
59 1 . × . . TABLE I. Benchmark points used in this study, where m Z (cid:48) = 1 . GeV and m ϕ (cid:48) = 1 . GeV always. Here, the V ij ≡ U ∗ D L i U D L j − U ∗ D R i U D R j are the mixing factors in Z (cid:48) N i ν j vertices, and α D = g X / π . Note that Z (cid:48) → ν ν is negligible for themixings considered. We refer to the MiniBooNE excess as MB, the BaBar excess as BB, and the accelerator experiments asAcc. The zeroes in BP-A are protected by a left-right symmetry ( Λ L = Λ R ). missing-mass resonance produced alongside initial-stateradiation (ISR), e + e − → γZ (cid:48) . In models where the Z (cid:48) de-cays semi-visibly inside the detector, B ( Z (cid:48) → vis + /E ) (cid:39) ,this limit can be relaxed. This was proposed in the con-text of inelastic DM models in Ref. [72], and later criti-cized in a more conservative analysis [73] (see also [74–76]).In our model, however, the mechanism put forwardin Ref. [72] is improved, as more visible energy is de-posited in the detector. For the bound to be relaxedabove the central value to explain the ∆ a µ anomaly, thedetection inefficiency for the Z (cid:48) decay products in ISRevents ought to be at most . . Note that in virtuallyall ISR monophoton events the produced Z (cid:48) promptly de-cays into N and/or N states, which subsequently leadto one or more e + e − + /E vertices. Such additional par-ticles are hard to miss in the barrel-like BaBar detec-tor, which operates with a . T magnetic field. In fact,after produced, all N states decay already inside thedrift chamber, while for BP-(A,B), we find that, for atypical . GeV N energy, (79 , of N states decaybefore the electromagnetic calorimeter (ECAL), followedby (11 , . inside the ECAL, and (8 . , . in themuon detection system. Fully invisible decays are rareand satisfy B ( Z (cid:48) → N N ) + B ( Z (cid:48) → N N ) × P escape N (cid:46) . × − for the BPs. Visible decays are also negligible, B ( Z (cid:48) → (cid:96) + (cid:96) ) (cid:46) O (10 − ) . Pseudo-monophotons at BaBar –
The dominant pro-duction of dark particles in e + e − colliders is s-channelpair production of HNLs due to the large values of α D ε .In particular, the process e + e − → Z (cid:48)∗ ( or Υ( nS )) → N ( N → N e + e − ) , (4)could fake monophoton signatures when the N decaysinside the BaBar ECAL. These events could contribute tothe large missing mass ( M ≡ s − √ sE CMγ ) monopho-ton sample at BaBar, where we point out that a mildexcess is observed in the GeV < M < GeV region.For an integrated luminosity at BaBar of 15.9 fb − in √ s = 10 . GeV and . fb − in √ s = 10 . GeV,BP-(A,B) predict a total number of single pseudo-monophoton events of (3 . , . × × P γN × ε B , (5)where ε B is the final detection and selection efficiencyof the monophoton analysis at BaBar, not including
25 30 35 40 45 50 55 60 M (GeV )10 E v e n t s / ( . G e V ) e + e − → N N N → N ( ee misID −→ γ ) fullbkg (611 events)sig (53 events) BaBar data (653 events) FIG. 1. BaBar monophoton data at large M = s − E ∗ γ √ s .The background prediction quoted by the collaboration (red)is added to the best fit prediction in our BP-B (blue) in thesolid black line. Event numbers are for entire HighM region( GeV < M < GeV ). the probability P γN that the N states decay inside theECAL and get reconstructed as a photon. For the ISRanalysis, ε ISR (cid:39) . − . , depending on M . In ourpseudo-photon case, however, it is impossible to estimate ε B without a dedicated detector simulation and the ma-chine learning algorithm utilized by BaBar. Nevertheless,we fit our model prediction to data, which will give anestimate of the value of P γN ε B required to explain theexcess in the model. Since backgrounds are much largerthan our signal above M > GeV , our fit uses onlythe data in GeV < M < GeV , where a to-tal of events are observed on top of a prediction of background events. Floating P γN ε B for BP-B, weminimize a binned Poisson likelihood, assigning a normalization uncertainty on the background model. Wefind a . σ (2 . σ ) preference for 53 signal events. Ourbest-fit point in BP-B is shown in Fig. 1, where eventswere selected if θ ee < ◦ , and the boost and azimuthalangle cuts were implemented as in Ref. [44]. This cor-responds to a total number of pseudo-monophotonevents, before any selection cuts. Finally, since bothBPs predict very similar shapes, we can make use ofEq. 5 to find P γN ε B (cid:39) (0 . , . . A dedicated anal-ysis at BaBar would be able to determine if such num-bers are experimentally justified. We also note that ourpseudo-monophoton rate is compatible with constraintson B (Υ(1 S ) → γ + /E ) < . × − at C.L. atBaBar [77], provided the e + e − → γ mis-ID rate is lessthan (100 , for BP-(A,B). MiniBooNE excess –
MiniBooNE is a mineral oilCherenkov detector in a predominantly ν µ beam with (cid:104) E ν (cid:105) (cid:39) MeV. Recent results with improved back-ground analysis and larger statistics [78] report an excessof . ± . ( . ± . ) e -like events in ν ( ν ) mode.Initially designed to search for short-baseline oscillationsreported by the LSND experiment [79], MiniBooNE re-ports a much more significant . σ excess. The largetensions with global datasets in oscillation models [80–82] (see also [83–85]) prompts new scenarios to explainthe excess.We propose that the MiniBooNE excess arises fromthe decay products of HNLs produced in ν µ upscatteringinside the detector, ν µ + H → ( N , → N + e + + e − ) + H , (6)where H = { C, p + } is the hadronic target. The e + e − pairs with small angular aperture or large energy asym-metry mimic a single EM shower in the Cherenkov de-tector. This is similar to the upscattering explanationproposed in Ref. [41], but successfully achieves fast HNLdecay without infringing upon any bounds .A prediction of our signal on top of MiniBooNE neu-trino data is shown in Fig. 2 for our BP-B. In our sin-gle generation approximation, the upscattering cross sec-tion is proportional to | V j | α D ( eε ) , where | V j | ≡| U ∗ D L U D L j − U ∗ D R U D R j | is the mixing factor in the ν N j Z (cid:48) interaction and takes O (10 − ) values. The scat-tering is predominantly electromagnetic via Z (cid:48) exchange,and due to the large values of α D ε , no interference withthe SM Z is observed. This, together with the purely vec-torial couplings of the Z (cid:48) , explains why the signal prefersto be forward with respect to charge-current quasi-elasticscattering. We note that scattering on protons is dom-inant, and that the angular spectrum predictions canimprove when nuclear effects and higher Q scatteringregimes are included. The produced e + e − that con-tributes to the excess has a small invariant mass, with m ee < m , − m . If m ee is too large, it contributes tothe NC π dataset, where an excess is also observed [95].We estimate the overall detection and signal selectionefficiency for our BPs to be (cid:39) . Although many up-scatterings lead to N → ( N → N e + e − ) e + e − , we donot include these double vertex events as a large fractionof them would be excluded by the MiniBooNE cuts. Old accelerator anomalies –
Many accelerator experi-ments in the 80s and 90s searched for ν µ → ν e transitions In Refs. [86–89], a similar idea was proposed in the context ofa transition magnetic moment, which closely resembles the lightdark photon models later studied in Refs. [30, 41, 90]. Suchscenarios predict exclusively forward signatures, cos θ > . .Other models with scalars decaying to e + e − have been discussedin Refs. [91–94]. at short-baselines, with some of them observing signifi-cant excesses. While a neutrino oscillation interpretationof these results is excluded, they can be explained withinour model, where the energy dependence and signal char-acteristics differ from those of oscillation. The largest de-viation was observed by the PS-191 experiment at CERNusing a E peak ν ∼ MeV ν µ beam and the fine-grainedECAL component of their detector. They observed anexcess of ± e -like events on a background of ± events, amounting to a σ significance [45, 96]. All ex-cess events contained a scattering vertex, followed by anelectromagnetic shower < mm away. A follow-up ex-periment, E-816 [46], was designed to test the PS-191anomaly at the Brookhaven National Laboratory (BNL)with a wide-band beam of mean energy (cid:104) E ν (cid:105) (cid:39) . GeV.E-816 also reported an excess of e -like events with a smallvertex-shower separation of < . mm, although at alower significance of (cid:38) σ due to larger systematic er-rors [46]. In our model, these excesses can be explainedby ν µ upscattering to N , which decays very fast to over-lapping or energy-asymmetric e + e − pairs, fitting the ex-ponential drop of events as a function of vertex-showerseparation. PS-191 and E-816 observed a larger excessthan MiniBooNE, which could be explained by the larger N upscattering rate (BP-B) or solely due to differentsignal reconstruction (BP-A).Other experiments with E peak ν (cid:38) GeV reported noexcess, namely E-734 [97] and E-776 [98, 99]. The strin-gent cuts against π backgrounds would veto most of our e + e − pairs and weaken the constraint. Another set ofbounds come from high energy experiments, such as NO-MAD [100], with (cid:104) E ν (cid:105) (cid:39) GeV, and CCFR [101] andNuTeV [102], both with (cid:104) E ν (cid:105) (cid:39) GeV. Their bounds,although very strong under the oscillation hypothesis, aremuch weaker for our model due to the log E ν growth ofthe Z (cid:48) mediated neutrino-nucleus cross-sections in com-parison to the linear E ν growth in the SM. Finally, wenote an unexplained excess of positron events observedat NOMAD [100] in a sideband sample of events con-taining showers far from the scattering vertex or thathad failed kinematic cuts. Such positrons are predictedin our model as coming from asymmetric e + e − pairs inthe late decays of our HNLs.We also note an intriguing excess reported by CCFRin the search for HNLs produced in scattering [47, 48,103, 104]. The experiment saw evidence for double-vertexevents with NC/NC events over an estimated overlaybackground of ± . stat. ) ± . syst. ) . A double-vertexevent was defined as one in which there were “two distinctand separate shower regions”, and NC/NC refers to twoneutral vertices, as opposed to NC/CC events, whereina second vertex contained a muon candidate. No excesswas observed in the NC/CC, which disfavored standardinterpretations with HNLs that have large branching ra-tios to muons. In our model, only NC/NC events appear,mainly from upscattering into N , which immediately de-cays into N e + e − , with the subsequent N → N e + e − decays typically happening after a few meters at CCFR . . . . . . E vis / GeV0255075100125150 E x ce ss e v e n t s m = 74 MeV, m = 146 MeV, m = 220 MeVcoh N → N (178 events)incoh N → N (334 events)coh N → N (26.90 events)incoh N → N (53.14 events) − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . θ E x ce ss e v e n t s m = 74 MeV, m = 146 MeV, m = 220 MeVcoh N → N (178 events)incoh N → N (334 events)coh N → N (27 events)incoh N → N (54 events) FIG. 2. MiniBooNE low energy excess and our model prediction in BP-B for ν µ upscattering into N → N e + e − (blue) and N → N e + e − (pink) in BP-B. The incoherent (filled) and coherent (hashed) scattering contributions are shown separately. energies. This leads to good agreement with the to m vertex-shower separation observed, given the typi-cal N energies of GeV. A naive scaling of the crosssections shows that the normalization is compatible withthe rate at MiniBooNE and PS-191.
IV. DISCUSSION AND CONCLUSIONS
Let us remark that our BPs satisfy all existing experi-mental constraints, including decay-in-flight bounds fromPS-191 [96, 105]. Searches for peaks in the muon spec-trum in π + /K + → µ + N j [106, 107] are also satisfied dueto strong vetoes against visible energy in the detector,as discussed in Ref. [32]. Intriguingly, the latest resultsfrom K + → e + N j searches at NA62 [108] indicate anexcess at m N = 346 MeV, with | U ej | (cid:39) . × − at2.2 σ (3.6 σ ) global (local) significance. Our model can ac-commodate this hint by identifying N with the requiredHNL and switching on the mixing with the electron neu-trinos. To take into account the visible decays of ourHNL, the required | U ej | is enhanced by a factor ∼ for ∼ ns lifetimes, as quoted by the experiment. For ourBPs, we also expect to see an excesses in the muon sector,depending on the K + → µ + N j efficiency at NA62.There is some freedom in the choice of the HNL pa-rameters while keeping the same key phenomenologicalfeatures, e.g. HNL decay length and Z (cid:48) branching ratios.For the dark photon parameters, the situation is moreconstrained. For instance, lower values of m Z (cid:48) , such as GeV with ε = 3 × − are possible, and decreasethe required α D | V j | couplings to explain MiniBooNE,PS-191, and BaBar by a factor of (1 . ∼ . . Goingmuch below m Z (cid:48) = 1 GeV leads to more forward angulardistribution at MiniBooNE and introduces tension withneutrino-electron scattering constraints [90]. A surveyof existing bounds and additional BPs are provided inAppendices B and A.We also want to highlight the left-right symmetry in BP-A, as in that case the lightest HNL ν has vanishinginteractions with the Z (cid:48) , except for the | V | vertex. In-cidentally, N could lie at the keV scale, and may be acandidate for non-thermal dark matter [109].Direct searches for our MiniBooNE explanation canbe performed at the Short-Baseline Neutrino program atFNAL [110, 111], which comprises three Liquid Argondetectors: SBND, µ BooNE, and ICARUS. Specifically,for BP-(A,B) we predict that µ BooNE [112] would see atotal number of ∼ neutrino upscattering events into N and (0 , events into N , before any efficienciesand for a total N POT = 13 . × . While the formerwould contain a single e + e − pair, the latter events wouldconstitute double vertex events with (cid:38) cm separa-tion. Around of the total number of events are dueto coherent scattering, and leave no visible proton tracks.Dedicated studies of the e + e − invariant mass, as well assearches for the double-vertex events would help discrim-inate our hypothesis from other dilepton MiniBooNE ex-planations. Other direct searches can be performed atthe NA62 kaon facility [113]. The decays of GeV/ckaons to K + → (cid:96) + α N i followed by N j → N k e + e − wouldconstitute a background-free signature, similar to the oneproposed in Ref. [32]. The new physics events would ap-pear as a displaced e + e − vertex with peaked kinematics,where ( p K − p (cid:96) ) = m j , ( p K − p (cid:96) − p ee ) = m k , and p ee = ( p e − + p e + ) ≤ ( m j − m k ) . The production rate iscontrolled by | U µj | , where for BP-(A,B) we predict a to-tal K + → µ + ( N → N e + e − ) event rate of (1970 , for N K = 2 . × fiducial kaon decays and an overall acceptance [114, 115].The dark photon can be searched for in the ISR eventsat BaBar, Belle-II [73, 116], and BESIII [117] by re-laxing the vetoes on additional e + e − pairs in the de-tector. The large value of ε required for the ∆ a µ ex-planation yields several hundred events at BaBar. Di-rect N j N k pair production, as well as Higgstrahlung e + e − → ϕ (cid:48) Z (cid:48) , would also appear as multiple displaced e + e − vertices at B -factories, and in the fixed-target ex-periments NA64 [118, 119] and LDMX [120], providing abackground-free signature for semi-visible dark photons.In summary, this letter provides an explanationto some of the most prominent low energy anoma-lies, including the MiniBooNE excess and the ∆ a µ anomaly. The phenomenological signatures we presentedare achieved in a renormalizable model which extendsthe SM by an anomaly-free U (1) (cid:48) gauge symmetry anda dark neutrino sector. The model is able to repro-duce the correct scale for the light neutrinos, albeit withsome level of fine tuning. Phenomenologically, our sce-nario only requires a semi-visible GeV-scale dark photonthat couples to O (100 MeV ) HNLs. We show that thedark photon not only evades sensitive searches for miss-ing mass resonances at BaBar, but can actually explain amild but continuous excess seen in the data thanks to thepseudo-monophotons from N → N e + e − decays. Dueto the large kinetic mixing required by ∆ a µ , such eventsnaturally arise from s-channel e + e − collisions producingHNLs. We point out that e -like events from upscatter-ing are better able to explain past anomalies reportedby PS-191 and E-816, compared to those from excludedoscillation hypotheses. Also curious is the prediction of O (2 cm ) lifetime for N , as it leads to double vertexevents at neutrino experiments and is compatible with asignificant excess reported by CCFR. The novel interplaybetween portal couplings and exotic decay signatures inour model offer striking signatures at current and up-coming experiments. Observations of displaced verticesat kaon and neutrino experiments, as well as the decays ofa semi-visible dark photon, would provide confirmationof our model. ACKNOWLEDGMENTS
It is a pleasure to acknowledge discussions withCarlos Argüelles, Martin Bauer, Evgueni Goudzovski,and Mike Shaevitz. We thank Maxim Pospelov fordiscussions and for pointing out additional constraints inan earlier version of this draft. This project has receivedpartial funding from the European Union’s Horizon2020 research and innovation programme under theMarie Sklodowska-Curie grant agreement No. 690575(RISE InvisiblesPlus) and No. 674896 (ITN Elusives)and the European Research Council under ERC GrantNuMass (FP7-IDEAS-ERC ERC-CG 617143). Theresearch at the Perimeter Institute is supported in partby the Government of Canada through NSERC and bythe Province of Ontario through Ministry of EconomicDevelopment, Job Creation and Trade, MEDT. AA isfunded by the UKRI Science, Technology and FacilitiesCouncil (STFC).
Appendix A: Details on Benchmark Points
In the main text we have focused only on the phe-nomenological aspects of our model, giving two BPs thatcan resolve the low energy anomalies. In this appendix,we offer more details on the model side, giving the vertexfactors for each relevant interaction that can be used tocompute physical observables. The BPs in the main textwere given in terms of a model with a single generationof active neutrinos, n = 3 sterile states and d = 1 darkvector-like fermions. We also present two additional BPsto illustrate the ranges of the HNL masses compatiblewith the phenomenology discussed. In particular, BP-C indicates the smallest scale of m and m which leadto sufficiently fast N and N decays. With BP-D, weillustrate the features of heavier masses.Following Eq. 2 in the main text, the full mass matrixis given as, ν cf M D M D M D M D M L Λ R M D M L Λ R M D M Λ L Λ R L Λ L Λ L M X R Λ R Λ R M X ˆ ν f , (A1)where now ˆ ν f ≡ (cid:16) ˆ ν cα ˆ ν cN ˆ ν cN ˆ ν cN ˆ ν cD L ˆ ν D R (cid:17) T . Thevalues for the mass matrix parameters used for our BPsare given in Table III.In the mass basis, HNLs mixing with the different fla-vors is given in Table IV. To clarify the nature of ourneutrino couplings to the neutral bosons, we write the ex-plicit vertices in the neutrino mass basis using the flavorgauge boson basis. To leading order in χ and taking lightdark photons: Z µ = Z µ + s W χX µ and Z (cid:48) µ = X µ − s W χZ µ .The interactions are given by L int ⊃ g c W Z µ ˆ ν m γ µ ( CP L − C † P R )2 ˆ ν m (A2) + g X X µ ˆ ν m γ µ ( V P L − V † P R )2 ˆ ν m + h ˆ ν m ( HP L + H † P R )2 √ ν m + ϕ ˆ ν m ( SP L + S † P R )2 √ ν m , where ˆ ν m is the mass eigenvector and P L,R = (1 ∓ γ ) / .The vertex factors are defined as C = U † α U α , (A3) V = U † D L U D L − U † D R U D R ,H = U TN Y U α + U Tα Y T U N ,S = U TN ( Y L U D L + Y R U D R ) + ( Y L U D L + Y R U D R ) T U N . We show the relevant vertex factors for dark bosons in ourBPs in Table V. For all BPs, we take m Z (cid:48) = 1 . GeV, m ϕ (cid:48) = 1 GeV and ε = 4 . × − . The mixing sin θ is assumed to be negligible for our BPs. The HNLs withmasses above m Z (cid:48) , namely N and N , are mostly in thesterile direction, with | V jk | (cid:28) , and | U N j | , | U N j | ∼O (1) for j = 7 , .The phenomenology of BP-C is similar to BP-B, with cτ (cid:39) . cm and cτ (cid:39) . mm. Notably, it repre-sents the smallest scale of HNL masses with lifetimesthat are compatible with the old accelerator anomalies,although it requires a slightly larger α D . This point alsoallows for the lightest scalar ϕ (cid:48) . On the other hand, BP-D features considerably larger masses with the largest N lifetime, cτ (cid:39) cm, and smallest N lifetime ofany point, cτ (cid:39) km. Displaced vertices would beslightly enhanced here, although the heavier masses re-sult in slightly worse distributions at MiniBooNE, morepeaked at lower energies.As mentioned in the main text, our model is also com-patible with hints of a mild excess at NA62 and we il-lustrate this with BP-D. We identify the MeV HNLas N , and turn on mixing with the electron neutrinos.Taking the Yukawa couplings in the electron sector as Y e (cid:39) . Y µ , or M eD i (cid:39) . M µD i for i = (1 , , , weobtain the mixings | U e | (cid:39) , | U e | (cid:39) . × − , | U e | (cid:39) . × − . (A4)It is important to note that the bounds from NA62 onboth | U e | and | U e | are weakened due to the fast de-cays of N and N . For N with lifetimes ∼ ns, theexperiment expects a weakening of the bound by a factor ∼ [108] implying an effective | U e | (cid:39) . × − .The electron mixing requires two active light neutri-nos, ˆ ν and ˆ ν . With our chosen Yukawas, ˆ ν is masslessand mostly in the ν e direction, with ˆ ν mostly in the ν µ direction. As we do not consider the full × mass ma-trix with three active light neutrinos, we do not attemptto reproduce the structure of the PMNS matrix, but notethat this can be achieved with appropriate choices ofthe Yukawa couplings in the active sub-block. The scat-tering cross-section at MiniBooNE is now proportionalto (cid:80) i =2 , | U µi V ij | α D ( eε ) (cid:39) | U µ | | V j | α D ( eε ) , sincethe | V j | mixings are negligible for massless ˆ ν . Appendix B: Survey of Existing Constraints a. Electroweak precision observables
An assessmentof the impact of kinetic mixing on electroweak precisionobservables (EWPO) requires a global fit to collider andlow energy data. This was performed in Ref. [121], wherea model independent bound on ε was derived. For m Z (cid:48) (cid:28) M Z , the authors find ε < . × − at C.L,just above our value of ε = 4 . × − . As a sanity check against more recent data, we also directly compute theoblique parameters S , T , and U [122] to leading order in ε = c W χ and µ ≡ g X v ϕ /M SM Z , neglecting the impact ofrunning in the dark couplings and corrections from darkfermion loops. For all our BPs, these are [123–125] S (cid:39) s W ε (1 + µ ) /α = 0 . , (B1) T (cid:39) − s W χ µ /α = − . × − , (B2) U (cid:39) s W ε /α = 0 . . (B3)Clearly, this is compatible with the current bounds of T < . and S < . at 95% C.L. [57]. The constraintson S can be much stronger when fixing T = 0 and U = 0 ,which would be mostly driven by the to σ discrepan-cies observed between direct M W measurements and theglobal best fit point. We plan to return to this issue infuture communication [56]. b. Deep-inelastic scattering constraints Recently,Ref. [126] appeared setting new model-independent con-straints on dark photons using ep + scattering data fromHERA [127]. At C.L., the authors find that ε (cid:46) . × − , in tension with our BPs. As the authorsdiscuss, inclusion of other datasets weakens the bound,which signals a mild tension between HERA and otherexperiments. Finally, we note that a naive rescaling ofthe constraints on contact interactions performed by theZEUS collaboration [128], where the probability distri-bution functions were allowed included in the fit, leadsto bounds that are weaker by a factor of ∼ than theones quoted by Ref. [126]. While HERA is certainly sen-sitive to our model at some level, we believe that it isnot excluding it at the C.L. Nevertheless, a trivialmodification to our setup to accommodate such bound isto lower m Z (cid:48) = 1 GeV. c. Z → invisible Dark fermions can be produced inthe decays of SM-like Z bosons via its couplings to thedark current. This is induced by kinetic mixing, and toleading order in χ it is L ⊃ Z µ g X s W χJ µX . (B4)This coupling is relevant in our model since g X ε is not sosmall. A constraint can be derived from LEP measure-ments of the Z boson decay width [129] and constrains Γ Z → inv < MeV. The largest new physics decay modeis Z → N j N k , for j, k > , which even without requiringthe HNLs to be invisible, yields Γ Z → N j N k (cid:39) | V jk | G F m Z √ π (cid:18) g X s W εg (cid:19) (B5) (cid:39) . MeV (cid:18) α D | V jk | ε . × − (cid:19) , safely below the current constraints even for the largest α D couplings, as it can be shown that (cid:80) j,k | V jk | = 2 .Another relevant process is Z → Z (cid:48) ϕ (cid:48) . Neglecting the BP MB ∆ a µ BB Acc α D m m m m | V | | V | | V | B ( Z (cid:48) → N j N k ) / % cτ /cm/eV /MeV / −
44 45 46 55 56 66 N N N A (cid:88) (cid:88) (cid:88) ( (cid:88) ) .
39 0 .
05 35 120 185 0 22 . . . × . . B (cid:88) (cid:88) (cid:88) (cid:88) .
32 0 .
05 74 146 220 13 . . .
15 11 0 .
48 1 . .
59 1 . × . . C (cid:88) (cid:88) (cid:88) (cid:88) .
76 0 .
05 62 110 180 13 . . . . .
019 0 .
23 70 0 .
15 1 . × . . D (cid:88) (cid:88) (cid:88) ( (cid:88) ) .
11 0 .
05 275 346 435 1 .
44 75 . . .
021 13 0 .
060 0 .
13 87 0 .
023 4 . × . . TABLE II. Illustrative benchmark points (BP-C and BP-D). For ease of comparison, we report also the values for BP-A andBP-B. For all points, m Z (cid:48) = 1 . GeV. Here, the V ij ≡ U ∗ D L i U D L j − U ∗ D R i U D R j are the mixing factors in Z (cid:48) N i ν j vertices, and α D = g X / π . Note that Z (cid:48) → ν ν is negligible for the mixings considered here. We refer to the MiniBooNe excess as MB,the BaBar excess as BB, and the accelerator experiments as Acc. The zeroes in BP-A are protected by a left-right symmetry( Λ L = Λ R ). Table of Theory ParametersA B C D m D / eV . − . . . m D .
278 1 . − .
635 6 . m D . − . − . − . M / eV − . − . − . . M .
10 6 .
00 5 .
07 6 . M − . − . − . − . L / eV − .
39 3 .
75 3 .
51 15 . L . . . . L .
00 0 .
00 12 . . R / eV − . − . − . − . R . . . − . R .
00 0 . − . . M X / eV − .
21 1 .
96 1 .
56 3 . TABLE III. Theory parameters for model. final state masses, we find Γ Z → Z (cid:48) ϕ (cid:48) = πα D t W ε M Z (cid:39) keV (cid:18) α D ε . × − (cid:19) , (B6)also satisfying the constraints independently of the fateof ϕ (cid:48) and Z (cid:48) in the detector. d. h → invisible Searches for Higgs decays to in-visible have been performed by CMS [130] and AT-LAS [131]. Latest preliminary results by ATLAS requirethat B ( h → invisible ) < . at 95% C.L. [132], which forthe SM value Γ SM h (cid:39) . MeV, implies Γ h → inv < . MeV. This constrains the Yukawas and scalar parame-ters of the theory.Firstly, we consider h → N i N j neglecting scalar mix-ing. Saturating the bound, we find Γ h → N j N k (cid:39) s θ | H jk | m h π = 0 . MeV (cid:18) | H jk | . × − (cid:19) (B7) Neutrino mixing A B C D | U µ | / − . . . | U µ | . . | U µ | .
28 14 . . . | U N | / − . . . . | U N | .
162 0 . . | U N | .
14 11 . . . | U N | / − . .
79 1 .
91 3 . | U N | . . . | U N |
398 12 . .
45 23 . | U N | / − .
65 0 | U N | .
75 0 | U N | .
51 0 | U D L | / − .
244 0 .
371 1 .
43 0 . | U D L | .
00 5 .
57 5 .
19 4 . | U D L | .
54 4 .
04 3 .
36 4 . | U D R | / − .
244 0 .
749 1 .
44 0 . | U D R | .
00 4 .
33 4 .
71 5 . | U D R | .
54 4 .
84 3 .
76 4 . TABLE IV. Neutrino mixing parameters for our BP-A, B,and C. Note that U D L = U D R for BP-A due to Λ L = Λ R . which does not lead to strong constraints on our modelgiven that the largest Yukawa we have is for BP-D, whereit is a few − .The parameters in the scalar potential are also sub-ject to constraints. We consider h (cid:48) → ϕ (cid:48) ϕ (cid:48) induced bythe scalar portal coupling λ Φ H . A direct computationneglecting final state masses leads to, Γ h → ϕ (cid:48) ϕ (cid:48) (cid:39) λ H v h πm h = 0 . MeV × (cid:18) λ Φ H . × − (cid:19) , (B8)which can be interpreted as a strong constraint onour scalar mixing angle θ (cid:39) ( λ Φ H / λ H ) × ( v Φ /v H ) < . × − ( v Φ / MeV ) . This value is currently ordersof magnitude below current sensitivity of K L → π ϕ (cid:48) Z (cid:48) vertex A B C D | V | / − . . . | V | . . . . | V | . . | V | / − .
43 0 . . | V | . . | V | .
60 0 .
184 0 . | V | . .
23 1 . | V |
909 869 702 899 | V | .
31 1 .
59 0 . ϕ (cid:48) vertex A B C D | S | / − .
205 7 .
87 2 .
17 1 . | S | / − . .
675 0 .
273 5 . | S | .
31 0 .
509 0 . | S | .
163 0 . . . | S | / − . . .
841 0 . | S | . .
394 0 . . | S | .
181 0 .
418 11 . . | S | .
20 1 .
58 0 . | S | .
444 0 .
744 50 . . | S | .
668 0 .
258 10 . . TABLE V. The vertex factors entering in Z (cid:48) ν i N j ( | V ij | ) and Z (cid:48) N j N k ( | V jk | ) interactions, as well as in ϕ (cid:48) ν i N j ( | S ij | ) and ϕ (cid:48) N j N k ( | S jk | ) interactions, as defined in A3. searches at KOTO or K + → π + ϕ (cid:48) searches at NA62.Decay to a pair of HNLs can also proceed via the darkYukawas if scalar mixing is present. Neglecting the finalstate masses, we find Γ h → N j N k (cid:39) s θ | S jk | m h π = 3 eV | S jk | (cid:18) s θ . × − (cid:19) , (B9)where s θ ≡ sin θ is chosen according to B8 for v ϕ =500 MeV. This clearly satisfies the bound as it can beshown that (cid:80) j,k | S j,k | = 4( | Y L | + | Y R | ) . Similarly,the decay h → Z (cid:48) Z (cid:48) is possible and may appear invisiblesome fraction of the time. Nevertheless, the tree-levelrate is Γ h → Z (cid:48) Z (cid:48) (cid:39) α D s θ m h π = 0 . eV α D (cid:18) s θ . × − (cid:19) , (B10)and loop-corrections from fermion loops are also negligi-ble. K → πϕ (cid:48) The KOTO experiment at J-PARC [133] hasset stringent constraints on B ( K L → π /E ) . Initial hintsof a signal in the latest unblinding [134] have later beenrevisited due to unexpected charged kaon backgrounds and signal mis-identification [135, 136]. The hinted val-ues led to branching ratios much larger than the SM pre-diction [137], and prompted several new physics stud-ies [35, 138–141]. In light of the new backgrounds andgiven stringent constraints on the scalar mixing foundabove, we refrain from trying to explain these events.We note that KOTO, as well as NA62, are only sensi-tive to singlet scalar emission in K → πϕ (cid:48) decays formixings of order s θ ∼ − , which for our small v Φ arealready excluded due to h → ϕ (cid:48) ϕ (cid:48) decays. A more ex-otic scalar sector could be invoked to explain this excess,where a new invisible real scalar S could be hidden underbackground at NA62 if m S ∼ m π [142], and could lead tosignatures at KOTO without violating the Grossman-Nirconstraints [143]. e. Meson → invisible We consider the decays of vec-tor meson states due to the vector nature of the dark pho-ton couplings. The best current bounds are at the level of B ( J/ψ → inv) < . × − at BES [144], and B (Υ(1 S ) → inv) < . × − at BaBar [145]. The branching ratiosinto HNLs, B ( V → N j N k ) , may be still be slightly abovesuch values, provided a sufficient number of the producedHNLs decay semi-visibily. In general, the branching ra-tio for the quarkonium states (V) used throughout ourarticle is B ( V → N j N k ) = α | V jk | ( g X εQ ) τ V M V f V ( M V − m Z (cid:48) ) = | V jk | α D × . × − for V = J/ψ . × − for V = Υ(1 S )1 . × − for V = Υ(2 S )1 . × − for V = Υ(3 S )0 . × − for V = Υ(4 S ) , (B11)where we neglected the final state masses, and took m Z (cid:48) = 1 . GeV and ε = 4 . × − . The decayconstants, f Υ( nS ) = 498 , , MeV for n = 2 , , ,were extracted from existing V → e + e − measurementsin Ref. [146]. For the fully invisible states N N (aswell as for light neutrinos) the mixing factor V jk is suf-ficiently small to avoid the constraints. Production of N N is the next largest contribution and it still satis-fies the most stringent limits from BES, since in all BPs | V | × P escape N < , where P escape N is the probabilityfor N to escape detection. f. Pseudo-monophotons As discussed in the maintext, s-channel e + e − collisions at BaBar can lead topseudo-monophoton events. It is not possible to ex-tract a constraint from this without a dedicated detec-tor simulation, as it relies on experimental details suchas the efficiency to reconstruct our e + e − as a photon,and on the specifics of the machine learning algorithm.In the main text, however, we proceeded to understandif it is at all feasible to explain a mild excess observedin the monophoton data. By finding a best-fit valuefor the normalization of events that are “photon-like",we have asked whether such rate is possible within our0model and whether the efficiencies it requires are reason-able. Here, photon-like refers to events where a N N pair is produced in the interaction point, followed by a N → N e + e − decay inside the ECAL. When plottingour prediction, we required that the angle of separationbetween the electrons be less than θ ee < ◦ , and selectevents within the angular acceptance of the ECAL detec-tor. For our total pair-production rate, we include HNLsproduced in Z (cid:48) mediated s-channel e + e − collisions, wherethe e + e − → ( Z (cid:48) ) ∗ → N i N j cross section was found to be d σ e + e − → N i N j d cos θ CM (cid:39) | V ij | αα D ε πs ( s − m Z (cid:48) ) (1 + cos θ CM )2 , (B12)neglecting final state masses and where θ CM is the centerof mass angle between N i and the collision axis. We alsoinclude a contribution from e + e − → Υ( nS ) ( n = 2 , , ),followed by decay into HNLs. We use σ ee → Υ(2 S ) ( √ s =10 . GeV ) (cid:39) nb and σ ee → Υ(3 S ) ( √ s = 10 . GeV ) (cid:39) nb as well as B11. g. Higgstrahlung Another source of HNL productionat e + e − colliders is dark higgstrahlung. Due to thelarge dark coupling, the process e + e − → ϕ (cid:48) Z (cid:48) is impor-tant [76], with a differential cross section d σ e + e − → ϕ (cid:48) Z (cid:48) d cos θ CM (cid:39) πα α D ε sin θ CM ( s − m Z (cid:48) ) + 8 m Z (cid:48) s s ( s − m Z (cid:48) ) , (B13)where θ CM is the center of mass angle between the Z (cid:48) andthe collision axis. This process is therefore comparableto direct HNL production. Remaining agnostic aboutthe decay products of ϕ (cid:48) , but requiring the decay Z (cid:48) → N j N k , the ratio between direct and higgstrahlung HNLproduction in our BPs is | V jk | s ( s − m Z (cid:48) )(( s − m Z (cid:48) ) + 12 s m Z (cid:48) ) × B ( Z (cid:48) → N j N k ) , (B14) (cid:39) . × | V jk |B ( Z (cid:48) → N j N k ) . For production of N N pairs in ours BPs A and B, thisconstitutes a ratio of just above . Given that ϕ (cid:48) decayspromplty and visibly, especially into N states, we do notinclude this contribution in our monophoton discussion,but emphasize that this offers yet more visible signaturesat e + e − colliders. h. Υ(1 S ) → invisible + γ Vector meson decays tophoton plus missing energy are direct probes of ourpseudo-monophoton events. The full process is
Υ(2 S ) → π + π − (Υ(1 S ) → γ + /E ) , where the π + π − kinematics canbe used to identify the Υ(1 S ) state. The current limitsare quoted in terms of the BR into an invisible pseu-doscalar, Υ(1 S ) → γ + A , and into a pair of invisiblefermions, Υ(1 S ) → γχχ . Most relevant to us are thethree-body decay limits taken at the smallest χ masses( m χ → ), where BaBar [77] constrains B (Υ(1 S ) → γχχ ) < . × − , (B15) which was improved by Belle [147] to B (Υ(1 S ) → γχχ ) < . × − , (B16)all at the 90% C.L. More recently BESIII [148] has setthe strongest limits on the two-body process B ( J/ψ → γA ) < . × − , (B17)but since the missing mass in this process is fixed, theconstraint does not apply to us.When implementing these bounds on our model, weused the BRs in B11. For comparing the Υ(1 S ) → γ + /E constraints to the BaBar pseudo-monophoton rate, onlythe BaBar limit is taken into account, as P γN is adetector-dependent quantity, and, under the simplifyingassumption that it is constant in energy and angle, it isthe same for the two processes.Since the efficiencies for our pseudo-monophotonevents are different than in the s-channel productionmode, we use the limits above to obtain an upper-boundon the detector-dependent quantity P γN . Neglecting thesub-dominant N N contribution, BaBar sets a limit of B (Υ(1 S ) → N N ) ×P γN < . × − at 90% C.L, whichimplies P γN < (18 , , . , for BP-(A,B,C,D). Sincethe probability for N to decay inside the ECAL is knownto be (14 , , , for BP-(A,B,C,D) ( E N (cid:39) GeVfor s-channel production), we use the limit on P γN to findthe largest allowed e + e − → γ mis-ID rate for our expla-nation of the BaBar excess not to be excluded. Under theapproximation that it is independent of the kinematics,this rate is bounded from above by (100 , , , .Clearly this allows to explain the monophoton excesswhile remaining consistent with the Υ(1 S ) → γ /E . i. NA64 searches The fixed-target NA64 [118, 119]experiment also sets stringent limits on invisible darkphotons. The
GeV electrons can produce dark pho-tons via bremsstrahlung in interactions with the densebeam dump material, e W → e W Z (cid:48) , with W a Tungstennucleus. The bound from . × electrons on target isshown in Ref. [118] for invisible dark photons with massesas large as m Z (cid:48) ∼ . GeV. For the latter dark photonmass, it constrains ε (cid:46) − at 90% C.L. As discussed inRef. [119], the semi-visible decays of the dark photon canweaken the bound. For our BPs ( m Z (cid:48) = 1 . GeV), wedo not have access to the exact value of the invisible- Z (cid:48) constraint, but under a conservative assumption of linearscaling with m Z (cid:48) , we find that NA64 does not constrainour BPs provided ∼ of Z (cid:48) particles produced arevetoed due to the subsequent semi-visible decays of N , .For the HNLs produced in the decay of the highest en-ergy dark photons ( E N , ∼ GeV), we have a typicaldecay length cτ of O (10 m ) and O (1 m ) for N and N ,respectively. Note that the dark photons are producedinside the beam dump and decay promptly. Under theseconservative assumptions, we find that most HNLs de-cay before or within the instrumented ECAL of NA64,assumed to be a total of ∼ m. The presence of oneor more vertices of e + e − pairs would be vetoed from the1invisible- Z (cid:48) search due to visible showers in the ECAL,as well as in the additional veto detectors. j. Beam dump and decay-in-flight searches HNLsheavier than N in our model are unconstrained by decay-in-flight searches due to their short lifetimes. On theother hand, N is longer-lived and faces strong con-straints from HNL searches at PS-191 [96, 105]. If N has new interactions, such as in BP-B, it decays fasterthan in the minimal HNL models, and the constraintsfrom decays in-flight are modified (see, e.g. , the discus-sion in Ref. [149]). While N is produced in π, K → µN decays, which are controlled by | U µ | , its subsequent N → νe + e − decays in BP-B proceed mainly through Z (cid:48) exchange, which is controlled by | V | . In that case,we require | U µ | | V | < | U ∗ µ U e | PS (cid:32) √ G F m Z (cid:48) eεg X (cid:33) × F , (B18)where | U ∗ µ U e | PS is the bound quoted by the PS-191experiment. The factor F = 3 . converts the boundon Dirac to Majorana HNLs, and takes into accountthat PS-191 assumed only charged-current decays in theiranalysis.We note that there are additional production chan-nels for N than in the minimal HNL models. On topof the standard meson decays π, K → (cid:96)N , HNLs canalso be produced via kinetic mixing in ρ, ω → N i N j and π , η, η (cid:48) → γN i N j , where the vector meson decays domi-nate. These channels have been explored in the context oftwo-component fermionic dark sectors in Refs. [150–153].In this context, limits on new dark sector fermions thatdecay to e + e − + /E have been set using CHARM [154]and NuCal [155] data. With the effective field theoryapproach of Ref. [153], we see that for cτ N i (cid:39) cmor smaller such constraints are safely avoided due tothe short lifetimes. For the longer-lived N , however,a simplified re-scaling of the constraints from Ref. [153],where g / Λ → | V || U µ | G F ( eεg X / m Z (cid:48) ) , and g / Λ →| V || V | ( eεg X / m Z (cid:48) ) for BPs B and C, shows that theconstraints are satisfied. A re-analysis of the PS-191 con-straints including these new production mechanisms for N , both from vector meson decays as well as from thesecondary decays of N , N could set stronger constraintsin our parameter space, but is beyond the scope of thiswork.We would like to highlight an event found in PS-191and shown in Fig. 9 of Ref. [156]: it has two tracks in theinitial decay detector which subsequently shower in theECAL. While it is unlikely to be due to photons, as theywould not be recorded in the flash tubes, it could be dueto two electrons coming from an N decay. We do notelaborate further on this intriguing event. k. Peak searches – Searches for a missing mass in π/K → µN j decays set stringent limits on | U µ | . In ourmodel, the N and N states can be produced, but wouldlead to visible signatures inside the two most relevant ex-periments, namely E949 [106] and NA62 [107, 108]. As pointed out in Ref. [32], the small probability to miss ad-ditional energy deposition in these experiments togetherwith the stringent vetoes against K → µνγ ( ∗ ) back-grounds would, in fact, veto most of our events. ForE949, the detection inefficiency is estimated to be largerthan . , the typical photon inefficiency, and so theconstraints would be weakened by factors of (cid:38) . Asimilar argument can be made about NA62, where the e + e − signatures would have to be missed by several dif-ferent detector components. In this way, our BPs are notexcluded by peak searches, in particular BP-B where werely on a relaxation by a factor of ∼ on the E949 limiton | U µ | . Note that the mass of both N and N is al-ways below the mass interval constrained by both E949and NA62. This is important as N has a larger proba-bility to escape these detectors due to its O (10 m ) decaylengths in the laboratory frame. l. Lepton number violating searches Due to electronmixing, we predict large rates for neutrinoless double-beta decay in BP-D. In addition to the light neutrinos,heavy N j states also contribute and may dominate. Theircontribution contains large uncertainties as the momen-tum dependence of the nuclear matrix element is impor-tant at the O (100) MeV mass scale. Nevertheless, a naiverescaling of the effective mass m ββ (cid:39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 m i U ei m i / (cid:104) p (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B19)leads to m ββ ≈ meV for (cid:104) p (cid:105) = (100 MeV ) whenall matrix elements U ei are real. This is to be comparedwith the current experimental sensitivity of m ββ < − meV at KamLAND-Zen [157]. We interpret this asa suggestion that, unless strong cancellations due to theMajorana phases are at play, we predict observable ratesof neutrinoless double beta within current experimentalreach. Loop contributions [51] and a full three activeflavor treatment of the mixing matrix are also importantand should be studied in more detail.Lepton number violating kaon decays of the type K + → µ + ( N j → µ + π − ) are important for generic heavyMajorana neutrinos [13, 114, 158]. This signature pro-ceeds via charged-current branching ratios of N j , and itis much suppressed in our model whenever j > , where B ( N j → µ + π − ) ∼ − − − . For N , such decays canin principle have large branching ratios, but the long-lifetimes of N renders the experimental searches insen-sitive. Appendix C: Old Accelerators Experiments –Additional Details
We now provide additional details regarding the accel-erator experiments. a. PS-191
The PS-191 detector was made of × mm flash tube chambers interleaved with mm thickiron plates giving the detector a very fine granularity,2 mm of iron or ∼ of a radiation length. Thiswas used to distinguish photons, whose showers startedfurther from the vertex due to conversions, from elec-trons, which showered immediately. As shown in Fig. 3of Ref. [45], most single-shower events started within thefirst chamber, which corresponded to ∼ mm. Theanalysis was restricted to events above MeV, to avoid π backgrounds. The initial e -like shower sample con-tained a total 57 events, which, after cuts on energy anddistance between vertex and shower start, left a pionbackground of ± events.In our model, the most frequent upscattering events atPS-191 would produce a N , which immediately decaysinto N e + e − . The electromagnetic (EM) shower createdby this decay is then close to the upscattering vertex,and would explain the sharp drop in number of eventsas a function of the shower-vertex distance. The issueof normalization between the number of events requiredat PS-191 versus those observed at MiniBooNE, whichis between a factor to larger, can be explained bythe fact that not all N events in MiniBooNE count assignal. This is due to the additional decay of N , whichyields a total of four charged leptons that are very rarelymis-reconstructed as a single EM shower. Therefore, byincreasing | V | in comparison to | V | , one can increasethe ratio of signal events between PS-191 and Mini-BooNE. This happens in BP-B, where | V | (cid:38) . | V | ,although it is forbidden in BP-A due to the left-rightsymmetry. It should be noted that the coherent cross sec-tion in Iron is larger than in Carbon, and that the mostenergy-asymmetric e + e − pairs may be reconstructed asa one track plus one shower events. b. E-816 The E-816 experiment used the same fine-grained ECAL as PS-191. The number of events wasquoted as a function of the scattering vertex and the startof the shower, allowing for e/γ differentiation. This wasmeasured in units of 0.25 radiation lengths ( ∼ . mm).Any photons converting before this would fake electrons,although the exponential nature of the conversion makessuch events unlikely. The experiment searched for ν e ex-cesses in the one track one shower ( T S) sample. Toreduce the π background, showers with E (cid:46) MeVwere cut from the analysis. According to their simu-lations, such cuts eliminated ∼ of π s while onlyremoving ∼ of ν e s. After cuts, the π backgrounddropped to ∼ . of the ν µ interactions and was of theorder of the ν e contamination in the beam. The electronexcess was then given by the subtraction of the T Sevents due to pions and those due to intrinsic ν e back-ground from the remaining T S events. They found anexcess of ± . (stat.) ± (sys.) and quoted a signif-icance of . ± . σ .Similar to PS-191, E-816 would also count upscatter-ing events into N as signal when the e + e − s are overlap-ping or highly energy-asymmetric. The ratio of ν e -likeevents to ν µ -like events, R = ( ν e + ν e ) / ( ν µ + ν µ ) , ob-served at E-816, R observed /R expected = 1 . ± . , is com-patible but somewhat smaller than the one at PS-191, R observed /R expected = (2 ± . / (0 . ± . . The collab-oration attributed this to unknown systematic errors inboth experiments. c. E-734 E-734 at Brookhaven National Laboratoryran with peak energy E peak ν ∼ . GeV at a baseline of ∼ m, and searched for ν µ → ν e transitions [97]. Theexperiment utilized a filter program to remove events notcontaining a single electromagnetic shower within an an-gular interval θ e < mrad relative to the beam di-rection, with the remaining events scanned by physiciststo remove events with more than one shower or addi-tional hadronic activity. It is interesting to note thoseevents with one shower and an associated upstream ver-tex were used as a control sample of photons. After a cuton the energy, . < E e ≤ . GeV, shower eventsremained. The main backgrounds were identified to bepion production in NC interactions, charged pion pro-duction in inelastic CC processes, and those from ν µ − e scattering. Of particular relevance is their cut on theshower energy of E e < . GeV, reducing the sample to events. The final sample contained events in theenergy range . < E e ≤ . GeV.While the experiment saw no excess, we note that mostevents in our model would not have passed the morestringent cuts. This is mainly due to the larger ener-gies required by the experiment, but also due to the cutsin energy loss, dE/dx, of the shower. Our events wouldmost likely resemble those of the upstream photon-likesample. d. E-776
E-776, running with both a narrow-(NBB) [98] and wide- (WBB) [99] band beam of meanenergy . GeV, searched for ν e appearance km fromthe target. A fine grained ECAL consisting of planesof proportional drift tubes interleaved with in. ( ∼ . the radiation length) thick concrete absorbers wasutilized. A total of . × events were in the full sam-ple, and shower events were selected in a scan of thesample. After cuts, which included a requirement that E e > MeV, only events remained. Further cutson EM shower identification were made, e.g. the numberof hits in a cluster and the length of the shower. This lefta sample of events. To eliminate the π background,the differences in shower profile of pions and electronswere accounted for - the former being wider and moreasymmetric. This cut was quoted to have an efficiency of ∼ for rejecting π s at GeV. The final sample con-tained electron shower-like events, with the remaining constituting the π s. Accounting for the probabilityof pion-electron mis-ID gave . events. The observed events was consistent with the background prediction of ± . (stat.) ± . (sys.) events ( . ± . (sys.) from π s and . ± . (sys.) from ν e s in the beam), and noexcess was reported by the experiment.Due to the cuts on energy and, in particular, showerprofile, a large number of our events would be removedin the analysis, weakening the constraint on our model. e. CCFR CCFR searched for production of HNLswith a magnetized toroidal spectrometer-calorimeter,3and studied double vertex events from ν − N interac-tions. The sample at CCFR was selected using a neu-tral current (NC) trigger, whose threshold for energy de-position in the calorimeter was GeV. To make surethe primary showers were indeed from an NC vertex, itwas required that no muons penetrated past the end ofthe showers. Subsequent cuts selected events with a sec-ondary shower downstream of the first, and further cutsbased on kinematical considerations ensured the primaryand secondary showers were separated by an angle rela-tive to the beam of < mrad. The remaining eventswere categorized by those containing two neutral currentvertices (NC/NC), and those with a neutral current ver-tex followed by a charged current vertex (NC/CC), withthe latter being accompanied by a visible muon track inthe secondary vertex. For the NC/CC events, cuts ondistance between the vertices were made and events withseparation > λ I were selected, where λ I is the nuclearinteraction length and λ I ∼ . cm in Iron. This left events, which was consistent with the estimated back-ground of . ± . ± . . For NC/NC events, the back-grounds depended on the shower separation. For longseparations, (cid:38) λ I , the major background was due torandom overlay events, events in which independent neu-trino interactions appeared correlated. 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