EExplaining the MiniBooNE Anomalous Excessvia Leptophilic ALP-Sterile Neutrino Coupling
Chia-Hung Vincent Chang, ∗ Chuan-Ren Chen, † Shu-Yu Ho, ‡ and Shih-Yen Tseng § Department of Physics, National Taiwan Normal University, Taipei 116, Taiwan Korea Institute for Advanced Study, Seoul 02455, Republic of Korea Department of Physics, Faculty of Science,The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
Recently, the MiniBooNE experiment at Fermilab has updated the results with increased dataand reported an excess of 560 . ± . . σ ) in the neutrino operation mode.In this paper, we propose a scenario to account for the excess where a Dirac-type sterile neutrino,produced by a charged kaon decay through the neutrino mixing, decays into a leptophilic axion-like particle ( (cid:96) ALP) and a muon neutrino. The electron-positron pairs produced from the (cid:96)
ALPdecays can be interpreted as electron-like events provided that their opening angle is suﬃcientlysmall. In our framework, we consider the (cid:96)
ALP with a mass m a = 20 MeV and an inverse decayconstant c e /f a = 10 − GeV − , allowed by the astrophysical and experimental constraints. Then,after integrating the predicted angular or visible energy spectra of the (cid:96) ALP to obtain the totalexcess event number, we ﬁnd that our scenario with sterile neutrino masses within 150 MeV (cid:46) m N (cid:46)
380 MeV (150 MeV (cid:46) m N (cid:46)
180 MeV) and neutrino mixing parameters between 10 − (cid:46) | U µ | (cid:46) − (3 × − (cid:46) | U µ | (cid:46) × − ) can explain the MiniBooNE data. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] F e b . INTRODUCTION Since the groundbreaking discovery of the neutrino oscillations at the Super-Kamiokandeexperiment in 1998 , numerous measurements have provided clear evidence that neutrinoshave non-zero masses and the mass eigenstates are an admixture of the ﬂavor eigenstates .Even though the mass generation mechanism and the mass ordering are still unknown, it iswell understood the assumption that neutrinos are of three diﬀerent ﬂavors ( ν e , ν µ , ν τ ) withtwo mass splittings and three mixing angles gives a good ﬁt to most of the neutrino data,including solar neutrinos, atmospheric neutrinos, long-baseline, and reactor experiments .On the other hand, there are some long-standing anomalies which suggest the existenceof non-standard neutrinos. For instance, an excess of ¯ ν µ → ¯ ν e appearance observed by theshort-baseline experiment LSND collaboration  indicates the presence of a fourth-ﬂavorneutrino, most likely a sterile neutrino ν s , participating in the neutrino oscillation scenariowith a much larger mass splitting of eV scale. With the similar design L ν /E ν ∼ / MeV,where L ν and E ν are the travel distance and energy of neutrino, respectively, MiniBooNEat FermiLab is built up to conﬁrm or disprove the anomaly reported by LSND. Based onthe ν e and ¯ ν e appearance data collected from 2002 to 2019, MiniBooNE reports excessesof 561 events in neutrino mode and 77 events in anti-neutrino mode, which corresponds to4 . σ eﬀect in total . Combining with the LSND result, the signiﬁcance even reaches 6 . σ .Assuming one sterile neutrino and applying a two-neutrino oscillation model, MiniBooNEreports the best-ﬁt point for data with the mass splitting ∆ m = 0 .
043 eV and the mix-ing angle sin θ = 0 .
807 , requiring ∆ m (cid:38) .
03 eV at 90% C.L., in agreement withLSND. However, the introduction of an eV-scale sterile neutrino, while generates ν µ → ν e appearance, also gives rise to the ν e disappearance at the short-baseline experiment, whichunfortunately is not observed. That is to say, the parameter region in the sterile neutrinoscenario favored by the MiniBooNE result is incomparable with the global ﬁt for the otherneutrino data [5–7].Therefore, there is a signiﬁcant interest in alternative explanations of the excess [8–10].Intriguingly, the MiniBooNE detector is unable to distinguish the single electron signal ofa ν e charged-current quasi-elastic scattering ( ν e n → pe − ) from photons or a collimated e + e − pair. Several eﬀorts have been devoted to the possibility of light exotic particles thatdecay inside the detector into photons or e + e − pairs to camouﬂage the electron signals. Forexample, a plausible alternative is a decaying sterile neutrino that is produced in chargedmeson decays. Because of the mixings with the active neutrinos, the sterile neutrino can beproduced, if kinematically allowed, in the decay of mesons when the proton beam hits thetarget. The sterile neutrino could decay into standard model (SM) particles, e.g. ν s → ν β γ with β = e, µ , and τ [11, 12]. However, its lifetime is usually long enough such that ν s canbe regarded as a stable particle in the short-baseline experiments. If new interactions areintroduced, the sterile neutrino would have more decay modes and could decay within thelength scale of the MiniBooNE experiment, even decay promptly. Then, the decay of sterileneutrino into photons or e + e − pairs inside the detector could possibly provide the excessreported by MiniBooNE .However, it has been pointed out in Ref.  that it is diﬃcult for them to ﬁt both theangular and energy distributions of the excess events. The key obstacle is that if the lightnew particle decays visibly, the total momentum of the ν e -like products will be equal to thatof the light new particle. For this new particle to enter the MiniBooNE detector, the trackangle must be small and thus the angular spectrum of the excess events is forward-peaked.2 IG. 1: The illustration of our setup to explain the MiniBooNE excess electron-like events inthe (cid:96)
ALP model, where L is the travel distance of the sterile neutrino produced by the chargedkaon decays, D is the diameter of the MiniNooBE detector, and θ a is the scattering angle of the (cid:96) ALP produced from the sterile neutrino decay. If the angular aperture of electron-positron pairsproduced from the (cid:96)
ALP decays is suﬃciently small, they can be treated as electron-like events.
Nevertheless, the MiniBooNE data has signiﬁcant excess even for cos θ e < .
8. This tensioncan be alleviated if the new particle decays semi-visibly since the invisible product couldtake away some transverse momentum. Following this strategy, Ref.  proposes a scenariowhere the sterile neutrino decays into a photon and a light neutrino, ν s → ν β γ . The angulardistribution is still more forward-peaked compared to data. It was also proposed in Refs. [8,14] that the ν µ may scatter with nucleons inside the detector via new physics to produce asterile neutrino, which subsequently decays into e + e − pairs, mimicking excess events. Thescenario seems to have a less forward-peaked angular distribution of the excesses.In this work, we tend to explain the MiniBooNE excess by a sterile neutrino N D of massaround 100 ∼
400 MeV and a O (10) MeV leptophilic axion-like particle ( (cid:96) ALP), a . Thesterile neutrino is produced in the decay of kaon from the target via its mixing with the ν µ . Then, it travels about 500 m and decays semi-visibly into a muon neutrino and a (cid:96) ALP,which in turn decays into an electron-positron pair in the detector, as sketched in Fig. 1.Our calculation shows that it is possible to obtain a rather mild forward-peaked angulardistribution of excess. In general, the mass of an axion-like particle and its couplings to theSM ﬁelds are strictly constrained by beam-dump experiments, astrophysical observations,and rare decays of mesons. However, most of the productions of axion-like particles in theaforementioned experiments rely on the couplings to the SM quarks. Since we consider a (cid:96)
ALP that interacts with the SM leptons only, as a result, the relevant bounds are placed bysupernova 1987A, electron beam-dump experiment E137, and electron ( g − e anomaly. Wewill discuss these constraints later.The structure of this paper is organized as follows. In the next section, we introduce theeﬀective Lagrangian of the (cid:96) ALP, focusing on the couplings to electrons and photons. Wealso discuss the decay modes of sterile neutrino and (cid:96)
ALP. Sec. III is the discussion aboutthe constraints of parameters in our model, including the supernova 1987A, E137, electronmagnetic dipole moment anomaly, and rare kaon decay. In Sec. IV, we demonstrate how weestimate the excess of ν e -like events and show our ﬁts to the MiniBooNE results. The lastsection is devoted to discussion and conclusions.3 I. THEORETICAL FRAMEWORKA. Sterile Neutrino and Leptophilic ALP
In our setup, we add one Dirac-type sterile neutrino, ν D , to the SM neutrino sector. Asusual, the neutrino ﬂavor eigenstates could be transformed into the mass eigenstates by aunitary matrix U . Explicitly, one can express the neutrino ﬂavor eigenstate ν β as a super-position of the neutrino mass eigenstates ν jL , N D  ν β = (cid:88) j =1 U βj ν jL + U β N D , (1)where β = e, µ, τ, D and j = 1 , , (cid:96) ALP), a , which only couples to the leptons, but not quarks. What is relevant for us is theinteractions of (cid:96) ALP with the sterile neutrino and electron. Assuming, for simplicity, theinteraction is diagonal in the ﬂavor eigenstates of leptons, the eﬀective Lagrangian densitycan be written as  L a(cid:96) = − ∂ µ a f a (cid:16) c N ν D γ µ γ ν D + c e eγ µ γ e (cid:17) , (2)where f a is the (cid:96) ALP decay constant, and c N and c e are dimensionless parameters of orderof unity. Notice that the diagonal (cid:96) ALP-vector current interactions give no physical eﬀectdue to the conservation of the vector currents, thereby we omit ∂ µ a ¯ (cid:96)γ µ (cid:96) interactions.Plugging Eq. (1) into Eq. (2), we then obtain a mixing between the mass eigenstates ofthe sterile and active neutrinos with the (cid:96) ALP coupling L a(cid:96) ⊃ − c N ∂ µ a f a (cid:16) U D j U ∗ D N D γ µ γ P L ν j + H.c. (cid:17) , (3)where P L = (1 − γ ) is the left-hand project operator. This term will be responsible for thedecay of the sterile neutrino into the (cid:96) ALP inside the detector.Besides the above couplings, the (cid:96)
ALP can also interact with photons via the one-looptriangle diagrams and chiral anomaly. We can rewrite the (cid:96)
ALP-electron coupling in Eq. (2)by applying the anomaly equation for the divergence of the axial-vector current c e ∂ µ a f a eγ µ γ e = − c e m e f a aeiγ e + c e α π af a F µν (cid:101) F µν , (4)where α (cid:39) /
137 is the ﬁne structure constant, F µν = ∂ µ A ν − ∂ ν A µ is the ﬁeld strengthtensor of photon, and (cid:101) F µν = (cid:15) µνρσ F ρσ is its dual tensor with (cid:15) = +1. From Eq. (4), theeﬀective interaction between the (cid:96) ALP and photons equals  L aγ = − g aγγ F µν (cid:101) F µν (5)with g aγγ the (cid:96) ALP-photon coupling of the form g aγγ = απ c e f a (cid:12)(cid:12)(cid:12)(cid:12) − F (cid:18) m a m e (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (6)4here m a is the (cid:96) ALP mass, the factor of 1 in the absolute value comes from the anomalyterm, and F ( z ) is the loop function whose form depends on the argument. To explain theMiniBooNE excess, we will assume that m a > m e . In this case, the loop function reads  F ( z >
1) = 1 z arctan (cid:18) (cid:112) /z − (cid:19) . (7)Note that we have g aγγ (cid:39) . × − ( c e /f a ) for m a (cid:29) m e , since F ( z (cid:29) →
0. We willuse this interaction to calculate the photophilic decay of the (cid:96)
ALP in the next subsection.
B. The decay width
We propose that at MiniBooNE, when the proton beam hits the target, the charged K meson produced decays into a sterile neutrino through its mixing with the muon neutrinoin Eq. (1). In our study, we assume that the sterile neutrino is much heavier than the (cid:96) ALP.Thus, the sterile neutrino can decay into a (cid:96)
ALP and a light neutrino as N D → a + ν jL .Using Eq. (3), the decay rate of the sterile neutrino into a + ν (cid:48) s is calculated asΓ N D → aν = Γ ¯ N D → a ¯ ν = (cid:88) j =1 Γ N D → aν jL = c N | U µ | m N πf a (cid:18) − m a m N (cid:19) (cid:39) . × − MeV (cid:18) | U µ | × − (cid:19) (cid:18) m N
380 MeV (cid:19) (cid:18) f a
100 GeV (cid:19) − , (8)where m N is the sterile neutrino mass. In Eq. (8), we have used the unitary condition andsymmetry property of U , and also assumed, for simplicity, | U e | , | U τ | (cid:28) | U µ | (cid:28)
1. Withthe mixing parameter U µ , the sterile neutrino can also decay into a muon plus a chargedpion or a muon neutrino plus a neutral pion : N D → µ ± π ∓ or ν µ π if it is kinematicallyallowed. The corresponding decay rates have been estimated in Ref.  asΓ N D → µ ( ν µ ) π = G F f π | U µ | m N π K (cid:2) m π , m µ (0) , m N (cid:3) (cid:39) . × − MeV (cid:18) | U µ | × − (cid:19) (cid:18) m N
380 MeV (cid:19) , (9)where G F (cid:39) . × − GeV − is the Fermi coupling constant, f π (cid:39)
130 MeV is the piondecay constant, and K is an order one dimensionless kinematical function . Apparently,these decay channels are subdominant in comparison with N D → aν unless f a (cid:38)
300 TeV.Hence, the dominant decay mode of the sterile neutrino after it arrives at the detector is a (cid:96)
ALP plus a light neutrino.Then, the (cid:96)
ALP can decay into electron-positron and photon pairs with the couplingsgiven in Eq. (2) and Eq. (5). The decay widths of the (cid:96)
ALP into e + e − and γγ are computedrespectively as Γ a → e + e − = c e m e m a πf a (cid:115) − m e m a , Γ a → γγ = g aγγ m a π . (10)5 - - - m a ( MeV ) ℬ ( a → XX ) ℬ ( a → e + e - ) ℬ ( a → γ γ ) FIG. 2: The decay branching fractions of the (cid:96)
ALP, which is independent of c e /f a . We show in Fig. 2 the decay branching ratios of the (cid:96)
ALP. As indicated, the (cid:96)
ALP mainlydecays into e + e − in the mass range we are interested in. The decay products e + e − inside thedetector, we propose, could possibly account for the excess reported by MiniBooNE. Noticethat the m a dependence in the Γ a → γγ can counteract the g aγγ suppression for heavy (cid:96) ALPwith m a (cid:38)
200 MeV, where the decay channel of a → γγ gives a non-negligible contributionto the total decay width of the (cid:96) ALP.Now, in order for the sterile neutrino and (cid:96)
ALP to both decay within the MiniBooNE de-tector, we have to examine the mean decay distances d N,a of both particles in the laboratoryframe. Using the results in Eq. (8) and Eq. (10), we obtain d N = γ N β N τ N (cid:39) . × m (cid:18) p N (cid:19)(cid:18) | U µ | × − (cid:19) − (cid:18) m N
380 MeV (cid:19) − (cid:18) f a
100 GeV (cid:19) , (11) d a = γ a β a τ a (cid:39) .
42 m (cid:18) p a (cid:19)(cid:18) m a
20 MeV (cid:19) − (cid:18) f a
100 GeV (cid:19) , (12)where γ N ( γ a ), β N ( β a ), τ N ( τ a ), and p N ( p a ) are the Lorentz boost factor, speed, lifetime, andmomentum of the sterile neutrino ( (cid:96) ALP), respectively. Therefore, with these ﬁducial valuesfor the masses and couplings, the sterile neutrino and (cid:96)
ALP can have the proper mean decaylengths which are consistent with the MiniBooNE experimental setup.
III. ASTROPHYSICAL AND EXPERIMENTAL CONSTRAINTS
In this section, we will scrutinize the astrophysical and experimental constraints of c e /f a and U µ in our (cid:96) ALP setup. For these two parameters, the associated astrophysical boundscome from celestial objects such as red giant, white dwarf, and supernova (SN), dependingon the mass scale of the (cid:96)
ALP. In our work, we will consider the (cid:96)
ALP with a few tens MeVmass, and so, the strongest limit is set by the SN1987A. On the other hand, there are severalterrestrial laboratories which can place constraints on these parameters as well, includingthe electron beam-dump experiment E137, electron magnetic dipole moment anomaly, andrare kaon decays, and so on. Finally, for an electron-positron pair to mimic a single electron-like event, we have to demand that the opening angle of an electron-positron pair is small6nough. This would truncate the momentum of the (cid:96)
ALP at a certain value, and then set alower bound on c e /f a . Here we brieﬂy discuss these constraints in the following. A. Supernova 1987A
The observed neutrino burst duration of SN1987A can impose a constraint for tens ofMeV axion-like particles (ALPs) . This is because the temperature of the proto-neutronstar (PNS) can reach of the order of 30 MeV. With these temperatures, the ALPs can beproduced inside the PNS and then carry away a lot of energy from it (which is known as thefree-streaming regime). This process would speed up the cooling rate of the PNS and shrinkthe period of neutrino burst. Since the energy loss rate due to the ALP should not exceedthe ones via the neutrinos, an approximate analytic bound on the energy loss rate throughthe ALP in the free-streaming regime is given by ˙ E a (cid:46) erg g − s − , (13)which is evaluated at the typical core density of 3 × g cm − and temperature of 30 MeV.It is worth mentioning that several numerical simulations demonstrated that the neutrinoburst duration would be roughly reduced by half when the limit of Eq. (13) is saturated .Given the couplings in Eq. (2) and Eq. (5), the primary production channels of the ALPinside the PNS are the electron-nucleus bremsstrahlung ( e − + N → e − + N + a ) and thePrimakoﬀ process ( γ + N → N + a ), respectively. For an ALP with mass of a few tens MeV,it has been estimated in Ref.  that the energy loss rate due to the former one is˙ E e N → e N a (cid:39) . × (cid:18) c e /f a GeV − (cid:19) erg g − s − , (14)where the Boltzmann suppression factor has been taken into account. On the other hand,with the help of Eq. (13) and the numerical results analyzed in Refs. [22, 23], the energy lossrate due to the Primakoﬀ process is estimated as˙ E γ N →N a (cid:39) . × (cid:18) g aγγ GeV − (cid:19) erg g − s − . (15)Then, as emphasized below Eq. (7), it follows that ˙ E e N → e N a / ˙ E γ N →N a (cid:39) O (10 ). Hence, theenergy loss of the PNS is mainly through the electron-nucleus bremsstrahlung in this model.Imposing Eq. (13) to Eq. (14), ˙ E e N → e N a (cid:46) erg g − s − , we yield c e /f a (cid:46) × − GeV − (free-streaming regime) . (16)Note that the above upper bound is no longer valid for suﬃciently large c e /f a . The reasonis that when the coupling strength of the ALP becomes too strong, it would be capturedwithin the PNS and cannot escape from it. This is the so-called trapping regime. To ﬁndout the lower bound of the trapping regime, one can require that the mean free path of theALP is smaller than the eﬀective radius of the PNS. Following Refs. [20, 23], the resultantlower bound of c e /f a is derived as c e /f a (cid:38) × − GeV − (trapping regime) . (17)7e present the SN1987A excluded range of c e /f a in the yellow shaded region of Fig. 3.The SN1987A can also give constraints for models with the sterile neutrino. For example,it has been considered in Ref.  to restrict U µ as the sterile neutrino can be produced inthe PNS by the Primakoﬀ upscattering via the photon exchange with nucleons ( ν + N →N + N D ) . However, since we assume the ALP in our model is leptophilic. Thus, it cannotbe produced via the (cid:96) ALP exchange with nucleons. Namely, we can evade the constraint of U µ from the SN1987A. B. Electron beam-dump experiment E137
There exist some experiments searching for long-lived light particles by impinging high-intensity proton or electron beams on the heavy materials, called beam-dump experiments.In the (cid:96)
ALP model, only electron beam-dump experiment such as E137  is relevant toour study since the (cid:96)
ALP only interacts with electron and photon. In the E137 experiment,a 20 GeV electron beam collides with plates of aluminum immersed in cooling water. With alarge number of electrons ( ∼ × ) cumulatively hitting on the target, many (cid:96) ALPs canbe produced from it. Once the (cid:96)
ALPs are generated, they would ﬁrst penetrate a shieldingabout 179 meters, and then reach an open-air decay region 204 meters long. At the end ofthe decay region, there is a detector which can receive visible signals from the (cid:96)
ALP decays.The main production mechanisms of the (cid:96)
ALP in the E137 experiment are the Primakoﬀeﬀect and bremsstrahlung from electrons , and as pointed out in Fig. 2, the (cid:96)
ALP decayspreferentially into an electron-positron pair. Notice that although g aγγ is much smaller than c e /f a in the (cid:96) ALP model, however, these two production mechanisms are comparable. To seethis, one can compare the scaling of the cross-section of these processes. In the former case,we have σ aγ ∝ αg aγγ , while for the latter one we have σ ae ∝ α ( m e /m a ) ( c e /f a ) . Then,by taking the ratio of them, we ﬁnd that σ ae /σ aγ ∼ O (1).The constraint of c e /f a by the E137 experiment is shown in the orange shaded region ofFig. 3, where the upper bound and lower bound correspond to the short-lived and long-lived (cid:96) ALP, respectively. In this region, no event has been seen by E137 .
C. Anomalous electron magnetic dipole moment
With the (cid:96)
ALP-electron interaction, there is a one-loop Feynman diagram involving the (cid:96)
ALP which contributes to the anomalous electron magnetic dipole moment, ( g − e . Thelatest measurement deviating from the SM value is given by [28, 29]∆ a e = a exp e − a SM e = − (8 . ± . × − , (18)corresponding to about 2 . σ tension with the SM prediction, where a e ≡ ( g − e /
2. On theother hand, the leading-order contribution to ∆ a e by the (cid:96) ALP can be found in Ref.  as∆ a (cid:96) ALP e = − c e m e π f a (cid:90) dx x x + (1 − x ) r , (19)where r = m a /m e . Notice that the (cid:96) ALP-photon interaction also gives a contribution to ∆ a e through the Barr-Zee diagram. However, this contribution is sub-leading since g aγγ (cid:28) c e /f a .8mposing Eq. (19) to Eq. (18), we ﬁnd that only the large values of c e /f a are subject to the( g − e anomaly, see the green shaded region in Fig. 3. D. Collimated e + e − pair as a single electron-like event For an electron-positron pair produced from the (cid:96)
ALP decay been identiﬁed with anelectron-like signal, we have to require that the opening angle of an electron-positron pair θ e + e − <
13 degrees . This upper bound of the opening angle can be translated into thelower bound of the momentum of the (cid:96)
ALP as  p a > p a, min ≡ (cid:112) m a − m e cot (cid:0) θ e + e − / (cid:1) (cid:39) . m a . (20)Now, in order for an electron-positron pair can be detected in the MiniBooNE experiment,the mean decay length of the (cid:96) ALP should be smaller than the diameter of the MiniBooNEdetector. Using Eq. (12) with Eq. (20) and requiring d a < D = 10 m, we arrive at c e /f a (cid:38) . × − GeV − (cid:18) m a
20 MeV (cid:19) − / . (21)Note that Eq. (21) is a conservative bound, below which the (cid:96) ALP still has a probability todecay within the detector. This factor will be considered in the computation of the excessevents in the next section. We show this bound as the black dashed line in Fig. 3.Based on the above constraints, we choose an optimistic benchmark point, m a = 20 MeVand c e /f a = 10 − GeV − (the red dot in Fig. 3) in our numerical calculation. Also, we havechecked that this benchmark point is far below the sensitivities of the current colliders and far above the limits from cosmology . E. Rare kaon decays
In our setup, the sterile neutrino is produced from the charged kaon decay, say K + → µ + N D , through the mixing between the sterile neutrino and the muon neutrino. With theneutrino mixing parameter U µ , the corresponding branching fraction is given by  B ( K + → µ + N D ) = B ( K + → µ + ν µ ) ρ µ ( m N ) | U µ | , (22)where B ( K + → µ + ν µ ) = 0 . ρ µ ( m N ) = x N + x µ − ( x N − x µ ) x µ (1 − x µ ) (cid:2) λ (1 , x N , x µ ) (cid:3) / (23)with x N ≡ m N /m K , x µ ≡ m µ /m K , and λ ( u, v, w ) = u + v + w − uv + vw + wu ).The E949  and NA62  are past and current kaon decay experiments searching for the A more recent measurement of the ( g − e anomaly can be found in Ref. , where ∆ a e = (4 . ± . × − (+1 . σ ). In this case, Eq. (19) can only contribute to the negative error bar, and it does not aﬀectour result too much. - - - - - - - - m a ( MeV ) c e / f a ( G e V - ) ( ℊ - ) e E137SN1987A d a > D ( θ e + e - < °) d a < D ( θ e + e - < °) FIG. 3: The various astrophysical and experimental bounds of c e /f a in the (cid:96) ALP model, wherethe yellow, orange, and green shaded regions are excluded by the SN1987A, electron beam-dumpexperiment E137, and ( g − e anomaly, respectively. In the gray shaded area, the number of eventsis suppressed as the (cid:96) ALP mostly decays outside the MiniBooNE detector. Our benchmark pointis indicated by the red dot right above the upper boundary of the E137 constraint.
150 200 250 300 350 40010 - - - - m N ( MeV ) U μ NA62 ( - ) K + →μ + N D preliminaryE949 ( ) K + →μ + N D FIG. 4: The upper limits of | U µ | from the kaon decay experiments at 90% C.L., where themagenta and blue shaded regions are excluded by E949 and NA62, respectively. sterile neutrino through this decay process. By measuring the muon momentum spectrum ormissing energy spectrum of the kaon decays, they can provide upper limits for | U µ | , whichare displayed as color lines in Fig. 4. In this ﬁgure, we take the data of the E949 experimentfrom Ref. , and for the NA62 ones, we adopt the preliminary updated result ( ∼ / | U µ | down to ∼ − for the sterile neutrino with masses between 175 and300 MeV. On the other hand, the NA62 experiment extends the search range of the sterileneutrino mass from 300 to 383 MeV, where | U µ | (cid:46) − . In the next section, we will take10 - - - p N ( GeV ) Φ N ( p N )( N / P O T / G e V / c m ) m N MeV ν mode ν mode FIG. 5: The ﬂuxes of the sterile neutrino as a function of p N for the neutrino (orange line) andanti-neutrino (blue line) modes with m N = 380 MeV, where the sharp peaks correspond to thestopped kaons. Note that N D and ¯ N D equally contribute to Φ N as the signal at the MiniBooNEdetector do not distinguish between them. m N = 380 MeV and | U µ | = 1 . × − as the benchmark point for our computations. IV. THE MINIBOONE EXCESS EVENTS AND OUR FITTING RESULTS
In this section, we will outline how we compute the excess event numbers in the (cid:96)
ALPmodel. Essentially, we follow the approaches given in Refs. [9, 41] with some modiﬁcations.They consider a sterile neutrino decaying inside the detector into an active neutrino and aphoton. The work reconstructs ﬁrst the kaon ﬂux from the given ﬂux of the muon neutrino.From the kaon ﬂux, they then derive the sterile neutrino ﬂux. We follow similar procedures.In comparison, however, we replace the massless photon with an unstable massive (cid:96)
ALP,and it is expected that the kinematics in our calculation is a little bit diﬀerent from theirs.Indeed, the condition of the opening angle of an electron-positron pair by the (cid:96)
ALP decayrequires a minimum of the (cid:96)
ALP momentum. In the computation, this may eliminate someof the events from contributing to excesses. In the following, we will ﬁrst write down all therelevant formulas for estimating the total number of events and then present our numericalresults before the end of this section.It is worth it to mention that the production of K + (or K − ) at the target can be param-eterized using the Feynman scaling . With the best-ﬁt parameters provided in Ref. ,one can generate the momentum distribution of K + . Then, it would be straightforward toget the kinematics of sterile neutrino, (cid:96) ALP and electron-positron pair in the decay chainof K + → µ + N D → µ + ν µ a , followed by a → e + e − . Given the total number of K + beingproduced at the target, we can estimate the excess of ν e -like events mimicked by collimated e + e − pairs. We adopt this method as a cross-check and obtain similar results. A. Angular and energy spectra of (cid:96)
In our model, the angle and energy of the signal are interpreted as the scattering angle θ a and energy E a of the (cid:96) ALP, which decays into a small opening angle electron-positronpair. To compute the distributions of the excess events, we integrate the decay spectra of11 p a ( GeV ) ℰ a p a ( GeV ) a m a MeV m N MeV
FIG. 6: Left panel : The MiniBooNE detection eﬃciency of the signal energy, here approximatedwith the momentum of the (cid:96)
ALP. Right panel : The momentum distribution of the (cid:96)
ALP with m a = 20 MeV and m N = 380 MeV. the (cid:96) ALP over the sterile neutrino ﬂux, Φ N ( p N ) (see Fig. 5), together with the probabilities P N, dec ( p N ) and P a, dec ( p N ) that the sterile neutrino and the (cid:96) ALP decay in the MiniBooNEdetector, respectively. Other necessary factors will be explained below. Since we constructthe sterile neutrino ﬂux from the muon neutrino ﬂux of the kaon decay, a normalized factor B ( K → µN D ) / B ( K → µν ) = ρ µ ( m N ) | U µ | should be included to account for the neutrinomixing U µ and diﬀerent kinematics of the muon neutrino and heavier sterile neutrino. Thepredicted spectrum S with respect to the variable Q : Q = cos θ a or E a can be written as amaster formula S ( Q ) = ρ µ ( m N ) | U µ | POT A MB (cid:90) dp N Φ N ( p N ) P N, dec ( p N ) W time ( p N ) × lab N d Γ lab N D → aν dQ P a, dec ( p a ) E a ( p a ) D a ( p a ) , (24)where POT denotes the number of protons on target, which is equal to 18 .
75 (11 . × for the neutrino (anti-neutrino) operation mode , A MB = π ( D/ is the eﬀective area ofthe MiniBooNE detector, W time ( p N ) is the timing-related weight due to the fact that thesterile neutrino arrives at the detector later than the light ones in a proton beam pulse  W time ( p N ) = ∆ t H (cid:0) ∆ t (cid:1) , ∆ t = t + δt − t N δt , H (cid:0) ∆ t (cid:1) = (cid:40) t >
00 if ∆ t < t = L/c (cid:39) . µ s ( t N = t /β N ) being the light (sterile) neutrino arrival time from thesource to the detector and δt (cid:39) . µ s being the time interval of the proton beam pulse, and E a ( p a ) and D a ( p a ) are the MiniBooNE detector eﬃciency  and the momentum distribu-tion of the (cid:96) ALP as functions of the (cid:96)
ALP momentum, respectively, which are displayed inFig. 6. Note that D a ( p a ) has to be normalized when performing the integral in Eq. (24).For the probabilities of the sterile neutrino and the (cid:96) ALP decaying inside the detectableregion of the MiniBooNE experiment, we have P N, dec ( p N ) = exp (cid:18) − L Γ N m N p N (cid:19)(cid:20) − exp (cid:18) − D Γ N m N p N (cid:19)(cid:21) , (26)12 a, dec ( p a ) = 1 − exp (cid:18) − D Γ a m a p a (cid:19) , (27)where Γ N (cid:39) Γ N D → aν and Γ a (cid:39) Γ a → e + e − . Now, in the case of the angular spectrum, S (cos θ a ),the normalized diﬀerential decay rate of the sterile neutrino respect to cos θ a in the lab frameis derived as 1Γ lab N d Γ lab N D → aν d cos θ a = 11 − m a /m N p a | p a E N − p N E a cos θ a | , (28)where the (cid:96) ALP momentum is given by  p a = (cid:0) m N + m a (cid:1) p N cos θ a + E N (cid:113)(cid:0) m N − m a (cid:1) − m a p N sin θ a (cid:0) m N + p N sin θ a (cid:1) . (29)For the energy spectrum, S ( E a ), one can use the chain rule and (29) to derive d Γ lab N D → aν /dE a .Note that what reported by the MiniBooNe experiment is the spectrum of the visible energy, E vis (or the reconstructed neutrino energy, E rec ν ). Since the (cid:96) ALP in our model decays visibly,thus, we can approximately take E a ≈ E vis .With the above tools, we can then compute the excess event numbers N cos θ a ,i and N E a ,i of i th-bin of the (cid:96) ALP angular and energy spectra respectively as N cos θ a ,i = (cid:90) cos θ a,i +1 cos θ a,i d cos θ a S (cos θ a ) , N E a ,i = (cid:90) E a,i +1 E a,i dE a S ( E a ) , (30)and the total events can be obtained easily by summing up the event numbers in each bin N cos θ a , total = (cid:88) i N cos θ a ,i , N E a , total = (cid:88) i N E a ,i . (31)Note that when evaluating the integrals in (30), the cut of the (cid:96) ALP momentum, p a > p a, min ,must be considered, see Eq. (20). B. Our ﬁtting results
Applying the formulas from Eq. (24) to Eq. (30), we present in Fig. 7 our ﬁtting results ofthe angular and visible energy spectra in the neutrino mode, including all of the excess dataand expected backgrounds reported by the most recent update analysis of MiniBooNE . Inboth ﬁgures, we assume m a = 20 MeV , m N = 380 MeV , c N = 0 . , c e /f a = 10 − GeV − , and | U µ | = 1 . × − as the benchmark point. One can see that our predictions of the spectrain the (cid:96) ALP model are consistent with the tendencies of the experimental data points. Also,the predicted total excess events are within the 1 σ range of the observed ones. The ﬁtting There is an ALP momentum conjugated to p a which is kinematically allowed in the lab frame. However,we have checked that this conjugate momentum is always far below p a, min , then it would not contributeto our calculation. - cos θ E v en t s m a MeV m N MeV c N c e / f a - GeV - U μ ⨯ - Data ( ν mode ) BackgroundOur fit
200 400 600 800 1000 12000100200300400500600 E vis ( MeV ) E v en t s m a MeV m N MeV c N c e / f a - GeV - U μ ⨯ - Data ( ν mode ) BackgroundOur fit
FIG. 7: Our numerical results for the angular spectrum (top panel) and visible energy spectrum(bottom panel) of the MiniBooNE experiment in the neutrino mode, where the black dots are theexcess electron-like events with errors and the green shaded region is the estimated backgrounds.With the benchmark point, the corresponding ﬁttings are shown as red dashed lines in the ﬁgures. result of the reconstructed neutrino energy spectrum is similar to that of the visible energyspectrum and is not shown here.We then use Eq. (31) to calculate the total excess events and depict the allowed regions ofparameter space that can explain MiniBooNE data. At the top panel of Fig. 8, we show the1 σ to 3 σ contours in the two-parameter plane of | U µ | versus m N with benchmark choice ofother parameters. Clearly, the neutrino mixing parameter within the range 10 − (cid:46) | U µ | (cid:46) − can account for the latest MiniBooNE results in a broad range of the sterile neutrinomass that is not excluded by the kaon decay experiments. Notice that there is a small partof the contours in the upper left corner of the ﬁgure, where 3 × − (cid:46) | U µ | (cid:46) × − with 150 MeV (cid:46) m N (cid:46)
180 MeV can ﬁt the data as well. We also draw the same contours inthe | U µ | versus c e /f a plane at the bottom panel of Fig. 8, where 9 × − GeV − (cid:46) c e /f a (cid:46)
50 200 250 300 350 40010 - - - - - - - m N ( MeV ) U μ NA62E949 Mean1 σ σ σ m a MeV c N c e / f a - GeV - - - - - - - - - - - c e / f a ( GeV - ) U μ Mean1 σ σ σ m N MeV m a MeV c N E137 NA62
FIG. 8: The parameter space of the | U µ | versus m N plane (top panel) and | U µ | versus c e /f a plane (bottom panel) satisfying the MiniBooNE data at 1 σ to 3 σ C.L. in the (cid:96)
ALP model, wherethe purple contour presents the mean value of the data. The shaded regions above the magentaand blue lines are excluded by the kaon decay experiments, E949 and NA62, respectively, and theshaded area left to the orange line is disfavored by the E137 experiment. × − GeV − with a similar range of | U µ | can explain the excess.In both ﬁgures, the upper (lower) portion of the contours corresponds to the short-lived15long-lived) sterile neutrino, where one can approximate the decay probability in Eq.(26) as P N, dec ( p N ) ≈ exp (cid:18) − L Γ N m N p N (cid:19) for the short-lived sterile neutrino limit D Γ N m N p N for the long-lived sterile neutrino limit . (32)Since Γ N ∝ | U µ | , then the number of excess events is increased as the | U µ | is decreased(increased) for the short-lived (long-lived) sterile neutrino, which explains the behaviors ofthe contours in these planes. Accordingly, one can also expect that the lower portion of thecontours is very sensitive to | U µ | , and this is because S ( Q ) ∝ | U µ | P N, dec ( p N ) ∝ | U µ | forthe long-lived sterile neutrino. V. DISCUSSION AND CONCLUSIONS
First, let us examine the feasibility of our (cid:96)
ALP model in more detail. In this model, weassume that the (cid:96)
ALP has a coupling to the sterile neutrino. Such coupling can be generatedby introducing a complex singlet scalar ﬁeld Φ and a pair of left and right chiral fermionicﬁelds ψ L,R , of the interactions as  L Φ ψ = − y N (cid:0) Φ ψ L ψ R + Φ ∗ ψ R ψ L (cid:1) , (33)and those ﬁelds are charged under a global axial U(1) symmetry. After symmetry breaking atthe energy scale υ Φ , where υ Φ is the vacuum expectation value of Φ, the angular componentof the complex scalar is identiﬁed with the (cid:96) ALP, Φ ⊃ υ Φ + ia/ √
2. Thus, Eq. (33) becomes L Φ ψ = − m N ψψ + 1 √ y N aψiγ ψ , (34)where m N = y N υ Φ . Comparing Eq. (34) to Eq. (2) and identifying ψ with ν D , we then obatin y N = c N m N /f a . For our benchmark point, it follows that y N (cid:39) − and υ Φ (cid:39)
250 GeV.Next, let us discuss the magnitude of the (cid:96)
ALP-lepton couplings in this model. For ourpurpose, we assume that the (cid:96)
ALP dominantly interacts with electrons. In other words, the (cid:96)
ALP couplings to the muon and tau, c µ,τ , and the (cid:96) ALP ﬂavor-changing couplings, say c eµ ,and so forth, are assumed to be negligibly small. Such hierarchy of the axion couplings canbe realized in the context of familon/ﬂaxion [15, 18, 45], a pseudo-Nambu-Goldstone bosonarising from the spontaneous breaking of a global Froggatt-Nielsen (FN) ﬂavor symmetry,U(1) FN . For example, one can consider the following eﬀective Yukawa interactions as L FN = − y jk (cid:18) ΦΛ (cid:19) n jk E Lj He Rk + h.c. , (35)where y jk is an order-one coeﬃcient, and Λ is the cut-oﬀ scale of the theory. Here E Lj , H ,and e Rk denote the left-handed lepton doublet, SM Higgs doublet, and right-handed chargedlepton singlet, respectively. The FN charge assignment for those ﬁelds is displayed in Tab. I,from which, n jk = [ E Lj ] + [ e Rk ]. Then, the breakdown of the FN and electroweak symmetriesleads to  L FN ⊃ − c jk ae jL iγ e kR + h.c. , c jk ≡ y jk n jk υ EW υ Φ (cid:18) υ Φ Λ (cid:19) n jk , (36)16 ield E Lj H e Rk ΦU(1) FN [ E Lj ] 0 [ e Rk ] − j, k = 1 , , here we have used γ P R,L = ± P R,L , where υ EW (cid:39)
174 GeV is the vacuum expectation valueof the SM Higgs ﬁeld. Therefore, with a proper FN charge assignment, we may achieve thehierarchy of the (cid:96)
ALP-lepton couplings in our (cid:96)
ALP model. However, the construction of aUV completion theory is beyond the scope of this paper, and we leave the detailed study ofthe model for future work.In this paper, we have shown that the collimated electron-positron pair produced fromthe sterile neutrino decay through the (cid:96)
ALP can account for the recently updated resultsof the MiniBooNE experiment. We ﬁnd that our resulting shapes of the distributions in theneutrino operation mode, especially the angular distribution, are in a good ﬁt with the data.Meanwhile, the total excess event numbers can be explained with the sterile neutrino masswithin 150 MeV (cid:46) m N (cid:46)
380 MeV (150 MeV (cid:46) m N (cid:46)
180 MeV) and the neutrino mixingparameter within 10 − (cid:46) | U µ | (cid:46) − (3 × − (cid:46) | U µ | (cid:46) × − ). Moreover, we havechecked that our benchmark choice can satisfy constraints from various astrophysical andterrestrial observations. The scenario could be tested by the searches of the (cid:96) ALP from thefuture colliders and by the sterile neutrino production from the kaon decay facilities.
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