A common leptoquark solution of flavor and ANITA anomalies

The ANITA experiment has seen anomalous Earth emergent showers of EeV energies which cannot be explained with Standard Model interactions. In addition, tests of lepton flavor universality in $ R(D^{(\ast)})$ and $R(K^{(\ast)})$ have shown significant deviations from theoretical predictions. It is known that, among single leptoquark solutions, only the chiral vector leptoquark $U_1 \sim (\mathbf{3}, \mathbf{1} , 2/3)$ can simultaneously address the discrepancies. In this letter, we show that the leptoquark motivated by flavor anomalies coupled to a sterile neutrino can also explain the ANITA Anomalous Events. We consider two scenarios, (a) the sterile neutrino, produced via resonant leptoquark mediated neutrino-nucleon interactions, propagates through the Earth without significant attenuation and decays near the surface to a $\tau$ lepton; and (b) a cosmogenic sterile neutrino interacts with the matter near the surface of Earth and generates a $\tau$ lepton. These two scenarios give significantly large survival probabilities even when regeneration effects are not taken into account. In the second scenario, the distribution of emergent tau energy peaks in the same energy range as seen by ANITA.


I. INTRODUCTION
The ANtarctic Impulsive Transient Antenna (ANITA) instrument is designed to detect interaction of ultra-high energy neutrinos via the Askaryan effect in ice.During its first and third flight, it also observed unexpected upward directed showers apparently emerging well below the horizon [1].The observed signal is consistent with τ induced showers.The essential details of the two Anomalous ANITA Events (AAEs) are given in Table I.The survival probability ( ) is estimated taking into account the neutrino regeneration effects and τ energy losses in [2]   The small survival probabilities within SM indicate that new physics scenarios should be invoked to explain these events.In the past, the AAEs have been explained in the framework of sterile neutrinos [3,4], Supersymmetry [2,5,6], and CPT symmetric universe [7].However, each of these explanation have their own limitations [5].Similarly, collider experiments such as LHCb, Belle, and BaBar have observed hints of Lepton Flavor Universality Violation (LFUV) in semi-leptonic decays of the B meson.The experimentally measured value of the observables R(D ( * ) ) [8,9] and R(K ( * ) ) [10,11] is consistently below SM prediction and together are dubbed as 'flavor anomalies' in this paper.These discrepancies can be explained in several extensions of SM, for example with leptoquarks [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].
It was proposed in [2] that a long lived BSM particle, which is produced in ultra-high energy (UHE) neutrino nucleon interactions, propagates freely through the chord of Earth, and decays to a τ near the surface can explain the AAEs.A natural candidate for this is τ (stau) in R-Parity conserving Supersymmetry [2] and neutralino (mostly Bino) in R-Parity violating Supersymmetry [5].In this paper, we consider two scenarios wherein a leptoquark, proposed as a resolution to flavor anomalies, also explains AAEs.In the first scenario, we extend the minimal leptoquark model of [12] with a heavy sterile neutrino (χ).The SM singlet χ is produced in UHE neutrino-nucleon interactions mediated by the leptoquark.
The sterile neutrino can travel inside Earth without significant attenuation and decays near the south pole.One of the decay products is the τ particle whose shower is seen by ANITA.
In the second scenario, an cosmogenic UHE sterile neutrino propagates freely through the chord of the Earth and produces a τ via leptoquark mediated interaction.Interestingly, the same leptoquark interaction also explains R(D ( * ) ) through b → cτ χ as shown in [15].
In Sec.II, we estimate the number of AAEs for isotropic and anisotropic flux.In Sec.
III, we provide details of the leptoquark model and discuss the two scenarios in detail before we conclude in Sec.IV.

II. ANITA ANOMALOUS EVENTS
One can estimate the number of EeV events seen by ANITA using where the effective area of ANITA A ≈ 4 km 2 is estimated using the Cherenkov angle [4], δT is the time period, δΩ is the solid angle.The neutrino energy (E ν ) is integrated over the range which gives correct range of shower energy and varies from model to model.In the above expression, is the survival probability and depends on E ν and model parameters.
One can get a crude estimate of the number of events expected from saturated Greisen- which gives N ≈ 500 .To get two events from isotropic GZK flux, one requires ∼ 4 × 10 −3 .Similar estimates were also obtained in [5].With the Standard Model interactions, the authors in [2] have estimated that SM ∼ 4.4 (0.32) × 10 −7 for neutrinos propagating 5740 (7210) km inside Earth.Thus the estimated number of anomalous events from GZK neutrinos with only SM interactions which makes observation of two events extremely unlikely.On the other hand, the upper limit on anisotropic flux of neutrinos is, which is several orders larger than the isotropic case.After accounting for the small solid angle (where δ θ ∼ 1.5 • is the angular uncertainty relative to parent neutrino direction [1]), one requires, aniso ∼ 2 × 10 −5 (6) which again cannot be explained via Standard Model interactions.One needs to look for Beyond Standard Model (BSM) scenarios to explain AAEs.

III. LEPTOQUARK RESOLUTION OF AAE
As has been discussed [15,16], a vector leptoquark U 1 with SU (3) C × SU (2) L × U (1) Y quantum numbers (3, 1, 2/3) can simultaneously explain the flavor anomalies.It is also one of the handful models that admit leptoquark coupling to a sterile neutrino [27].The interaction of U 1 with fermions in the mass basis is, where V is the CKM matrix and contribution of PMNS matrix is ignored.

A. Heavy Sterile Neutrino
In this section, we assume that the sterile neutrino is sufficiently heavy so that its contribution to semi-leptonic B decays is kinematically forbidden.Even though the interaction of up-type quarks with a sterile neutrino can generate dangerous scalar and pseudo-scalar operators, their contributions can be neglected and the conclusions in [16] remain unchanged.
The required texture of coupling matrices is, The left-handed coupling (g L ) generates the desired Wilson coefficients (i.e.δC 9 = −δC 10 with the correct sign for b → sµµ and g V L > 0 for b → cτ ν).In this way, U 1 is one of the rare solutions that can simultaneously address both the anomalies.The right-handed coupling (g R ) is severely constrained as it generates scalar and pseudoscalar operators that are disfavored.The sterile neutrino χ can also couple to other up-type right handed quarks, but we have neglected those couplings and their constraints for simplicity.In this section, we will assume the mass of leptoquark U 1 to be M U = 1.5 TeV and the couplings to be, which can explain the flavor anomalies.Such a choice is within the reach of future LHC searches but allowed from present limits [16,17].We treat the coupling g x and mass of the sterile neutrino (M χ ) as free parameters of the theory.The singlet is produced near the surface of Earth through neutrino-nucleon interaction mediated by the leptoquark.It is assumed that the cross section for the process is dominated by the resonant s-channel neutrino-quark interactions.It has been pointed out in [28] that the gluon initiated process can also give significant contributions.However, this will give an O(1) correction to survival probability and has been neglected for the heavy sterile neutrino case.The production cross section can be approximated in the narrow width limit as, where f q is the parton distribution function (PDF) of We have used ManeParse [29] and NNPDF3.1(sx)[30,31] datasets for the PDFs.The numerical factors (1.08, 0.11, and 0.5) are obtained using the central value of CKM parameters [32].Note that the PDFs are evaluated at small-x where the quark and anti-quark PDFs are similar and hence neutrino and anti-neutrino have similar cross section.
The interaction length is estimated as, LQ = (ρN A σ LQ ) −1 where we have used ρ ≈ 4 and N A = 6.022×10 23 cm −3 in water-equivalent units.Even though the density is larger near the center of Earth, the approximation for density is valid for the chord lengths relevant for AAE.
As opposed to previous studies with three body decay of a singlet [5], in this paper we estimate the two body decay width of the sterile neutrino to a pseudoscalar meson and the tau lepton.Since the decay width is being estimated in the rest frame of sterile neutrino of mass few GeV, one can integrate out the heavy leptoquark and write the effective Lagrangian as, where q ∈ {s, b} and ∈ {µ, τ }.We also use the expression, where P is a pseudoscalar meson of mass M P and f P is the associated form factor.The rest frame decay width of the sterile neutrino is, where the phase space factor is, For numerical estimation we use, and the quarks and lepton masses used are M c = 1.29 GeV, M s = 95 MeV, M µ = 105.66 MeV, and M τ = 1.77GeV respectively.The singlet can also decay into ν τ which produces a τ through SM interactions.This possibility has been ignored in this paper as it is less probable.The decay length of χ in Earth's frame is estimated as, where the last approximation is true for the range of energies involved.In this scenario, where l ⊕ is the length of path traversed by neutrino inside Earth and for EeV neutrinos, 0 ∼ 275 km [2].However, this is severely modified when one takes neutrino regeneration effects during propagation.In [2], the probability is obtained using simulations and mentioned in Table I.We denote these probabilities with SM .In presence of BSM interactions, the survival probability of the neutrino flux can be estimated using, The above expression can be understood as follows.The parentheses denote the fraction neutrinos that survives SM interactions after propagating a distance l 1 .These neutrino undergo leptoquark interactions with the matter and produce a sterile neutrino.The sterile neutrino propagates a distance of l ⊕ − l 1 before it decays near the surface of Earth in the δ ≈ 10 km window that will produce an observable τ .In Fig. 2 we have shown the parameter space that gives LQ > SM , and LQ > 1 × 10 −6 for the two values of l ⊕ .We find that the maximum survival probability in this scenario is of the order 4 × 10 −6 .It is understood that neutrino regeneration effects can dramatically increase LQ similar to SM.However, complete estimation requires simulation of neutrino propagation which is beyond the scope of this work.Moreover, we find that the precision measurement of B + c decay modes can probe the most interesting part of the parameter space.We evaluate the branching fraction B = Br(B + c → + χ) for ∈ {µ, τ } to be, where f B + c = 0.43 GeV [15] and M B + c = 6.275GeV [32].Since the typical branching ratio of leptonic mode is very small, we take the conservative limit of B = 10% for both µ and τ modes to constrain our parameter space.We also show limits for B = 1% which will be accessible in future B-factories and can test the model.
To estimate the number of events, we consider the benchmark scenario M χ = 4.0 GeV g x = 0.8 (19) for which the survival fraction is BSM ∼ (1.5 − 2.0) × 10 −6 .This gives the expected number of AAE per direction to be 0.03 using the saturated anisotropic flux.Given that IceCube exposure is around 10 times larger than ANITA, one expects 0.3 events to be seen by Ice-Cube.It was pointed out in [2] that the three through-going track like events in IceCube may be attributed to such new physics scenarios.A detailed estimate for IceCube must also account for muon and neutrinos in the final state.
In this scenario, larger values of the coupling g x seem to be preferable.However, they would be constrained from future measurements of B µ .One can avoid these constraints if ).If one is willing to give up simultaneous explanation of both flavor anomalies, another interesting possibility opens up i.e. light sterile neutrino.

B. Light Sterile Neutrino
In [15], it is shown that U 1 leptoquark coupled to a light sterile neutrino can also explain the flavor anomalies.However, as opposed to [16], R(D ( * ) ) is explained via right-handed couplings and R(K ( * ) ) via left-handed ones.It is seen that a simultaneous explanation in this scenario is in tension with big bang nucleosynthesis but R(D ( * ) ) can be explained successfully.The Lagrangian for the leptoquark is, where g s is the strong coupling constant and κ = 0(1) for a minimally-coupled (gauge) theory.The excess can be explained with the following choice of coupling and leptoquark mass, Considering the LHC constraints on the model, we chose M U = 1.5 TeV which is close to the lightest allowed mass for κ = 1.To a good approximation, g bτ ∈ {1.1, 1.4} which translates to g x ∈ (1.0, 1.25) using (21).In this limit, the model has signatures in future 300 fb −1 analysis.These limits are considerably weakened for κ = 0.One can refer to [15] for detailed discussion of the model and other constraints.
To explain AAE, we assume anisotropic flux of light sterile neutrinos incident on Earth.
Such neutrinos can either be produced via the leptoquark interactions or via oscillation of active neutrinos near the source.The sterile neutrinos can pass through the Earth almost unattenuated, however, a fraction of them can interact with the matter in Earth and produce a τ near the surface.In this section, we consider both neutrino-quark and neutrino-gluon interactions.The relevant Feynman diagrams are shown in Fig. 3 χ χ χ c g g The neutrino-quark interaction is dominated by the s-channel resonant contribution and the cross section can be estimated by The difference in y-dependence is due to the RR nature of interaction as opposed to LR in the previous case.On the other hand, the neutrino-gluon interaction cross section can be estimated using, We implemented the model in FeynRules [33,34] and the cross section is calculated using CalcHep [35].As was shown in [28], the gluon initiated process are significant for large energies and of the same order of magnitude as the quark initiated processes.The cross section depends on κ as evident from Fig. 3(b).In Fig. 4, we show the variation of σ q and σ g with incident sterile neutrino energy.We also show the relative strength for κ = 0 and 1.
FIG. 4. The variation of cross section σ q (σ ) with incident sterile neutrino energy is shown in blue (red).The inset shows the difference in magnitude of σ g for κ = 0 and 1 in arbitrary units.
The fraction of incident neutrinos that interact with matter in Earth is given by, where q/g = (ρN A σ q/g ) −1 .One must note that, for neutrino-quark interactions E τ = E χ /2 whereas for neutrino-gluon interaction E τ = E χ /4.By uniformly varying incident sterileneutrino energy, we show the variation of = q + g with energy of emergent tau in Fig. 5 . An interesting result of this scenario is that the distribution peaks for tau energy in the same range as seen by ANITA.
To calculate N in this model, one needs an estimate of the incident flux of sterile neutri- nos.If the sterile neutrinos are produced due to oscillation from the active ones, then the flux is proportional to the square the mixing angle.For large mixing, the cross section will dominated by SM interactions and the sterile neutrino will be significantly attenuated by Earth.For small mixing, albeit the sterile neutrino propagates freely, the incident flux is smaller and → × θ 2 ∼ 10 −7 .We emphasize again that this estimate is obtained without taking regeneration effects into account and we anticipate that it to improve significantly.

IV. CONCLUSION
Since the observation of AAEs, many BSM scenarios have been invoked to explain the discrepancy.In this letter we have proposed two models that can significantly enhance the τ survival probability while simultaneously addressing the flavor anomalies.In the first scenario, we have extended chiral vector leptoquark model which explains R(D ( * ) ) and R(K ( * ) ) [12] by a sterile neutrino.The cosmogenic UHE neutrinos interact with the matter in Earth and produce a sterile neutrino that propagates freely inside Earth and decays near the surface to a τ .The precise measurement of Br(B c → τ χ), which is possible in upcoming B factories, will provide a good test of this model.
In the second scenario, a cosmogenic UHE sterile neutrino passes through the Earth almost unattenuated and interacts with the matter in Earth to produce an observable τ .
The same interactions and parameters also explain R(D ( * ) ) anomaly [15].The interesting result is that the distribution of emergent τ energy peaks in the same regime as observed by ANITA.This model has observable signatures in 300 fb −1 LHC searches.
In summary, the observation of lepton flavor universality violation and Earth emergent τ with EeV energy can be explained in a common framework.Moreover, it has testable signatures in upcoming experiments.

1 FIG. 1 .
FIG. 1.The Feynman diagrams for the process involved in Model A. Left: The s-channel neutrino quark interaction mediated by leptoquark U 1 that produces sterile neutrino in final state is shown.Right: The decay mode of sterile neutrino to charged lepton and D + s is shown.The shaded circle represents the effective vertex.

4
and hence for shower energy ∼ 0.5 EeV, one requires the incident neutrino to have energy E ν ∼ 2 EeV.With only SM interactions, one can estimate the bare survival probability 0 = e −l ⊕ / 0

1 FIG. 2 .
FIG.2.The parameter space that gives LQ > SM (blue), and LQ > 1 × 10 −6 (dark blue) for l ⊕ = 7210 km is shown.Similar projections for ⊕ = 5740 km is shown by red curves.The gray shaded region is conservatively ruled out from B + c decays and the limits for various B are shown.The top part is excluded using the perturbativity limit g x ≤ √ 4π.The neutrino energy is fixed to be 2 EeV.The benchmark point considered in the text is shown.

FIG. 5 .
FIG.5.The variation of q , g , and is shown in blue, red, and black respectively.The solid curve is for κ = 1 and the dashed curve for κ = 0.The chord length l ⊕ is fixed to be 5740 km (left) and 7210 km (right).We fix g x = 1.2 for both the plots.The region shown in green is the observed shower energy for the two events.

TABLE I .
Properties of the anomalous events.