Zee-Burst: A New Probe of Neutrino Non-Standard Interactions at IceCube

We propose a new way to probe non-standard interactions (NSI) of neutrinos with matter using the ultra-high energy (UHE) neutrino data at current and future neutrino telescopes. We consider the Zee model of radiative neutrino mass generation as a prototype, which allows two charged scalars -- one $SU(2)_L$-doublet and one singlet, both being leptophilic, to be as light as 100 GeV, thereby inducing potentially observable NSI with electrons. We show that these light charged Zee-scalars could give rise to a Glashow-like resonance feature in the UHE neutrino event spectrum at the IceCube neutrino observatory and its high-energy upgrade IceCube-Gen2, which can probe a sizable fraction of the allowed NSI parameter space.

Introduction.-The observation of ultra-high energy (UHE) neutrinos at the IceCube neutrino observatory [1][2][3][4][5][6] has commenced a new era in neutrino astrophysics. Understanding all aspects of these UHE neutrino events, including their sources, energy flux, flavor composition, propagation, and detection, is of paramount importance to both astrophysics and particle physics communities [7,8]. A simple, single-component unbroken powerlaw flux Φ(E ν ) = Φ 0 (E ν /100 TeV) −γ gives a reasonably good fit to the high-energy starting event (HESE) component of the IceCube data, with the latest best-fit values of Φ 0 = (6.45 +1. 46 −0.46 ) × 10 −18 GeV −1 cm −2 s −1 sr −1 and γ = 2.89 +0. 20 −0.19 at 1σ significance [9]. Any anomalous features in the observed event spectrum could potentially be used as a probe of fundamental physics. One such anomalous feature could be in the form of a new resonance. The purpose of this Letter is to show that such a new resonance can arise naturally in the popular Zee model of radiative neutrino masses [10,11], which contains two charged scalars. We refer to this Zee-scalar resonance as the 'Zee-burst'.
Within the SM, the only resonance IceCube is sensitive to is the Glashow resonance [12], where electron anti-neutrinos hitting the target electrons in ice could produce an on-shell W -boson:ν e e − → W − → anything. The energy of the incoming neutrino required to make this resonance happen is fixed at E ν = m 2 W /2m e = 6.3 PeV. One candidate Glashow event was identified in a partially-contained PeV event (PEPE) search with deposited energy of 5.9 ± 0.18 PeV [5,13], but has not been included in the event spectrum yet [6]. The nonobservation of Glashow events might be still consistent with the SM expectations within the error bars, given the uncertainty in the source type (pp versus pγ), as well as (ν e , ν µ , ν τ ) flavor composition (1:2:0 vs 0:1:0) [14][15][16][17][18]. On the other hand, the possibility of observing a Z-boson resonance (Z-burst) at IceCube due to UHE anti-neutrinos interacting with non-relativistic relic neutrinos [19] is bleak, as the required incoming neutrino energy in this case turns out to be E ν = m 2 Z /2m ν 10 23 eV, well beyond the Greisen-Zatsepin-Kuzmin cut-off energy of ∼ 5×10 19 eV for the UHE cosmic rays [20,21]the most likely progenitors of the UHE neutrinos (for related discussion, see Ref. [22]). An interesting alternative is the existence of secret neutrino interactions with a light (MeV-scale) Z [23][24][25][26] or light neutrinophilic neutral scalar [27][28][29], in which case the resonance could again fall in the multi-TeV to PeV range which will be accessible at IceCube. Heavy (TeV-scale) resonances induced by neutrino-nucleon interactions mediated by exotic charged particles, such as leptoquarks [30][31][32][33], or squarks in Rparity violating supersymmetry [34][35][36][37]. have also been discussed. In this Letter, we propose the possibility of light charged scalar resonances at IceCube, which are intimately related to neutrino mass generation [10], as well as observable non-standard interactions (NSI) [38] (for a recent update, see Ref. [39]).
As a prototypical example, we take the Zee model [10] -one of the most popular radiative neutrino mass models, which contains an SU (2) L -singlet charged scalar η ± and an SU (2) L -doublet scalar H 2 , in addition to the SMlike Higgs doublet H 1 . The original version of the Zee model [10] is fully consistent with neutrino oscillation data [40] (for explicit neutrino mass fits, see Ref. [11]), although the Wolfenstein version of the model [41] which assumes a Z 2 symmetry, thus making the diagonal entries of the neutrino mass matrix vanishing, is excluded by oscillation data [42,43]. Furthermore, it was pointed out in Ref. [11] that both the singlet and doublet charged scalar components can be as light as ∼ 100 GeV, while satisfying all existing theoretical and experimental constraints in both charged and neutral scalar sectors. More interestingly, such light charged scalars can lead to sizable diagonal NSI of neutrinos with electrons, with the maximum allowed values of the NSI parameters (ε ee , ε µµ , ε τ τ ) = (8%, 3.8%, 43%). We show here that the possibility of having a resonance feature with these light charged Zeescalars ('Zee burst') provides a new probe of NSI at highenergy IceCube, complementary to the low-energy neutrino oscillation and scattering experiments.
Light charged scalars in the Zee model.-In the Higgs basis [44] with the mixing angle ϕ given by where µ is the dimensionful coefficient of the cubic term µH i 1 H j 2 ij η − in the scalar potential, with {i, j} being the SU (2) L indices and ij being the SU (2) L antisymmetric tensor.
The leptonic Yukawa couplings are given by the Lagrangian where {α, β} are flavor indices, c denotes the left-handed antilepton fields, and H a = iτ 2 H a (a = 1, 2) with τ 2 being the second Pauli matrix. The neutrino mass is generated at one-loop level and is given by where M = Y v/ √ 2 is the charged lepton mass matrix and κ is a one-loop factor given by According to Eq. (4), the product of the Yukawa couplings f and Y is constrained by the neutrino oscillation data, which allows for only one of these couplings to be of order one. We will adopt the choice Y ∼ O(1) and f 1, which maximizes the neutrino NSI in the model [11].
For the IceCube phenomenology, we are specifically interested in the light charged scalar scenario. This is confronted with several theoretical and experimental constraints, such as charge breaking minima, electroweak precision tests, charged lepton flavor violation (cLFV), collider constraints from LEP and LHC, lepton universality tests and monophoton constraints. It was shown [11] that both h + and H + charged scalars can be as light as 100 GeV, while satisfying all these constraints. The main constraints for light charged scalars come from direct searches at LEP, which are applicable as long as Y αe = 0 for any flavor α. More stringent limits from lepton universality tests in W decays [45] will apply if Y ee = 0, restricting the charged scalars masses to above 130 GeV [11]. In what follows, we will consider the scenario where Y τ e = 0 and Y ατ = 0 for α = e or µ, which satisfies all constraints for m h + = 100 GeV, and at the same time, allows for the largest NSI effect.
Signature at IceCube.-Expanding the last term in Eq. (3), we get For β = e, this will induce neutrino-electron interactions mediated by the charged scalars h − and H − .
There is no interference with the SM Glashow process (even for α = e), because the Zee burst involves only right-handed electrons. Thus, depending on the mass spectrum of h − and H − , we would expect either one or two additional resonance peaks in the IceCube energy spectrum. We will consider two benchmark scenarios: (i) m h − ≈ m H − , so that the two peaks are indistinguishable, i.e. contribute to the same energy bin, and (ii) GeV, so that the two peaks are distinguishable (i.e. their dominant contributions fall in different energy bins).
To estimate the modification to the event spectrum, we compute the number of events in a given energy bin i as Here T is the exposure time for which we use T 0 = 2653 days, corresponding to 7.5 years of live data taking at IceCube [6]; Ω is the solid angle of coverage and we integrate over the whole sky; E is the electromagneticequivalent deposited energy which is an approximately linear function of the incoming neutrino energy [46]; the limits of the energy integration E min i and E max i give the size of the ith deposited energy bin over which the expected number of events is being calculated; Φ να (E) is the differential astrophysical neutrino+anti-neutrino flux for flavor α, for which we use a simple, singlecomponent unbroken power-law, isotropic flux Φ(E ν ) = Φ 0 (E ν /E 0 ) −γ with the IceCube best-fit values of Φ 0 = 6.45 × 10 −18 GeV −1 cm −2 s −1 sr −1 and γ = 2.89 [9]; and A να is the effective area per energy per solid angle for the neutrino flavor ν α , which includes the effective neutrinomatter cross section, number density of target nucleons/electrons and acceptance rates for the shower and track events. In presence of new interactions as in Eq. (6), only the neutrino-electron cross section gets modified, which in turn affects the effective area. For the SM interactions only, we use the publicly available flavordependent effective area integrated over solid angle from Ref. [5] (for 2078 days of IceCube data), along with a 67% increase in the acceptance (for 2653 days of data) [47]. In presence of non-SM interactions as in Eq. (6), we rescale the effective area accordingly by taking the ratio of the cross sections, assuming that the acceptance remains the same.
In the SM, neutrinos interact with nucleons via charged-and neutral-current processes. In the energy range of interest, the corresponding deep inelastic scattering cross sections can be approximated by [48] In addition, there are subdominant antineutrino-electron interactions, except in the energy range of 4.6-7.6 PeV, when theν e -e − interaction becomes important due to the Glashow resonance [12]. In the vicinity of the resonance, the dominant piece of the cross section can be expressed by a Breit-Wigner distribution as [49]: where s = 2m e E ν and Γ W is the total width of the W boson with BR(W − →ν e e − ) = 10.7% and BR(W − → had) = 67.4% [50]. At resonance, Eq. (9) gives σ Glashow (E ν = 6.3 PeV) = 3.4 × 10 −31 cm 2 , about 240 times larger than σ CC ν(ν)N (E ν = 6.3 PeV) ≈ 1.4 × 10 −33 cm 2 . However, due to the narrowness of the resonance and the E −γ ν nature of the astrophysical neutrino flux, the ratio of the reconstructed events between the resonance-inducedν e -e and non-resonant ν(ν)-N interactions is not so pronounced in the event spectrum, as shown by the red-shaded histograms in Fig. 1. For instance, for E ν > 4 PeV, N Res /N non−Res ∼ 2.05 giving a total of about 0.3 events in the Glashow bin for the IceCube best-fit flux. Also shown in Fig. 1 (gray shaded) are the total expected atmospheric background (from atmospheric muons and neutrinos, as well as the charm contribution) and the 7.5 year IceCube data [9]. The vertical line at 60 TeV denotes the low-energy cutoff for the HESE analysis, i.e. the bins below this energy are not considered in the fitting process. Now in presence of light charged scalars, we expect a new resonance forν α e − → X − → anything (where X − = h − , H − for the Zee model) with a cross section similar to Eq (9): where Γ X = αβ |Y αβ | 2 sin 2 ϕ m X /16π is the total decay width of X. The factor of 1/3, compared to Eq. (9), is due to the difference in the degrees of polarization between scalar and vector bosons. In Fig. 1, we consider a benchmark case with m h − ≈ m H − = 100 GeV, so that the two new resonances due to h − and H − coincide, and thus, maximize the effect in the bin containing the resonance energy E ν = m 2 h − /2m e , as shown by the light, medium and dark blue-shaded histograms corresponding to three illustrative values of Y τ e = 1, 0.5, 0.25 respectively. The excess events due to this new resonance mostly populate the energy bins between 7.6-12.9 PeV, distinguishable from those dominated by the Glashow resonance bin (4.6-7.6 PeV), and the effect is more pronounced for larger Yukawa couplings, as expected from Eq. (10). Here we have taken the maximal mixing ϕ = π/4 and BR(h − →ν τ e) = 60%, BR(h − →ν β τ ) = 40% (with β = e or µ) for a fixed Y τ e given above and accordingly chosen Y βτ , while all other Yukawa couplings Y αβ are taken to be much smaller than one to satisfy the cLFV constraints [11]. Note that as we increase the mass difference ∆m h ≡ m H − − m h − , the two peaks start populating different bins, but because of the falling power-law flux, the effect is more pronounced in the smallest resonance energy bin. Also note that we cannot make ∆m h exactly zero, otherwise the neutrino mass vanishes [cf. Eq. (5)]. From Fig. 1, it is clear that for a given charged scalar mass m h − , the Yukawa coupling Y τ e cannot be made arbitrarily large without spoiling the best-fit to the observed IceCube HESE data. We can use this fact to derive new IceCube constraints in the m h − − Y τ e plane, as shown in Fig. 2 by the thick black contours. The curve labeled 'IC 1T 0 ' represents the parameter set which would give rise to one event when summed over the last three bins considered by IceCube best-fit (4.6 < E ν /PeV < 10) with the current exposure T 0 = 2653 days [9], and the other curves are with increased exposures of 2T 0 , 4T 0 , 10T 0 and 50T 0 respectively, keeping the other parameters in Eq.  Probing NSI.-The same Yukawa interactions in Eq. (6) lead to neutrino NSI with electrons, given by [11] where G F is the Fermi coupling constant. In Fig. 2, we show the predictions for ε τ τ by thin black dotted contours. Here again we have taken the maximal mixing case with ϕ = π/4 to get the largest possible NSI. The shaded regions are all excluded: blue shaded by direct LEP searches [51,52] and lepton universality (LU) tests in tau decays [50]; green shaded by LEP dilepton searches [45,53]; purple shaded (dashed) by LEP monophoton searches off (on) Z-pole [54,55]; red shaded by BOREXINO [56], orange shaded by global fit to neutrino oscillation plus COHERENT data [57], and brown shaded by IceCube atmospheric neutrino data [58,59]. For more details on these exclusion regions, see Ref. [11]. Note that the atmospheric neutrino data only constrains |ε τ τ − ε µµ | < 9.3% [58,59], which in the Zee model is equivalent to a bound on ε τ τ itself, because both ε τ τ and ε µµ cannot be large simultaneously due to stringent cLFV constraints. One can do similar analysis for other ε αβ , which are however restricted to be less than a few % [11], and hence, are not so promising for IceCube. We should comment here that the LEP dilepton constraints [45] shown in Fig. 2 (green shaded region) are equally applicable to the extra neutral CP-even and odd scalars (H, A) present in the Zee model, since they could modify the e + e − → + α − α cross section via t-channel mediation through the Yukawa couplings Y αe . Moreover, these neutral scalars are required to be quasi-degenerate with the doublet charged scalar H + in order to satisfy the electroweak T -parameter constraint [11].
From Fig. 2, we see that the existing constraints on NSI are stronger than the current sensitivity of highenergy IceCube data. However, the (non)observation of a resonance-like feature in the future IceCube HESE data could provide a complementary probe of the allowed NSI parameter space, which can even supersede the future DUNE sensitivities (shown by the upper and lower blue solid lines for 300 and 850 kt.MW.yr exposures, respectively [60]). We note here that an exposure of 10T 0 does not necessarily require 75 years of IceCube running, as a number of factors could improve the conservative projected IceCube limits shown here in a non-linear fashion. For instance, the future data in all the bins may not scale proportionately to the current data and may turn out to be in better agreement with the SM prediction, thus restricting even further any room for new physics contribution. Similarly, the energy-dependent acceptance rate might improve in the future (as it did by 67% from two to seven years of data [47]), thereby increasing the effective area, and hence, the 'effective' exposure time defined here at a rate faster than linear. Finally, the proposed IceCube-Gen2 with 10 km 3 detector volume [61] could increase the total effective exposure by about an order of magnitude. At the very least, combining IceCube data with the future KM3NeT data [62] could increase the effective exposure by a factor of two.
Before concluding, we remark that for heavier charged scalars, the resonance energy will be shifted to higher values at which IceCube will become less sensitive, given an isotropic power-law spectrum. However, if there exists powerful transient sources of UHE neutrinos, then IceCube, as well as current and next-generation radio-Cherenkov neutrino detectors, such as ARA [63], ARI-ANNA [64], ANITA [65], GNO [66] and RNO [67], could be sensitive to electrophilic charged scalars up to a TeV or so (corresponding to the resonance energy of EeV), as might occur e.g. in left-right symmetric model [68]. The possibility of a larger flux at higher energies, together with better energy resolution of the IceCube detectors, might help distinguishing the degenerate versus non-degenerate charged-scalar mass spectrum by exploiting the 'dip' features in Fig. 2.
Conclusion.-We have proposed a new way to probe light charged scalars using a Glashow-like resonance feature in the ultra-high energy neutrino data at IceCube and its future extensions. The same interactions that lead to the new signature at IceCube also give rise to observable non-standard interactions of neutrinos with matter, so that the UHE neutrinos provide a complementary probe of NSI. Taking the popular Zee model of radiative neutrino mass as a prototypical example, we have provided an explicit realization of this idea.