Inferring astrophysical neutrino sources from the Glashow resonance

We infer the ultrahigh energy neutrino source by using the Glashow resonance candidate event recently identified by the IceCube Observatory. For the calculation of the cross section for the Glashow resonance, we incorporate both the atomic Doppler broadening effect and initial state radiation ν¯ee−→W−γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\nu}}_e{e}^{-}\to {W}^{-}\gamma $$\end{document}, which correct the original cross section considerably. Using available experimental information, we have set a generic constraint on the ν¯e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\nu}}_e $$\end{document} fraction of astrophysical neutrinos, which excludes the μ-damped pγ source around 2σ confidence level under the assumption that neutrino production is dominated by the ∆-resonance. While a weak preference has been found for the pp source, next-generation measurements will be able to distinguish between ideal pp and pγ sources with a high significance assuming an optimistic single power-law neutrino spectrum. The inclusion of multi-pion production at very high energies for the neutrino source can weaken the discrimination power. In this case additional multimessenger information is needed to distinguish between pp and pγ sources.


Introduction
The IceCube Observatory has successfully established the observation of ultrahigh energy (UHE) neutrino flux below a few PeV energies [1][2][3][4][5][6][7][8].However, it remains a mystery as to where those neutrinos come from.One of the most popular mechanisms rests on accelerated cosmic rays colliding with ambient targets around the source [9][10][11][12][13][14].There is a variety of source models for UHE neutrinos [15][16][17][18] which can usually be classified into the pγ and pp types depending on whether the target particle is a photon or a proton.
For both pγ and pp sources, after traveling an astronomical distance the fluxes of three neutrino flavors strongly mix with each other due to neutrino oscillations, which ends up with a nearly democratic flavor composition ϕ ⊕ νe + ϕ ⊕ νe : ϕ ⊕ νµ + ϕ ⊕ νµ : ϕ ⊕ ντ + ϕ ⊕ ντ ≈ 1 : 1 : 1 at Earth § .It is unlikely to disentangle those two sources by traditional flavor ratio measurements [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].The difference between those two sources lies in the composition of neutrinos and antineutrinos.For the pγ neutrino source, cosmic rays collide with photons to produce charged pions (mostly π + below a certain energy threshold) followed by the decays π + → µ + ν µ and µ + → e + ν µ ν e , which results in more neutrino flux than antineutrino flux, i.e., ϕ S ν : ϕ S ν = 2 : 1.In comparison, the pp source will give rise to nearly equal fractions of π + and π − , which leads to ϕ S ν : ϕ S ν = 1 : 1.The key to distinguishing those two sources is by measuring the ν e fraction f νe ≡ ϕ νe /(ϕ νe + ϕ νe ), thanks to the Standard Model process ν e e − → W − → X predicted by S. L. Glashow [36].Due to the resonance enhancement, the cross section of ν e e − scattering around E ν ≈ 6.3 PeV is larger than that of the deep inelastic scattering (DIS) by more than two orders of magnitude.This promises us an excellent channel to differentiate between the ideal pp (with f ⊕ νe ≈ 0.5) and pγ (with f ⊕ νe ≈ 0.23) sources, as continuously § Throughout this work, we use the superscript '⊕' to denote the quantity at Earth and 'S' to denote that at source.anticipated in previous works [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53].It is worthwhile to emphasize that the above argument holds only for ideal pp and pγ cases.Other possible neutrino sources such as neutron and charm decays can give rise to a ν e fraction different from the typical pion decays.Moreover, if the multi-pion channel in the pγ scattering dominates the neutrino production when the collision energy is very high, more multimessenger information about the source in addition to the ν e fraction is necessary to disentangle the source degeneracy.In practice, the overall diffuse neutrino flux might be contributed by different types of sources, and the ν e fraction f ⊕ νe can take any reasonable values in between.In this regard, we shall first treat the ν e fraction as a free parameter to be determined by the experimental probes regardless of the model assumptions.The experimental information about this parameter can then be used for theoretical interpretations under specific source assumptions.
Excitingly, with its unprecedented detection volume, the IceCube Observatory has collected one candidate event with an energy deposition E dep = 6.05 ± 0.72 PeV in the sample of partially contained events [1].The probability that this event stems from the Glashow resonance (GR) is high, around 99 % by using the best-fit neutrino flux taken from ref. [54].In this work, a timely quantitative assessment is carried out to infer the ν e fraction by taking f ⊕ νe as a free parameter and to explain the level that we can differentiate between pγ and pp sources.We have included both the radiation of initial photons [55][56][57] and the Doppler broadening effect [58] while calculating the GR events.Using the updated cross section, we investigate both the results for the current GR candidate in IceCube as well as the prospects of next-generation experiments.

A full treatment of Glashow resonance
As more and more UHE neutrino data have been accumulated, it becomes increasingly important to take into account the subleading effects for the theoretical evaluation of the GR.There are mainly two effects that should be emphasized: (i) the initial state radiation (ISR) [55,56]; (ii) the Doppler broadening effect [58].At the leading level, the cross section for the process ν e e − → W − → X reads [1] where M W ≈ 80.433 GeV is the mass of the W boson, Γ W ≈ 2.09 GeV is the total decay width and Br W − →νee − ≈ 10.7 % is the branching ratio of the channel W − → ν e e − .The ISR and the Doppler broadening effect are found to considerably modify the above picture and should be included for completion.Let us start with the ISR.This effect becomes increasingly notable when the centerof-mass (COM) energy is much higher than the mass of the initial charged lepton, for which the collinear emission of photons is significant.For instance, in the Large Electron-Positron Collider (LEP), the ISR should be taken into account when analyzing the Z boson peak [59].For UHE neutrino telescopes like IceCube, the ISR cross section near the GR will receive a large enhancement factor of ln(M W /m e ) ≈ 12 on top of the fine structure constant α.
The ISR can be consistently included by using the structure function approach in analogy with the DIS off hadrons.The modified cross section will be [56] σ where Q represents the energy scale, x is the longitudinal momentum fraction of the electron after the photon radiation, σ (0) is the cross section without the initial-state photon, and Γ e/e is the structure function of the electron.We take the structure function from ref. [60] which includes soft photons resummed to all orders and hard photons up to O(α 3 ).
The second effect of interest is the Doppler broadening due to the motion of atomic electrons [58].The velocity of atomic electrons β is typically of the order O(α c).A simple estimation shows that this velocity will shift the COM energy square from s = 2E ν m e to 2E ν m e (1 − β cos θ), where θ is the angle between the electron velocity and the incoming neutrino in the laboratory frame.This broadens the COM energy by around 0.6 GeV in comparison to the W decay width Γ W = 2.09 GeV.Non-relativistic electrons in the atom have the four-momentum (m e + |k| 2 /(2m e ), k), where |k| ≈ m e β.By integrating over the electron wave function, one can arrive at the total cross section [58] σ where ϕ represents the azimuth angle, F (β) is the velocity distribution of electrons and x ′ = cos(θ).Since the calculation framework was already outlined in ref. [58], we give more details about the updated calculation in appendix A.
Those two effects can be combined, and their joint result is shown as the red curve in fig. 1 for the H 2 O target, along with the cross sections without (solid black curve) or modified by only one (blue and orange curves) of those effects.In comparison, the charged-current (CC) and neutral-current (NC) interactions are depicted as dashed and dotted black curves, respectively.Some remarks on the results are given below.
• The ISR will reduce the peak at the resonance energy E ν ≈ 6.3 PeV by almost 20 %.
Furthermore, the cross section above the resonance energy is enhanced by a factor of more than two.This is due to the radiative return phenomenon, for which the photon in the process ν e e − → W − γ carries away some energy such that the W production will be made on shell even if √ s > M W . • The Doppler broadening effect for the H 2 O target is small compared to the ISR in the logarithmic scale.To see the detailed impact we also show the result in a flat scale as fig.4. The resonance peak is reduced slightly, while the width is broadened due to the motion of atomic electrons.• The combined result of the ISR and the Doppler broadening is obtained with a convolution, which reduces the peak by around 30 %.However, we should note that those effects will be partly smeared by the finite energy resolution of the IceCube detector.
We have checked that the eventual effect can decrease the events within the energy window near the GR by almost 10%.With the full GR cross section, we are able to calculate the event rate in IceCube and compare it to both experimental data available now and those from future experiments.

Analysis framework
In order to constrain the ν e fraction in the total diffuse neutrino flux, we calculate the likelihood by fitting models with different values of f νe = ϕ νe /(ϕ νe + ϕ νe ) to the available IceCube data.The reason why we use f νe to measure the ν e fraction is that it almost solely determines the spectrum of single cascade event topology at PeV energies in the IceCube detector.
The observed GR candidate in IceCube belongs to the PeV energy partially contained events (PEPEs), in comparison to the high energy starting events (HESEs) where the shower is fully contained inside the fiducial volume.Even though the PEPE effective volume is nearly twice the volume of HESE at PeV energies only one event with an energy deposition E dep = 6.05 ± 0.72 PeV has been observed within the energy window 4 PeV < E dep < 10 PeV.For HESE three PeV events have been collected [61], nicknamed Bert, Ernie and Big Bird.However, all of them have energies below 3 PeV, which are most likely contributed by the DIS.Even though the GR has not significantly arisen in the HESE sample, HESE is useful to fix the normalization and shape of UHE neutrino flux which are crucial for our extraction of the ν e fraction.
In ref. [6], the IceCube Collaboration has analyzed the overall UHE neutrino flux with HESEs collected over 7.5 years, assuming a flavor ratio ϕ ⊕ νe + ϕ ⊕ νe : ϕ ⊕ νµ + ϕ ⊕ νµ : ϕ ⊕ ντ + ϕ ⊕ ντ = 1 : 1 : 1.During our analysis we will use the HESE results including uncertainties from ref. [6] to set the spectrum of neutrino flux and use PEPE to extract the ν e fraction f ⊕ νe .Note that a more thorough analysis would assume a completely free flavor ratio.However, on the one hand, the latest IceCube HESE fit available has fixed the flavor ratio [6].On the other hand, ideal pp and pγ astrophysical models reasonably prefer such a democratic ratio after neutrino oscillations over an astronomical distance.
For demonstration, we choose two benchmark flux models in our analysis: (i) the unbroken single power-law model; (ii) the single power-law model with an exponential energy cutoff.The former one reads which represents models consistent with the Fermi acceleration mechanism and extends to infinite energies.In practice, the reachable energy of astrophysical accelerators always features a cutoff due to the Hillas criterion [62].For the cutoff model, the flux in eq.(3.1) will be multiplied by a suppression factor exp (−E ν /E cutoff ).To confine the flux parameters, we construct a likelihood based on the results in ref. [6]: with the best-fit values Φ bf 0 = 6.37 and γ bf = 2.87, as well as the 1σ errors σ(Φ 0 ) = 1.54 and σ(γ) = 0.2.For the cutoff model we further derive the likelihood for E cutoff from fig.VI.9 of ref. [6] where the test-statistic has been marginalized.Note that in this case we have ignored possible correlations among Φ 0 , γ and E cutoff , which are not provided.Nevertheless, such a choice will be more conservative because less information is utilized in our analysis.
After the prior knowledge of {Φ 0 , γ(, E cutoff )} has been established by HESE, we continue with fitting f ⊕ νe to PEPE.The task is to calculate the likelihood L νe (f ⊕ νe ) with the GR candidate we have.The joint likelihood can then be obtained with L tot = L 6ν × L νe for the parameter set Θ ≡ {Φ 0 , γ(, E cutoff ), f ⊕ νe }.In the frame of extended likelihood analysis of unbinned data [63], the likelihood is calculated with where µ DIS and µ GR are the expected event numbers within the energy window E dep ∈ [4,10] PeV for the DIS and the GR, respectively, and # i represents in general all possible GR candidates.Moreover, P DIS/GR ( # i|Θ) is the normalized probability to have an event at # i's energy for the given model parameter set Θ. Since there is only one GR candidate so far we have n = 1 in eq.(3.3).The event numbers can be obtained by integrating the flux and cross sections with the detector configuration.

Likelihood
Single power law with energy cutoff The likelihood (in blue) or posterior (in brown) of the ν e fraction f ⊕ νe inferred from the Glashow resonance event in IceCube with 4.6 years of data taking.The upper panel assumes a single power-law flux model with central values and uncertainties from ref. [6], while the lower one has incorporated an exponential cutoff E cutoff in the neutrino spectrum.The expected ν e fractions of three representative ultrahigh energy neutrino source models, including the ideal pp (f ⊕ νe ≈ 0.5), the ideal pγ (f ⊕ νe ≈ 0.23) and the ideal µ-damped pγ (f ⊕ νe ≈ 0) sources, are indicated by the dashed vertical lines.When the pγ collision energy at the source is very high, we may expect deviations from those ideal source models.By assuming an equal mixture of single-pion and multi-pion production at the source, we find the expected value f ⊕ νe ≈ 0.36 at Earth, shown as the dotted vertical lines.If multi-pion production is more dominant than this assumption, the expected f ⊕ νe should move to even larger values.The sensitivity of the future IceCube-Gen2 project with an effective exposure of ten (fifty) years is shown as the dashed (dotted) blue curves, assuming that the pp source is dominant with Φ 0 = 6.37, γ = 2.7 and E cutoff = 5 PeV.

Main results
With the framework above, we can compute the total likelihood L tot as a function of the parameter set {Φ 0 , γ(, E cutoff ), f ⊕ νe }.The likelihood can then be used for either frequentist or Bayesian interpretations.For the frequentist interpretation, we obtain the likelihood maximum L max tot (f ⊕ νe ) by marginalizing over the other parameters.For the Bayesian interpretation, we need to derive the posterior distribution of f ⊕ νe by integrating over the likelihood and priors.We choose flat priors on Φ 0 , γ, f ⊕ νe and ln E cutoff for illustration.VI.9 of ref. [6], while the solid curve is derived from the Glashow resonance candidate event by marginalizing over the other model parameters.
Our main results are given in fig.2, which shows the likelihood function (in blue) or posterior distribution (in brown) of the ν e fraction f ⊕ νe inferred from the IceCube 4.6-year data.Uncertainties from neutrino flux parameters have been systematically included and marginalized when we constrain f ⊕ νe .The upper and lower panels stand for the assumptions of an unbroken single power-law flux model and a single power-law model with a varying exponential energy cutoff, respectively [6].For blue curves, the horizontal lines with −2 ln L = 1 and 4 roughly set the 1σ and 2σ confidence levels.For brown regions, the 1σ and 2σ credible intervals have been covered from dark to light colors.
We find that for all cases, the µ-damped pγ source with f ⊕ νe ≈ 0 (single-pion production via the ∆-resonance for the ideal scenario) is excluded by around 2σ level.The current IceCube 4.6-year data weakly favor the pp source but are not able to exclude the ideal pγ source considerably (only at 1σ or so); see the dashed vertical lines.While interpreting the above results, one must keep in mind that neutrinos may not only be produced by the ideal ∆-resonance of the pγ scattering, but also by other possible effects that can dominate at high energies [64,65], such as multi-pion production, higher resonances, and the direct (t-channel) production of pions.Note that the above considerations do not affect our model-independent results of f ⊕ νe extracted from experimental data.For those cases, the theoretically expected value of f ⊕ νe for the pγ source will shift towards larger values.The actual magnitude of the deviation depends on the details of the π + and π − mixture at the source.For demonstration, we assume that the single-pion and multi-pion channels have the same production rate at the source and draw the expected value f ⊕ νe ≈ 0.36 as the dotted vertical lines in fig. 2. If the multi-pion channel contributes more, this vertical line should move even further to the right.On the other hand, for the pp source the multi-pion contribution does not change the expected value of f ⊕ νe .Last but not least we should emphasize that the GR event can also constrain the pos-sible energy cutoff E cutoff in the neutrino spectrum.The original best-fit value of E cutoff without GR is around 5 PeV in ref. [6], with a 2σ lower boundary at 0.5 PeV.The presence of the GR candidate event will push the 2σ lower boundary to 2.2 PeV, as illustrated in fig. 3.

Outlook
Using the recent GR candidate event identified by IceCube, we have performed an analysis to infer the ν e content in UHE astrophysical neutrinos.We treat the ν e fraction as a model-independent free parameter and have set a generic constraint on it by including the uncertainties in the UHE neutrino flux.From the candidate event measured so far, we find a weak preference for the pp source under the ideal assumption.The situation will be greatly improved by the upcoming next-generation neutrino telescopes.If the neutrino production in the pγ source is dominated by other channels at higher energies such as multi-pion production, we would need additional information from the multi-wavelength observations of the source to distinguish between pp and pγ sources.
In the future, there are many projects such as IceCube-Gen2 [66,67], Baikal-GVD [68], KM3NeT [69], P-ONE [70], TAMBO [71], TRIDENT [72] and so on, which will provide very valuable sensitivities to PeV astrophysical neutrinos [73][74][75][76].We take IceCube-Gen2 for demonstration by rescaling the current IceCube target mass by ten times, and perform a count analysis in the energy window of [4,10] PeV.The sensitivity for ten (fifty) years of effective exposure is shown as the dashed (dotted) curves in fig. 2. Because the flux parameters {Φ 0 , γ(, E cutoff )} can be very precisely determined in the future [67], we choose a reasonably optimistic spectrum as Φ 0 = 6.37, γ = 2.7 and E cutoff = 5 PeV in making the forecast; see fig.16 of ref. [67] for example.It is worth noting that the tau neutrino telescope TAMBO can be sensitive to the Glashow resonance event by searching for the tau-induced showers from W decays.Unfortunately, with a dedicated simulation close to the TAMBO setup, we find that the event number is too small, e.g., only two events with an optimistic flux and ten years of exposure, compared to the DIS background of O(100).We will elaborate on the related analysis in a future work.
Assuming the pp type as the true source, i.e., f ⊕ νe = 0.5, we expect eleven GR events in IceCube-Gen2 with ten years of exposure for the optimistic single power-law model.If we take an exponential cutoff E cutoff = 5 PeV in the spectrum, the event expectation would be reduced to three.The expected number of events is still diverse due to low statistics of events at PeV energies.For the single power-law model, IceCube-Gen2 with ten years of exposure can already differentiate ideal pp from pγ sources with a 2σ confidence level.However, if there is an exponential cutoff at 5 PeV, an effective exposure of fifty years would be required to reach the 2σ level.Those results can also be applied to other telescopes by adjusting the effective exposure.By measuring the spectrum precisely in the future, one may go beyond the assumptions of single power-law flux model (with cutoff) and take the spectrum with a general energy dependence.
The hybrid cascade and early muon reconstruction in IceCube can already greatly improve the angular resolution of the GR shower.In case of the increased statistics, GR events detected in future experiments can also be used to produce a map of the sky and identify associated PeVatrons [77][78][79].Our main point is that knowledge about neutrino sources will be significantly improved by those upcoming facilities with large statistics, which also guarantees a robust frontier for possible new physics studies [80][81][82][83][84][85][86][87][88][89].We follow the procedure outlined in ref. [58] to include the Doppler broadening effect of atomic electrons.By integrating over angular variables in eq. ( 2.3), we arrive at where Now the problem is attributed to the integration over the averaged electron velocity distribution F (β).In terms of the wave function of an electron with quantum numbers n and l, the distribution reads dr r n+1 e −µr j l (kr) , where k = m e β and µ n l = ξ n l /a 0 .Here, a 0 denotes the Bohr radius, ξ n l = Z eff /n = (Z−σ n l )/n, and σ n l accounts for the screening of the nuclear charge by the other electrons in the atom.
After the integration, we can get the velocity distribution for atoms up to Z = 26 [58]: Note that we have checked the expressions in ref. [58] and corrected possible discrepancies as in our eqs.(A.5) and (A.10).We take the ice molecule H 2 O as an example.For oxygen, µ 1s = 7.6579, µ 2s = 2.2458 and µ 2p = 2.2266 [90], and for hydrogen µ 1s = 1.We weigh the distribution functions by averaging over the electron numbers: Using eq.(A.1) together with eq.(A.11) we get the Doppler broadened cross section for ice as the target, which is depicted in fig. 4. The effect reduces the peak by about 12 %.Even though the total cross section integrated over the initial neutrino energy is barely altered, the broadening effect will make a difference when a non-uniform neutrino spectrum is considered.

Figure 3 .
Figure3.The log-likelihood of the energy cutoff E cutoff .The dashed curve is taken from fig.VI.9 of ref.[6], while the solid curve is derived from the Glashow resonance candidate event by marginalizing over the other model parameters.

2 )Figure 4 .
Figure 4. Cross section for the Glashow resonance process νe + e − → W − → X with and without Doppler broadening and assuming ice (H 2 O) as the target.The black curve represents the cross section without Doppler broadening and for the orange curve Doppler broadening is included.

F
ice (β) = 2F H (β) + 8F O (β) 10 .(A.11) Cross section for the Glashow resonance process ν e + e − → W − → X with and without the initial state radiation and Doppler broadening effect.The black curve shows the cross section without initial state radiation and Doppler broadening, the blue dotted one includes initial state radiation and the orange dotted one includes Doppler broadening.The red curve is the cross section with both Doppler broadening and initial state radiation effects, and the tabulated result of this curve is given in our supplementary material.Both the broadening and the radiative return are visible.For the Glashow resonance curves we averaged over the electrons in H 2 O for the target.