High-energy cosmic neutrinos as a probe of the vector mediator scenario in light of the muon $g-2$ anomaly and Hubble tension

In light of the recent Muon $g-2$ experiment data from Fermilab, we investigate the implications of a gauged $L_{\mu} - L_{\tau}$ model for high energy neutrino telescopes. It has been suggested that a new gauge boson at the MeV scale can both account for the Muon $g-2$ data and alleviate the tension in the Hubble parameter measurements. It also strikes signals at IceCube from the predicted resonance scattering between high-energy neutrinos and the cosmic neutrino background. We revisit this model based on the latest IceCube shower data, and perform a four-parameter fit to find a preferred region. We do not find evidence for secret interactions. The best-fit points of $m_{Z'}$ and $g_{\mu\tau}$ are $\sim10$~MeV and $\sim0.1$, respectively, depending on assumptions regarding the absolute neutrino masses, and the secret interaction parameter space allowed by the observed IceCube data overlaps with the regions of the parameter space that can explain the muon $g-2$ anomaly and Hubble tension as well. We demonstrate that future neutrino telescopes such as IceCube-Gen2 can probe this unique parameter space, and point out that successful measurements would infer the neutrino mass with $0.06~{\rm eV}\lesssim \Sigma m_\nu\lesssim 0.3~{\rm eV}$.

In light of the recent Muon g − 2 experiment data from Fermilab, we investigate the implications of a gauged Lµ − Lτ model for high energy neutrino telescopes. It has been suggested that a new gauge boson at the MeV scale can both account for the Muon g − 2 data and alleviate the tension in the Hubble parameter measurements. It also strikes signals at IceCube from the predicted resonance scattering between high-energy neutrinos and the cosmic neutrino background. We revisit this model based on the latest IceCube shower data, and perform a four-parameter fit to find a preferred region. While the data are consistent with the absence of resonant signatures from secret interactions, we find the preferred region consistent with the muon g − 2 anomaly and Hubble tension. We demonstrate that future neutrino telescopes such as IceCube-Gen2 can probe this unique parameter space, and point out that successful measurements would infer the neutrino mass with 0.05 eV Σmν 0.3 eV.

I. INTRODUCTION
The recent data from the Fermilab Muon g − 2 Collaboration indicates that muon magnetic moment may disagree with phenomenological predictions from the Standard Model (SM) [1], consistent with the earlier E821 experiment at Brookhaven [2]. Although it can be explained by the SM physics through the hadronic vacuum polarization [3], this may indicate beyond the Standard Model (BSM) physics coupled to the muons. As a consequence of electroweak gauge symmetry, modifications to muon physics would imply modified neutrino physics as well given that charged leptons and neutrinos come together in SU (2) doublets. In this paper, we explore an example of this in the context of gauged lepton number. A very large number of possible interpretations of the new Muon g-2 results have already appeared .
As is well known, gauging the lepton number combination L µ − L τ is anomaly free. It is is also experimentally challenging to probe given that its main effects are to modify the interactions of unstable charged leptons and neutrinos. Intriguingly, this combination of gauged lepton numbers can both explain (g − 2) µ and be consistent with the constraints from null experiments [42][43][44]. In principle, this scenario can be tested at NA64µ, the European Spallation Source, and DARWIN [45]. This work focuses on a different probe of gauged L µ − L τ , involving only neutrinos. We study the current and future sensitivity of the IceCube neutrino telescope to such new gauge interactions, by examining in detailed the modifications to the spectrum of high-energy cosmic events [46,47]. Such interactions can also alleviate the tension in Hubble parameters between the local value and cosmic microwave background (CMB) data, through delaying the neutrino free-streaming by self-interactions [48][49][50][51][52][53] or adding the effective number of relativistic (neutrino) species [54][55][56][57]. We find that the data from IceCube and the Muon g − 2 collaboration can be combined to yield a non-trivial determination of neutrino masses. Earlier work has also examined the impact of gauged L µ − L τ at IceCube [58][59][60]. We also note that extended gauge symmetries (e.g., different baryon and lepton number combinations) may allow one to also connect to the LMA-Dark solution of neutrino oscillations for gauge boson masses in the range we are considering < l a t e x i t s h a 1 _ b a s e 6 4 = " y X m j y z I Z N 9 L P k s W o f + Z 7 7 v o P 0 g t q a Q B 6 E 6 S H Z P i d a v 0 c D 9 U 9 k m D M / X 7 R E I D r Y e B b 5 M B N V f 6 t z c S / / L a s e n v d R I u o 9 i A Z J + L + r H A J s S j Z n C P K 2 B G D C 2 h T H H 7 V 8 y u q K L M 2 P 6 K t o S v S / H / p F W t u D u V 6 u l W u X a Y 1 1 F A q 2 g N b S A X 7 a I a O k F 1 1 E Q M 3 a J 7 9 I i e n D v n w X l 2 X j 6 j Y 0 4 + s 4 J + w H n 9 A P n C m u 8 = < / l a t e x i t >

Cosmology
(g-2)µ + H0 tension < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 X D 3 c w P J t Y y t G N Y v 9 f Z s 9 8 g N c 6 8 = " FIG. 1: Preferred and excluded regions in the mediator and neutrino mass plane. The IceCube data prefers parameters shown in the red curves, while the combination of muon (g−2) and Hubble tension prefer the region between the blue curves. The combination of data sets can be used to infer non-trivial bounds on the absolute neutrino masses (see text for details).
In Fig. 1 we show the IceCube preferred region in red in the plane of the unknown gauge boson mass and the unknown neutrino mass. The shaded area shows the region of neutrino masses excluded by cosmology [63], while the region between m Z ∼ 10 − 17 MeV shows the range of the gauge boson mass that may explain the (g−2) µ observations while remaining consistent with the null results from CCFR [43,64] and Borexino [59,65,66]. For simplicity we have assumed the natural level of loop-induced kinetic mixing for the Borexino constraint, but in principle this can be relaxed by allowing model-dependent additional particles in the loop. This would only allow for slightly lighter gauge boson masses. Similar masses and gauge couplings can also explain the Hubble tension via the extra contribution to the radiation density from the light vector particle (e.g., Ref. [57]).
The remainder of this paper is as follows. In the next section, we introduce the model and the neutrinoneutrino cross section the model predicts. In Sec. III we consider implications of the present IceCube shower data can provide, being careful to allow for fairly weak priors on both particle and astrophysical parameters. We discuss the potential of the next generation detectors in confirming or excluding the model in Sec. IV and we conclude in Sec. V.
We consider a model of the gauged L µ − L τ number, where the Langrangian includes the terms where g µτ is the gauge coupling, Z is the new gauge boson with mass m Z , and the current associated with the new symmetry is where L i is the lepton doublet of the i th generation. This new gauge interaction allows for high-energy neutrinos to scatter on the neutrinos of the cosmic neutrino background (CνB). The most significant effect is the schannel scattering cross section, which in terms of mass eigenstates can be written as, where for a given incoming neutrino energy E ν the Mandelstam variable s j is s j ≈ 2m j E ν , where {m 1 , m 2 , m 3 } are the masses of the mostly active neutrinos, and the width is Γ Z = g 2 µτ m Z /(12π). We have also defined the effective charge Q ij in the above for scattering of the mass eigenstates, which can be written as Q ij = U † GU ij , where G = diag(0, +1, −1), and U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. In principle, the t-channel contributions can also be relevant at large couplings [67][68][69], but we have checked numerically that we can neglect it here.
The neutrinos scattering off each other can cause the depletion of astrophysical neutrinos at the resonant energies E j = m 2 Z /(2m j ) [46,47,70]. Because of Γ Z m Z , the cross section would be localized around E ν = E j : We can calculate the neutrino optical depth as [71]: 1 and n(t) is the neutrino number density.
Here H(z) is the Hubble parameter at redshift z and Θ(x) is the Heaviside function. We can calculate the flux of the astrophysical neutrinos ν i as (e.g., [72]) where H(z) is the Hubble parameter and R(z)(dN ν /dE ν ) is the differential rate density of the astrophysical neutrinos. For the redshift evolution of sources, R(z), we assume that they are distributed according to the starformation rate [73]. Finally, the fluxes of the neutrino flavors ν α are given by φ α = i |U αi | 2 φ i .

III. IMPLICATIONS OF HIGH-ENERGY NEUTRINO DATA
Secret neutrino self-interactions lead to striking spectral distortions at IceCube [46,47,58,67,70]. To find this spectral feature, it is crucial to use a data sample with a good energy resolution. High-energy starting events (HESEs), including showers and starting tracks, are often used, in which the deposit energy distribution is calculated [67,74]. For this purpose, we consider the 6yr shower data sample which is dominated by the electron and tau contributions to the data sample [75]. This has an advantage of having more events especially at lower energies, which enables us to determine the spectral index better. (Note that the 6-yr shower analysis result, s ν = 2.53 ± 0.07, is consistent with the 7.5 yr HESE analysis result, s ν = 2.87 +0. 20 −0.19 [76], within ∼ 1.5σ.) For simplicity, we construct a simple log-likelihood function to do the analysis at the level of the binned flux shown in Fig. 3 of Ref. [75]. The ordinary SM fit that the Ice-Cube collaboration performs, fits to an unbroken powerlaw with the slope s ν Φ(E ν ) = 3 × 10 −18 GeV · s · cm 2 · sr where Φ(E ν )(100 TeV) ≡ 3 × 10 −18 GeV · s · cm 2 · sr −1 Φ WB is the all-flavor flux at 100 TeV. The IceCube collaboration officially found the best-fit values, s ν = 2.53 ± 0.07 and Φ WB = 1.66 +0.25 −0.27 . We perform a log-likelihood analysis considering neutrino energies 4.7 TeV ≤ E ν ≤ 9.1 × 10 4 TeV, sorted into 13 log-spaced bins as in Ref. [75]. Instead of the calculation of the expected number of events we calculate E 2 ν Φ(E ν ) at each bin of energy and compare it with the flux data given in Fig. 3 of Ref. [75]. We caution that our approach is only approximate. At present, detailed information is not publicly available for the shower data, and we do not take into account details such as effects of the energy smearing due to neutral-current interactions and systematic errors from the atmospheric background. Nevertheless, we confirm that our analyses are broadly consistent with the IceCube results, giving the best fit values s ν = 2.49 and Φ W B = 1.65 with an allowed region similar to Fig. 2 of Ref. [75], so the method is accurate enough for the purpose of this work.
In Fig. 2 we display the marginalized best-fit regions. In each panel, the two parameters not shown have been marginalized over. We consider two cases for the neutrino mass spectrum. Case I fixes the neutrino masses to Another parameter we do not marginalize over is the redshift evolution of astrophysical sources. To fit our model to the IceCube data we assume that R(z) follows the star-formation rate, which is reasonable for most astrophysical models [77]. We fit the data to both (Φ W B , s ν ) as the collaboration does, but also the two new particle physics parameters associated with the new gauge symmetry, g µτ and m Z .
Let us first discuss the upper left panel, which displays the best-fit region in the mass-coupling plane. We see that both Cases I and II prefer a relatively narrow range of vector masses. This fits the expectation from the schannel resonance cross section which is sharply peaked around At present, the IceCube shower and HESE data lack statistics in the 0.2 − 1 PeV range [75,76]. This possible dip-like feature has been paid attention to for several years [58]. The self-interaction cross section around these resonance energies can induce significant depletion  [64], the blue region is excluded by Borexino [66], the green region is a limit from the IceCube HESE data [78] and the yellow region is bounded by cosmology [57]. The dashed curves are the preferred regions that explain or alleviate the anomalies: the purple band is the region favored by the (g − 2)µ discrepancy, the green band is the region which alleviates the tension in the Hubble parameter measurements [57], the two cyan and red regions represent the two cases considered in this work as regions preferred by the IceCube shower data.
of neutrino flux, and produce better fits to the shower data.
The fact that there appears to be no upper bound on the coupling in the upper left panel of Fig. 2 can be understood as a result of its partial degeneracy with the normalization of the source flux. Larger couplings result in greater flux depletion, but this can be partially offset with larger normalization (and to some extent other slopes). This explanation is confirmed in the bottom middle panel, which displays the best-fit regions in the coupling-normalization plane. Let us also mention depending on how exactly we calculate the contained energy information in each bin there could be less than 20% of analysis related uncertainties which could result in moving the mass regions slightly to the left, which accepts larger values for the normalization. The conclusions of this work are however unchanged.
We show in Fig. 3 the preferred regions by IceCube we find in this work in the mass-coupling plane, accompanied by the excluded region by the CCFR experiment using the measurement of the neutrino trident crosssection [64] (gray shaded region), the blue shaded region is the excluded region by Borexino [66], while the purple band represents the preferred 2σ region from the (g − 2) µ discrepancy [45]. It is also important to note that such leptophilic interactions mentioned in this work can also affect the relativistic degrees of freedom of neutrinos and so to avoid tension with cosmology it requires that m Z 10 MeV so that ∆N eff < 0.5 [57]. We show the excluded region in yellow. It was also mentioned in Ref. [57] that an additional Z boson can also alleviated the Hubble tension, for the mass-coupling region shown with the green band. Last but not least, one could see the all the favored regions cross each other at m Z = 10 − 17 MeV and g µτ = (4 − 6.5) × 10 −4 , shown in the hashed red region. Note that the cosmological limit used here considers the kinetic mixing, which is stronger than limits only with neutrino self-interactions although it depends on ∆N eff [59,79,80].
The preferred regions are intriguing because results obtained from three independent measurements meet each other. On the other hand, we stress that the IceCube data have not shown evidence for secret neutrino selfinteractions, by which we can place an upper limit on the coupling rather than the preferred region. We also show the previous results by Ref. [78] in Fig. 3. The constraints are weaker than the limits from Borexino as well as other laboratory experiments such as the kaon-decay measurement implying g µτ 0.01 [67]. Let us next turn to the future prospects for confirming or refuting the secret neutrino scattering interpretation of IceCube data.

IV. FUTURE PROSPECTS
Lastly, we consider the impact of next-generation detectors such as IceCube-Gen2 [81] on the gauged L µ − L τ scenario considered in this work. In particular, we are interested in the unique parameter space, where the muon g − 2 anomaly solution, Hubble tension alleviation, and IceCube preferred region are overlapped. For demonstration, we adopt m Z = 15 MeV and g µτ = 5 × 10 −4 as the fiducial scenario (shown as the black cross in Fig. 3). Using the zenith-angle-averaged effective areas for showertype events based on Fig. 25 from Ref. [81] we estimate the number of events coming from a given neutrino flux. As in the analysis in the previous section, this approach here is different from those in Refs. [67,74] that used the energy deposited in the detector. In Fig. 4 we compare neutrino spectra with and without BSM neutrinoneutrino scatterings in red and black data points respectively, assuming 10 years of IceCube-Gen2 data and the neutrino spectrum with s ν = 2.53 and Φ WB = 1.66. It shows that with statistics expected in IceCube-Gen2, the dip feature will be evident if it exists. We also compute the resulting χ 2 , and find that our fiducial scenario would be ∼ 5σ discrepant with the SM case without secret interactions. Although results depend on our understanding of astrophysical components, this demonstrates that such model can be probed by the IceCube telescopes.
In Fig. 4, only statistical errors are considered. In reality, there are other systematics, which need to be taken into account. As noted above, the deposited energy is smaller than the neutrino energy, which can make  the expected dip broader. The atmospheric background gives additional systematics in the analysis. On the other hand, this analysis only used the shower data. Muon track data including starting and through-going events should also give us information. One may able to further uncover the nature of the preferred model of secret self-interactions by combining spectral and flavor modifications [74], and global analyses as in Ref. [82] will be more powerful.

V. SUMMARY AND DISCUSSION
It has been suggested that the gauged L µ − L τ model accounts for the muon g − 2 anomaly. High-energy neu-trino data provide an independent test for this model through dip signatures caused by secret neutrino selfinteractions. We showed in this work that the current 6yr shower data of IceCube prefers couplings and masses consistent with the Muon g − 2 data, which also overlaps with the parameter space alleviating the Hubble tension [57]. This is because the current IceCube shower data have a paucity in the 0.2 − 1 PeV range, although this may simply reflect the change of astrophysical components [72,83]. If this result is confirmed, the combination of the data may not only reveal the existence of a new fundamental symmetry, but also uncover the neutrino mass spectrum. We found that the required parameter space is narrow, and the total neutrino mass has to range from 0.05 eV to 0.3 eV. This is also encouraging for future neutrino mass measurements (e.g., [84][85][86][87]).
We also demonstrated that the gauged L µ − L τ scenario for the muon g − 2 anomaly and Hubble tension can critically be tested by IceCube-Gen2. This result is consistent with that of the previous work that showed that next-generation neutrino telescopes can reach the limit expected in the mean free path limit [74].
Another important test with high-energy neutrinos is to utilize multimessenger observations from individual neutrino sources [88][89][90]. In particular, BSM neutrino echoes -delayed neutrino emission through secret neutrino-neutrino scatterings provide a test that is insensitive to the unknown astrophysical spectrum. Ref. [89] showed that IceCube-Gen2 can reach g µτ ∼ 10 −4 − 10 −2 for the vector mediator scenario.