Influences of non-standard interactions on PeV neutrino events with and without a $L_{\alpha}-L_{\beta}$ symmetry

Searching for signals of physics beyond the standard model (SM) is an active field in particle physics, which is mainly focused on the precision or energy frontier. Recent observed astrophysical neutrino events in the TeV-PeV energy rangeAartsen et al. (2013a, b, 2014, 2016); Abbasi et al. (2022) provide a probe to explore new physics such as Lorentz invariance violationColladay and Kostelecky (1998); Coleman and Glashow (1999); Kostelecky and Mewes (2004); Roberts (2021); Zhang and Yang (2022), neutrino decayBeacom et al. (2003); Meloni and Ohlsson (2007); Baerwald et al. (2012); Pagliaroli et al. (2015); Bustamante et al. (2017); Denton and Tamborra (2018); Huang and Zhou (2024); Valera et al. (2024), pseudo-Dirac neutrinosKobayashi and Lim (2001); Beacom et al. (2004); Carloni et al. (2024); Rink and Sen (2024); Franklin et al. (2023), non-unitary leptonic mixing matrixAntusch and Fischer (2014); Blennow et al. (2023); Aloni and Dery (2024), non-standard interaction (NSI)Huitu et al. (2016); Adamson et al. (2017); Esteban et al. (2018); Agarwalla et al. (2020); Masud et al. (2021); Abbasi et al. (2021a); Bakhti and Rajaee (2021); Brahma and Giri (2022); Lazo Pedrajas (2024). Based on a direct modification of the Lagrangian of SM, NSIs of neutrinos could change their flavor oscillation. The NSI parameters are constrained by atmospheric neutrinos Fornengo et al. (2002); Abbasi et al. (2021a), long-baseline oscillationAdamson et al. (2017); Bakhti and Rajaee (2021), and the coherent elastic neutrino-nucleus scattering experiments like COHERENTLiao et al. (2024). In this paper, we investigate the constrains on NSI parameters with the updated data of IceCubeAbbasi et al. (2023) and examine the effects of NSI on astrophysical neutrino events at PeV energies.

The influence of NSI on neutrino oscillation is parameterized by a $3\times 3$ Hamiltonian matrix. The matrix contains 9 real parameters, which makes predictions on the deviation from the standard oscillation complex at high energies. In the case where the NSI satisfies some symmetry, the form of the Hamiltonian can be simplified. A well-known example is the gauge symmetry $L_{\alpha}-L_{\beta}$He et al. (1991), where $\alpha,\beta$ represents $e,\mu,\tau$. Furthermore, if the mass of the extra gauge boson is tiny, the propagation of neutrinos could be impacted by a long-range potentialBustamante and Agarwalla (2019); Coloma et al. (2021); Agarwalla et al. (2023, 2024). In this case, the Hamiltonian matrix of NSI is diagonal in the flavor bases. The long-range interaction parameters are constrained by the results of neutrino oscillations experiments and IceCube observations. For astrophysical neutrinos, the effect of the long-range potential on the flavor ratio at Earth is mainly considered. However, because of the limitation of the resolution of neutrino flavor at IceCube, the constraint on the flavor ratio is loose at present. In this paper, we examine effects of the NSI satisfying a $L_{\alpha}-L_{\beta}$ symmetry on the PeV astrophysical neutrinos to complement the work on the long-range interaction influences. To be specific, we study the impacts of the NSI potential on the Glashow resonance eventsGlashow (1960); Barger et al. (2014); Huang and Liu (2020); Xu and Rong (2023) and $\nu_{\tau}$ eventsAbbasi et al. (2024). Although the number of the events is rare by now, several PeV events in the upcoming experiments (e.g., P-ONE)Agostini et al. (2020), IceCube-Gen2Aartsen et al. (2021)) may give strong restrictions on the NSI potential. In addition, we consider the effects of NSI without a $L_{\alpha}-L_{\beta}$ symmetry on the events as a comparison.

The paper is organised as follows. In Sec. II, the neutrino flavor transition probability on the bases of NSI is introduced. In Sec. III, the impacts of NSI on PeV neutrinos, including the flavor ratio, the flux, and the events number, are examined. Finally, a conclusion is given.

The effective neutrino-matter interaction leading to a long-range potential is subject to three contributions, namely the Standard Model (SM) term mediated by the Z boson, the interaction of the gauge symmetry $L_{\alpha}$-$L_{\beta}$ mediated by a novel boson $Z^{{}^{\prime}}_{\alpha\beta}$, and the contribution of mixing between the Z and $Z^{{}^{\prime}}_{\alpha\beta}$Agarwalla et al. (2023). It reads as follow,

The first item on the right-hand is the contribution from the SM

The second term describes neutrino-matter interactions through the new mediator $Z^{{}^{\prime}}_{\alpha\beta}$Agarwalla et al. (2023), i.e.,

It shows the contribution from the $L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$ gauge symmetries, and the corresponding interaction is noticeable due to a source rich of electrons. The third term generated from the mixing of the bosons $Z$ and $Z^{{}^{\prime}}$ is written asAgarwalla et al. (2023)

It only affects the results under the $L_{\mu}-L_{\tau}$ symmetry due to a neutron sourceAgarwalla et al. (2023). On the bases $L_{eff}$ and the assumption that the matter is isoscalar and electrically neutral, the total potential of the Earth, Moon, Sun, Milky Way and Cosmological should be consideredBustamante and Agarwalla (2019), namely

Since the number density of electrons and neutrons in the universe changes as the universe expands, the redshift-averaged potential is employed. In this paper, we take a simplified view-point, and treat the total $V_{\alpha\beta}$ as a new-physics parameter to constrain by observations, irrespective of its specific origins.

Accordingly, the Hamiltonian matrix of neutrinos during their propagation is generalised as

The first term on the right-hand determines oscillations in vacuum and the standard neutrino-matter interactions, i.e., Wolfenstein (1978); Mikheyev and Smirnov (1985)

where $U$ is the leptonic mixing matrix in vacuum, $V_{CC}=\sqrt{2}G_{F}N_{e}$ is the standard matter potential. The second term working as the long-range potential is due to tiny masses of mediators of the new neutrino-matter interactions.

According to the analysis of the referenceAgarwalla et al. (2023), the potentials of the form $L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$ bring similar results of flavor conversion probability. So the forms of $L_{e}-L_{\tau}$ and $L_{\mu}-L_{\tau}$ are considered in the following sections. As $\alpha,\beta=e,\tau;\mu,\tau$, the expressions of the term $\mathbf{V_{\alpha\beta}}$ are respectively

Considering antineutrinos, $V^{ant}_{\alpha\beta}$=$-V_{\alpha\beta}$.

The $\mathbf{V_{\alpha\beta}}$ shown here is a special case derived from a NSI between neutrinos and matter. In order to compare the effects of NSIs with and without a $L_{\alpha}-L_{\beta}$ symmetry, we introduce the non-diagonal potential from general NSIs, namely

For simplicity, $V_{\alpha\beta}$ still takes real values in this case.

Considering the limited energy resolution of neutrino telescopes which are only sensitive to the average probability, the $\nu_{\alpha}\rightarrow\nu_{\beta}$ transition probability reads

where $U^{N}$ is derived from the diagonalization of the total Hamiltonian.

Note that the influence of $V_{CC}$ can be neglected in the case that the range of the new interaction is larger than the size of the Earth. Accordingly, we focus on the case that $V_{\alpha\beta}$ is dominant in the $\mathbf{H}$ for the PeV neutrinos.

Now we examine the constraints on the NSI parameters. As for the data sample, 4318 days of high-energy start-up events are used for the analysis, with a total of 164 updated eventsAbbasi et al. (2023). Considering that there may be background events in the sample, namely atmospheric $\mu$, atmospheric neutrinos, and transient neutrinosAartsen et al. (2016); Halzen and Wille (2016a, b), we need a clean set of events to obtain more reliable analysis results. Based on the reason, the maximum likelihood analysis considers the shower events mainly generated by the interaction of $\nu_{e}(\overline{\nu}_{e})$ and $\nu_{\tau}(\overline{\nu}_{\tau})$Aartsen et al. (2015a, b); Kopper (2018); Denton and Tamborra (2018), and sets the energy range of neutrinos to 60TeV-3PeV with a total of 64 data samples, see Fig.1.

We employ a binned maximum likelihood method to constrain the parameters of the NSI potential in the neutrino propagation path. The expected number of events at each energy bin k is

where T=4318 days is the time of data-collecting, $\int_{k}$ represents the integral over each interval k. $A_{\alpha,~{}k}(E)$ represents the effective area of the IceCube Collaboration corresponding to the $\alpha$-flavor neutrino at the k energy intervalAartsen et al. (2013a), and $\Phi_{\nu_{\alpha}+\overline{\nu}_{\alpha}}(E,V_{\alpha\beta})=\Phi_{\nu+\overline{\nu}}(E)(\sum_{\beta}\overline{P}^{N}_{\alpha\beta}(E,V_{\alpha\beta})\phi^{s}_{\beta})$ with $\phi^{s}_{\beta}$ being the flavor ratio at the source of astrophysical neutrinos. $\Phi_{\nu+\overline{\nu}}(E)$ indicates the entire diffuse flow, and its specific expression is as follow

The best fit values and 68% confidence level of energy-spectrum parameters in the all-sky model are listed in Table IAbbasi et al. (2021b).

The number of shower events in the energy interval k is denoted by $\overline{N_{k}}$. It is further assumed that the expected number of shower events $N_{k}({V_{\alpha\beta}})$ in different energy intervals k obey the poisson distribution, namely

Each likelihood function is

Taking the logarithm of both sides of the above equation, we can obtain the following expression

The value of the function $\ln[L(V_{\alpha\beta})]$ is dependent on the leptonic mixing parameters and the energy-spectrum parameters of the diffuse flux. However, we find that the mixing parameters and the spectrum index $\gamma$ moderately impact the magnitude of $\ln[L(V_{\alpha\beta})]$, and the normalization parameter $\phi$ determines the characteristic of dependence of $\ln[L(V_{\alpha\beta})]$ on $V_{\alpha\beta}$. Therefore, the best fit data of the mixing parameters with the normal mass-ordering (NO) from the global fit analysis NuFIT5.0 published by PDGEsteban et al. (2020) and the best fit value of the spectrum index in Tab.1 are taken here. The influence of normalization parameter $\phi$ on $\ln[L(V_{\alpha\beta})]$ is averaged with a Monte Carlo integration method. Based on the set-up of the nuisance parameters, the behaviors of $-\ln[L(V_{\alpha\beta})]$ of the models are shown in Fig.2-3. Accordingly, the upper limits (95% C.L.) on the potentials are obtained, see Tab.2-3. We note that although the statistic procedure employed here is simple, the updated sample data of IceCube can set more stringent constraints on the potential parameters in comparison with the results from the long-baseline experimentsAgarwalla et al. (2024).

We examined the effects of NSI with and without a $L_{\alpha}-L_{\beta}$ symmetry on the HAN events at PeV energies. The constraints on potential parameters of NSI are obtained with the updated IceCube shower data. On the bases the given parameters, the impacts of NSI on the flavor ratio of HANs at Earth, the flux $\phi_{\nu}$, and expected events number at IceCube Observatory are shown. In general, the considered NSI with a $L_{\alpha}-L_{\beta}$ symmetry can notably reduce the Glashow resonance events for neutrinos from $\mu^{\pm}$ damping source and the $\nu_{\tau}$ events for neutrinos from $\pi^{\pm}$ decay source. Considering the upcoming neutrino observatories, the observations of PeV events may give more stringent constraints on the NSIs with a $L_{\alpha}-L_{\beta}$ symmetry.