Testing Lorentz Violation with IceCube Neutrinos

: Lorentz violation (LV) induced by Quantum Gravity has been tested at much lower energies than the Planck scale with more and more observational evidence. In recent studies, the time of ﬂight difference between the TeV neutrino and MeV photon from Gamma Ray Bursts (GRBs) have been used to constrain the LV energy scale, based on the energy-dependent speed variation. Here, we performed a correlation study between the updated 7.5 year high-energy starting events (HESE), neutrino alert events detected by IceCube, and a full sample of more than 7000 GRBs, and we found six GRB-neutrino candidates, including four alerts and two track events. We obtained the ﬁrst order energy scale of quantum gravity, namely E QG = 8 + 15 − 5 × 10 17 GeV, which was consistent with other authors‘ work. We suggest that neutrinos and anti-neutrinos can be identiﬁed, respectively, due to the delay or advance of the observed time. For future point source search study of neutrinos, the arrival time difference of different particles may have to be taken into account.


Introduction
Lorentz invariance (LV), as one of our most fundamental symmetries, lies at the roots of both the standard model and general relativity and has been tested in various regimes with a tremendous amount of effort over the past decade [1][2][3][4][5]. Theories such as strings [6], spacetime foam [7], and loop quantum gravity [8] could provide the solutions to the violation at extremely high energy of about 10 19 GeV, far beyond the capabilities of observation. Even though some models indicate that certain effects can be tested at much lower energies, it is still not accessible with human-made accelerators. The alternative approach to investigate the characteristics of quantum gravity is with cosmic messengers. Even their energies are far below the Plack energy scale, their cosmologcial distances can amplify LV effects. Thanks to the development of instruments and technologies, groundbased and space-based experiments are detecting more and more high-energy gamma rays, neutrinos, and cosmic rays from various extragalactic sources, such as Gamma Ray Bursts (GRBs), Active Galactic Nuclei (AGN), etc. Therefore, the methods using multiastrophysical messengers have been widely applied and become more favorable than ever before.
Gamma-ray bursts (GRBs), as the most luminous explosions in the universe, are supposed to be the dominant sources of ultra-high-energy cosmic rays (UHECRs). They have particularly interesting features for an LV test, such as a short time structure, highenergy radiation, and long cosmological distance. On the other hand, according to the most popular models, GRBs are believed to be one of the best candidates for multi-messenger study. During the prompt emission, relativistic protons will interact with low energy photons and/or low energy protons to produce TeV neutrinos and MeV gamma rays by internal shocks. However, there has been no significant association found between GRB and neutrinos so far. Most of the early quantum gravity (QG) studies focused on the dispersion relation of electromagnetic radiation from GRBs [3,[9][10][11]. As it is believed a particle with higher energy from further distance is more suitable and preferred for the study, the TeV-PeV neutrino as a better probe compared to MeV-GeV gamma ray is expected to play a key role in answering the questions. The relevant studies of the time delays between IceCube neutrinos and GRB photons have been performed by a few groups [4,[12][13][14], while the observed time delay includes at least the effects of the intrinsic time delay, the QG time delay, and other terms due to the nonzero rest of mass, as well as the Shapiro effect [15]. The scale of the intrinsic time delay is estimated at the level of thousands of seconds according to the duration of GRBs, which is much smaller than our QG time delay resulting from the propagation of high energy neutrinos. The other effects are also negligible compared to the QG time delay of PeV neutrinos; so, we simply treat the obsevred time delay as the QG time delay. In this work, we perform the analysis for ∼PeV alert events and ∼100 TeV track events from IceCube.
IceCube Neutrino Observatory has been successfully operating for more than 10 years. With tens of events observed every year, no significant association has been found so far [16], which might be proof of the LV effect on astrophysical observations and may provide a hint on the analysis of IceCube neutrinos. There have been more than 7000 GRBs observed by various instruments up to now, and they are still being updated. In this work, to find the time difference of photons and neutrinos we performed the correlation search of high-energy IceCube neutrino and GRBs, calculated the LV parameters, and obtained the typical QG energy scale E QG at the level of 10 17 GeV.
In this paper, we describe the neutrino data sample from IceCube and the GRB catalog adopted in this work in Section 2. In Section 3, we discuss the modified dispersion relation for energetic particles, which leads to the time-of-flight difference. In Section 4, we present our analysis methods for the correlation search and show our results in Section 5. In the end, we provide our conclusion and discussion in Section 6.

IceCube Neutrino Events
Since the first announcement of astrophysical neutrinos at the IceCube Neutrino Observatory in 2013, IceCube has detected tens of high-energy starting events (HESE) each year, with the interaction vertices inside the detector fiducial volume. So far there have been 102 events in the 7.5 years from 2010 to 2017 [17][18][19][20][21], including track-like events mainly from charged-current interactions of muon neutrinos, shower-like events from other interactions, and coincident events from cosmic-ray air showers. The recent updated 102 events from the 7.5-year dataset were constructed with a new method, where a minimizer was used to determine the best-fit neutrino direction with more accuracy. However, among these 102 events reported, only 82 of them had arrival time information. Hereafter, we call these 82 events HESE events, including 58 shower-like, 22 track-like, and 2 coincident events. Shower events are likely to be associated with a few GRBs due to the relatively poor angular resolution shown in Figure 1, so they are not included in this analysis. Track events have a much better resolution around 1 • , which makes them ideal for the correlation search. The energy uncertainty of all the shower and track events was around 10% to 15%.
In April 2016, the first HESE alert was sent to the network, since then the alert frequency has been about three to four times per year. Another trigger system is extreme high energy (EHE) alerts, which are neutrino events coming from the Northern Hemisphere, passing through the Earth with higher energies. Both of them, instead of providing the reconstructed energy, give the estimated number of photoelectrons in the PMTs, which can not be used easily to estimate the true energy. Therefore, we abandon these two catalogs in the analysis. After 2019, HESE and EHE alerts were replaced by GOLD/BRONZE alerts [22], with the probability of at least 50%/30% astrophysical origin, respectively. We call this the alert catalog in this work. Up to 1 April 2022, there have been 31 GOLD and 45 BRONZE alerts. As can be seen in Figure 1, in general, GOLD alerts have better angular resolution and higher energies than BRONZE.

GRB Catalog
Gamma Ray Bursts (GRBs) are believed to be very promising candidates for producing neutrinos and photons. TeV neutrinos and MeV photons can be generated from the photomeson interaction. The time delay or advance between neutrinos and photons will be key to setting limits on the LV parameters. The theoretical study has shown that for the concerned neutrino and photon energy regime, they are most likely produced simultaneously, or neutrinos are emitted seconds after the prompt emission.
In the correlation study, we used a GRB catalog [23] that merged the observations from a wide range of resources including GCN-circulars [24], Fermi-GBM, Fermi-LAT [25], Swift [26], IPN [27], BeppoSAX [28], and BATSE [29]. This GRB catalog contained 7799 sources from 1991 to 1 April 2022, thanks to the contribution from global observatories. Among them, only 573 GRBs had redshift, which is an essential factor for the analysis. In Figure 2, the histograms show the redshift distribution of short/long GRBs, which were symmetrical about the average value of 0.6/1.4, respectively, and the standard deviation was 0.47/0.42. So we estimated the redshifts of the unknown ones as 0.6 for the 1118 short GRBs and 1.4 for the 5159 long GRBs. This assumption is consistent with that in previous studies [4,12].

Theoretical Model
The Lorentz violation has been widely studied in many quantum gravity (QG) models in recent years. The typical scale at which the Lorentz invariance would be strongly violated is the so-called Planck energy, ∼10 19 GeV. It is beyond the accessibility of the current experiments on Earth and can not be tested with astronomy observation. However, it is very likely that LV violation at the Planck scale could have some relic effects at much lower energies. Many attempts have been made to place constraints on high-energy deviations from the Lorentz invariance, which enters through modified dispersion relations. Here, we considered a simplified model with single symmetry-breaking energy scale of E QG for both neutrino and anti-neutrino [4]. For a particle propagating in the quantum spacetime with energy E E QG , the LV modified dispersion relation can be written in a general form with only the leading-order correction as in [30], where n accounts for the order expansion of the leading term, s n = +1(−1) presents a decrease (increase) in particle speed with energy, E QG,n is the nth-order QG energy scale to be determined by experiments, and m is the rest mass of the particle. Here, the value of s n depends on the type of particles, such as 1 for neutrino and −1 for anti-neutrino [31,32].
Considering the current status of the experiments, we only focus on the first order; so, we set n as one and dismiss higher orders. Both GRB photons and high energy neutrinos are relativistic particles; hence, it is reasonable to set the rest mass m = 0. Using the relation v = ∂E/∂p, we obtained the modified propagation velocity [7] as Such energy-dependent speed variation can cause a time difference between particles. By taking into account the cosmological expansion, the QG time correction of neutrino and photon with energy E ν and E p respectively can be written as [4,30,31] where z is the redshift of GRBs, and s ν and s p are the sign factors of neutrino and photon.
As the studied neutrino energy between 10 TeV and 100 PeV is much higher than that of the GRB photon (below 1 GeV), s p E p is negligible with respect to s ν E ν . We where t ν and t p represent the arrival times of neutrino and GRB photon. Taking Equation (3) into this formula, Equation (4) could be rewritten as where K can be expressed as According to Equation (5), there would be a linear relation between ∆t obs /(1 + z) and sK, if the energy dependent speed variation does exist.

Analysis Methods
We searched through the neutrino and GRB datasets with the two conditions below to select the GRBs-neutrino association.

•
The angular distance between neutrino and GRB events should not be larger than 3 σ 2 p + σ 2 ν , where σ p and σ ν are the angular uncertainties of GRB and neutrino events separately.
• ∆t obs /(1 + z) should be within a specified selection window, which is calculated based on the initial value of E 0 , as described below.
For the second condition, we calculated the bound K bound according to the maximum energy of neutrinos, E ν, max , and the maximum redshift of GRBs, z max , which is We assigned E QG an initial value obtained from the work of [4], which is E 0 = 6.5 × 10 17 GeV.
With E 0 and limits of K, we set the bound for ∆t obs /(1 + z) as the gray region in Figure 3, which means ∆t obs /(1 + z) should be within the range of [−K bound /E QG + ∆t in , K bound /E QG + ∆t in ]. For the intrinsic time delay, ∆t in , which is due to the physical process at the source, our current understanding is not sufficient to provide any robust estimation. We assumed that ∆t in was comparable to the duration of GRB up to 1000 s, and ∆t in was negligible compared to ∆t obs in the analysis. So we assigned ∆t in as 0 in the analysis.
As ∆t in = 0, E QG for each GRB/neutrino association according to Equation (5) can be found from With these two conditions, there are 73 remaining neutrinos, including 39 BRONZE alerts, 31 GOLD alerts, and 3 track events. Among these events, most were associated with more than one GRB, which is because the observed time delay of neutrinos can be up to tens of days, and GRBs occur about once per day, even if the angular resolution of our neutrino sample was good enough. There were still some neutrinos associated with only one GRB, which could provide a more intriguing estimation of E QG .  The gray region set the bound according to Equation (7). The blue lines show the linear relation of E QG = E 0 . In panel (a), the green circles and orange stars are associations with neutrino energy below 680 TeV and between 680 TeV and 2 PeV. In panel (b), the purple triangles, brown squares, and pink pentagons are three high energy GOLD alerts with their associated GRBs, respectively. In panel (c), the yellow diamonds are associations with neutrino energy below 1 PeV, and the cyan star is the association with 6.0702 PeV neutrino energy. In panel (d), the two red stars are associations of a track event and GRB, while the gray cross is the one with a much lower E QG and is treated as outlier.

Results
In Figure 3, we showed the ∆t obs /(1 + z) versus K for GRB-neutrino associations satisfying our conditions. (1) In Figure 3a, there were two clusters around the y-axis contributed from neutrinos with energies below 680 TeV. These clusters gave a E QG much lower than E 0 , and no strong linear correlation among the clusters was found. So we excluded them from the analysis. There were three GOLD alerts with energies between 680 TeV and 2 PeV shown as orange stars with number 3, 5, and 6 labeled, which only associated with one GRB each. Their calculated E QG were in agreement with E 0 . (2) In Figure 3b, we show the three GOLD alerts above 2 PeV, associated with more than one GRB each, meaning the unreliable association between neutrino and GRBs. (3) In Figure 3c, the BRONZE alerts below 1 PeV with associated GRBs are presented as yellow diamonds, with no linear correlation. One BRONZE alert with remarkably high energy 6070.2 TeV was associated with only one GRB shown as a blue star with number 4 labeled. (4) In Figure 3d, three track events were associated with one GRB each, and two of them provided a calculated E QG in agreement with E 0 , while the other one was an outlier with a much lower E QG . These two are shown as red stars with number 1 and 2 labeled. Overall, there were six neutrino and GRB associations selected for our analysis, and their labeled numbers were the same as shown in Table 1. Based on the selected six GRB/neutrino associations in Table 1, we fitted the points and found the best fit value of E QG as shown in Figure 4, which was log 10 (E QG / GeV) = 17.9 ± 0.5. Therefore, the QG energy scale was found to be E QG = 8 +15 −5 × 10 17 GeV. The uncertainty of E QG was mainly due to the uncertainty of the redshift and neutrino energy.   Table 1

Discussion and Conclusions
With the most recent neutrino events from IceCube and a complete sample of GRBs, we searched for the association candidates. The track and alert neutrino events were adopted in the analysis, thanks to their better angular resolution and higher energies. With our selection conditions, six GRB/neutrino associations listed in Table 1 implied a linear correlation, as plotted in Figure 3.
We found that for TeV to PeV neutrinos, the observed time delays were up to tens of days, which may explain why no significant correlation has been found between astrophys-ical objects and IceCube neutrinos so far. For later neutrino study and analysis, these QG effects have to be taken into account.
The work by other groups so far has not showm strong evidence of LV effects, and different methods lead to contradictory conclusions about the absence or presence of observable QG effects in time delays. The tightest bound of first order effective QG energy scale obtained with photons from GRB090510 was E QG > 9.3 × 10 19 GeV [9]. Another limit of E QG > 1.09 × 10 17 GeV was obtained assuming a physical association between a PeV neutrino and the blazar PKS B1424-418 [34]. Some other studies adopted statistical approaches combining data from multiple sources, applied to both GRB photons [3,10,11] and GRB-neutrinos [4,12,13], as well as the combination of the two [14]. Their results suggested a correlation between time delay and neutrino energy with E QG ∼ 5 × 10 17 GeV. In this work, we analyzed the updated data with an improved statistical method, yielding a QG energy scale E QG = 8 +15 −5 × 10 17 GeV, in agreement with the work in [4]. Our QG energy scale was much lower than the limits from photon analysis, which may indicate different critical energy scales for different particles, as also proposed by Coleman and Glashow [35].
In summary, establishing the association between neutrino and GRB is even more challenging with LV than in a no-LV scenario. To solve this problem, we need more highenergy observations of neutrinos and GRBs with better angular and energy resolution. Early high-energy observations of short GRBs with redshift could also be helpful. Meanwhile, according to the information of delay or advance time, the identification of the neutrino and anti-neutrino could be possible, which is not accessible for now.
Author Contributions: H.Z. and L.Y. conceived and designed the investigation; both contributed equally to analyzing the data; to performing the modeling; and to writing the paper. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement:
The data used to support the findings of this study are included within the article.