Consistent Lorentz violation features from near-TeV IceCube neutrinos

TL;DR

This paper finds evidence of Lorentz violation in IceCube neutrino data associated with gamma-ray bursts, indicating new physics at a high energy scale and an earlier neutrino emission compared to photons.

Contribution

It identifies a consistent Lorentz violation scale from neutrino-GRB correlations at different energies, suggesting CPT violation and earlier neutrino emission.

Findings

Lorentz violation scale around 6.4×10^17 GeV

Neutrinos emitted earlier than photons from GRBs

Potential CPT violation in ultrahigh-energy neutrinos

Abstract

A recent proposal to associate 60~TeV to 2~PeV IceCube neutrino events with gamma-ray bursts~(GRBs) indicates the Lorentz violation of cosmic neutrinos and leads further to the $CPT$ symmetry violation between neutrinos and antineutrinos. Here we find that another 12 northern hemisphere track events possibly correlated with GRBs from three-year IceCube data satisfy the same regularity at a lower energy scale around 1~TeV. The combined fitting indicates a Lorentz violation scale ${E}_{\rm LV}=(6.4\pm 1.5)\times10^{17}~{ \rm GeV}$ and an intrinsic time difference ${\Delta {t}_{\rm in}=(-2.8\pm 0.7)\times10^2~{\rm s}}$, from which we find an earlier emission of neutrinos than photons at the GRB source. We also suggest analyzing neutrino events detected a few minutes before the GRB trigger time to test the $CPT$ violation of ultrahigh-energy neutrinos.

Tables

Table 1. Table 1: Parameters of TeV and PeV GRB neutrinos. The 13 GRB candidates are suggested from the associated GRBs of IceCube neutrinos with energies above 60 TeV by the maximum correlation criterion  Amelino-Camelia:2016fuh ; Amelino-Camelia:2016ohi ; Huang:2018 . 12 of the 13 events are “shower” events and the last 2.6 PeV one is “a track” event. The event serial numbers here are provided by the IceCube database, except the last ATel id # ​ 7856 # 7856 \#7856 . The mark ∗ represents the estimated value of the redshift.

Event	GRB	$z$	$\Delta t_{\mathrm{obs}}\left({10}^{3}s\right)$	$E$ (TeV)
$\#2$	100605A	${1.497}^{\ast }$	$-$113.051	117.0
$\#9$	110503A	1.613	80.335	63.2
$\#11$	110531A	${1.497}^{\ast }$	185.146	88.4
$\#12$	110625B	${1.497}^{\ast }$	160.909	104.1
$\#14$	110725A	${2.15}^{\ast }$	1320.217	1040.7
$\#19$	111229A	1.3805	73.960	71.5
$\#20$	120119C	${2.15}^{\ast }$	$-$1940.176	1140.8
$\#26$	120219A	${1.497}^{\ast }$	229.039	210.0
$\#33$	121023A	${0.6}^{\ast }$	$-$171.072	384.7
$\#35$	130121A	${2.15}^{\ast }$	$-$2091.621	2003.7
$\#40$	130730A	${1.497}^{\ast }$	$-$179.641	157.3
$\#42$	131118A	${1.497}^{\ast }$	$-$146.960	76.3
$\#7856$	140427A	${2.15}^{\ast }$	3827.439	$2.6\times {10}^3$

Table 2. Table 2: Parameters of northern hemisphere track events. The 19 northern hemisphere track events with associated GRBs are suggested by the IceCube Collaboration  Aartsen:2014aqy ; Aartsen:2017wea . The first one marked by ⋆ is based on the first four-year search  Aartsen:2014aqy and another 18 are from the search extended to three additional years from May 2012 to May 2015  Aartsen:2017wea . The mark † represents 12 events falling within the error band in Fig.  2  (b).

GRB	$\Delta t_{\mathrm{obs}}\left(s\right)$	$E$ (TeV)	$K$ (s$\cdot$TeV)	$\frac{\Delta t_{\mathrm{obs}}}{1+z}\left(s\right)$
100718A	15⋆	10	3.36483	4.7619
120612B	47.71†	0.54	0.1817	15.146
120911A	120.94†	0.98	0.32975	38.3937
130116A	69.25	2.1	0.70661	21.9841
130318A	29.83†	0.46	0.15478	9.46984
	44.58†	0.32	0.10768	14.1524
130925B	108.8†	0.7	0.23554	34.5397
131029B	50.49†	0.68	0.22881	16.0286
131202B	85.18	1.7	0.57202	27.0413
140404B	$-$38.49†	1.1	0.37013	$-$12.219
140521B	98.37†	0.79	0.26582	31.2286
140603A	41.35	1.5	0.50472	13.127
	$-$33.78	2.1	0.70661	$-$10.7238
141029B	$-$10.33†	0.7	0.23554	$-$3.27937
	$-$80.99†	0.45	0.15142	$-$25.7111
150428B	71.35	3.2	1.07675	22.6508
150428D	$-$43.69†	0.54	0.1817	$-$13.8698
150507A	58.24	1.8	0.60567	18.4889
	$-$74.44†	0.69	0.23217	$-$23.6317

Equations

$$v(E)=c\left[1-s_n\frac{n+1}{2}{\left(\frac{E}{E_{\mathrm{LV},n}}\right)}^n\right],$$

$$\Delta t_{\mathrm{LV}}=s\cdot (1+z)\frac{K}{E_{\mathrm{LV}}},$$

$$K=\frac{E_{\mathrm{h}}-E_{\mathrm{l}}}{H_0}\frac{1}{1+z}{\int }_0^z\frac{(1+z^{\prime })\mathrm{d}{z}^{\prime }}{\sqrt{{\Omega }_{\mathrm{m}}(1+z^{\prime }{)}^3+{\Omega }_{\Lambda }}},$$

$$\frac{\Delta t_{\mathrm{obs}}}{1+z}=\Delta t_{\mathrm{in}}+s\cdot \frac{K}{E_{\mathrm{LV}}}.$$

$$|\frac{\Delta t_{\mathrm{obs}}}{1+z}-\Delta t_{\mathrm{in}}^{\prime }|=\frac{K}{E_{\mathrm{LV}}^{\prime }},$$

$$1/E_{\mathrm{LV}}^{\prime }$$

$$\Delta t_{\mathrm{in}}^{\prime }$$

$$1/E_{\mathrm{LV}}^{\prime \prime }$$

$$\Delta t_{\mathrm{in}}^{\prime \prime }$$

$$1/E_{\mathrm{LV}}$$

$$\Delta t_{\mathrm{in}}$$

$$E_{\mathrm{LV}}=(6.4\pm 1.5)\times 10^{17}\mathrm{GeV},$$

$$\Delta t_{\mathrm{obs}}<\Delta t_{\mathrm{in}}(1+z)=-882s.$$

Full text

I Introduction

Cosmic neutrinos from astrophysical sources have the potential to reveal novel features of the Universe. The IceCube Neutrino Observatory has observed plenty of neutrino events, including dozens of neutrinos with energies above 30 TeV, with also four PeV scale neutrinos Aartsen:2013bka ; Aartsen:2013jdh ; Aartsen:2014gkd ; Kopper:2015vzf ; Aartsen:2016ngq ; Aartsen:2018vtx . It has long been expected that gamma-ray bursts (GRBs) can be one class of sources for ultrahigh-energy cosmic neutrinos Eichler:1989ve ; Paczynski:1994uv ; Waxman:1995vg ; Waxman:1997ti ; Piran:1999bk . However, the IceCube Collaboration examined GRBs within a short temporal window of a few hundred seconds and suggested possible correlations between GRBs and some lower-energy neutrinos with energies around 1 TeV (we call “near-TeV” later), with the conclusion that these near-TeV events are also consistent with atmospheric backgrounds Aartsen:2014aqy ; Aartsen:2016qcr ; Aartsen:2017wea .

The situation changes significantly when combining the association between neutrinos and GRBs with studies on Lorentz-invariance violation (LV) AmelinoCamelia:1997gz ; Jacob:2006gn . The long propagation distance and the high energy of GRB neutrinos can produce an observable difference between GRB trigger time and the arrival time of neutrinos due to the LV effect Jacob:2008bw . Amelino-Camelia and collaborators associated IceCube 60-500 TeV shower neutrino events with GRB candidates within the time range of three days, and revealed roughly compatible speed variation features between GRB neutrinos Amelino-Camelia:2015nqa ; Amelino-Camelia:2016fuh ; Amelino-Camelia:2016ohi and photons Shao:2009bv ; Zhang:2014wpb ; Xu:2016zxi ; Xu:2016zsa ; Xu:2018ien ; Liu:2018qrg . In a recent study Huang:2018 , all four PeV scale IceCube neutrino events are associated with GRBs by extending the temporal window to a longer range of three months, and such four events are found to be consistent with TeV scale events for an energy-dependent speed variation. Such findings indicate the Lorentz violation of cosmic neutrinos. It is also found that there are both time “delay” and “advance” events, which can be explained by different propagation properties between neutrinos and antineutrinos Huang:2018 . This leads further to the charge conjugation, parity transformation, and time reversal ($CPT$) symmetry violation between neutrinos and antineutrinos, or an asymmetry between matter and antimatter Zhang:2018otj . As these results are of fundamental importance, it is thus necessary to check whether additional IceCube neutrino events still support the revealed regularity Amelino-Camelia:2016ohi ; Huang:2018 or not.

In this work, we provide a reexamination of these near-TeV neutrino events which are possibly associated with GRBs by the IceCube Collaboration Aartsen:2016qcr ; Aartsen:2017wea . We find that all of these near-TeV northern hemisphere “track” events with associated GRBs during 2012–2015 can satisfy the regularity obtained from the 60 TeV to 2 PeV neutrinos with associated GRBs Amelino-Camelia:2016ohi ; Huang:2018 . By performing the combined linear fitting of the 13 high-energy TeV and PeV events Amelino-Camelia:2016ohi ; Huang:2018 (with energies from 60 TeV to 2 PeV) together with the 12 lower-energy near-TeV northern hemisphere events Aartsen:2014aqy ; Aartsen:2016qcr ; Aartsen:2017wea (with energies around 1 TeV), we find that these lower-energy GRB neutrinos are consistent with the TeV and PeV GRB neutrinos for an energy-dependent speed variation at $E_{\rm LV}=(6.4\pm 1.5)\times 10^{17}~{}{\rm GeV}$. We also find that GRB neutrinos are emitted $280\pm 70$ seconds before GRB photons at the source. As a prediction, we suggest analyzing neutrino events observed several minutes before the GRB trigger time to test the revealed regularity.

II Model

The LV physics can be determined or limited by the energy-dependent speed variation of GRB photons and neutrinos AmelinoCamelia:1997gz ; Jacob:2006gn . For a particle propagating in the quantum spacetime with energy $E\ll E_{\rm Pl}$ (the Planck scale $E_{\rm Pl}\approx 1.22\times 10^{19}~{}{\rm GeV}$), the leading terms in a Taylor series expansion of the LV modified dispersion relation can be written as

[TABLE]

where $n=1$ or $n=2$ corresponds to linear or quadratic dependence of the energy, $s_{n}=\pm 1$ is the sign factor of LV correction, and $E_{{\rm LV},n}$ is the $n$th-order LV scale to be determined by experiments. Such a speed variation can cause a propagation time difference between particles with different energies. By taking into account the cosmological expansion, the LV time correction in the $n=1$ case of two particles with energies $E_{\rm h}$ and $E_{\rm l}$ , respectively, can be written as Jacob:2008bw ; Ellis:2002in

[TABLE]

where $z$ is the redshift of the GRB source, $s=\pm 1$ is the sign factor and

[TABLE]

is the LV factor. We adopt the cosmological parameters Agashe:2014kda $[\Omega_{\rm m},\Omega_{\Lambda}]=[0.315^{+0.016}_{-0.017},0.685^{+0.017}_{-0.016}]$ and the Hubble expansion rate $H_{0}=67.3\pm 1.2~{}{\rm km\cdot s^{-1}\cdot Mpc^{-1}}$. By taking the intrinsic time difference $\Delta t_{\rm in}$ into consideration, the observed arrival time difference $\Delta t_{\rm obs}$ between two particles detected on the Earth is

[TABLE]

For neutrinos emitted with associated GRBs, $\Delta t_{\rm obs}$ can be represented by the difference between the arrival time of the neutrino and the trigger time of the given GRB. According to Eq. (4), there would be a linear relation between $\Delta t_{\rm obs}/(1+z)$ and $K$, if the energy-dependent speed variation does exist.

III TeV and PeV scale GRB neutrinos

A regularity of energy-dependent speed variation was revealed from the studies on TeV and PeV scale IceCube neutrino events with associated GRBs Amelino-Camelia:2016ohi ; Huang:2018 . Amelino-Camelia and collaborators Amelino-Camelia:2016fuh ; Amelino-Camelia:2016ohi selected 9 GRB candidates from the associated GRBs of IceCube TeV neutrino events with the time mismatch of three days. These 9 TeV events are selected from the shower events detected from 2010 to 2014 and with energies between 60 and 500 TeV. In a recent study Huang:2018 , all 4 PeV neutrinos are found to be associated with GRB candidates in the time range within three months. The associations of these 13 neutrino events with GRBs are based on the maximum correlation together with direction and time criteria Amelino-Camelia:2016ohi ; Huang:2018 . These 13 events with the associated GRBs, the redshift values $z$, the observed time differences $\Delta t_{\rm obs}$, and the neutrino energies $E$ are listed in Table 1. The observed time difference $\Delta t_{\rm obs}$ could be positive or negative, just like the LV time correction $\Delta t_{\rm LV}$. Following the convention in Refs. Amelino-Camelia:2016ohi ; Huang:2018 , we call $\Delta t_{\rm obs}>0$ (or $<0$) events “late (or early) neutrinos,” and call $\Delta t_{\rm LV}>0$ (or $<0$) cases “time delay (or advance).”

Since the redshift measurement is not available for most GRBs, we use “most likely ?estimated values of the redshift for those GRBs without redshift measurement Amelino-Camelia:2016ohi ; Huang:2018 . We find a well linear correlation between the observed time difference and the LV factor, which can be described by a pair of inclined lines Huang:2018

[TABLE]

with the slope and the intercept

[TABLE]

The correlation coefficient is $r=0.978$, which implies a relatively strong linear correlation between $\Delta t_{\rm obs}$ and $K$. Such regularity indicates a Lorentz violation scale $E^{\prime}_{\rm LV}=(6.5\pm 0.4)\times 10^{17}~{}\rm{GeV}$, which is comparable with the results determined by GRB photons Zhang:2014wpb ; Xu:2016zsa ; Xu:2016zxi ; Amelino-Camelia:2016ohi ; Xu:2018ien . The error range of $\Delta t_{\rm in}$ covers the zero point; hence, whether these neutrino events are emitted before or after the GRB photons cannot be determined by only analyzing TeV and PeV events. By taking into account the error ranges of $E^{\prime}_{\rm LV}$ and $t^{\prime}_{\rm in}$, we can draw an error band on the $\Delta t_{\rm obs}/(1+z)$ versus the $K$ plot, as shown in Fig. 1. The opposite signs of LV correction also indicate that neutrinos and antineutrinos have different propagation propertie; i.e., one is superluminal and the other is subluminal. This can be explained by the Lorentz violation due to the $CPT$ odd feature of the linear Lorentz violation Huang:2018 , and leads further to the $CPT$ violation between neutrinos and antineutrinos Zhang:2018otj .

IV Consistency in Northern Hemisphere track events

The IceCube Collaboration Aartsen:2014aqy ; Aartsen:2016qcr ; Aartsen:2017wea also suggested associations between GRBs and lower-energy neutrino events with closer temporal coincidence. Among these events, 5 shower events are from the three-year search for all flavors of neutrinos Aartsen:2016qcr , one northern hemisphere track event is from the first four-year search Aartsen:2014aqy , another 18 are from the search extended to three additional years from May 2012 to May 2015 Aartsen:2017wea , and 13 southern hemisphere track events are from the five-year search from May 2010 to May 2015 Aartsen:2017wea . The southern hemisphere track events are difficult to disentangle from the atmospheric muon and $\nu_{\mu}+\bar{\nu}_{\mu}$ produced by cosmic-rays, but the atmospheric muon background from the northern hemisphere can be reduced by the Earth. Therefore in most cases, southern hemisphere events are excluded and only northern hemisphere events are analyzed Aartsen:2017wea .

According to Eq. (3), we can get the LV factor $K$ of the 37 lower-energy GRB neutrino events. As recommended by the IceCube Collaboration, if the redshift value is unmeasured, we use 0.5 for short bursts and 2.15 for long bursts. We draw the $\Delta t_{\rm obs}/(1+z)$ versus the $K$ plot for all-sky shower events, northern hemisphere track events, and southern hemisphere track events in Fig. 1. It is remarkable that almost all of the northern hemisphere track events fall within the error band of the energy-dependent speed variation that was revealed by the analysis on TeV and PeV GRB neutrinos Amelino-Camelia:2016ohi ; Huang:2018 . The parameters of these 19 events with associated GRBs are listed in Table 2.

To check the consistency more specifically, we do a combined fitting of 13 high-energy GRB neutrinos Amelino-Camelia:2016ohi ; Huang:2018 and 18 northern hemisphere track events from 2012 to 2015 Aartsen:2017wea . The only event from the first four-year search and marked by ⋆ in Table 2 is excluded in our analysis, because the IceCube detector was not completed during that observation time. The error estimation is according to the Methods in Ref. Huang:2018 . The fitting result shows a well linear correlation between the observed time difference and the LV factor, as shown in Fig. 2. The fitting parameters are

[TABLE]

and the correlation coefficient is $r=0.98$. If zooming in to the low-energy region of Fig. 2 (a), as shown in Fig. 2 (b), we can find that 12 of 18 northern hemisphere track events are within the error range of the combined fitting. Considering that sources of IceCube neutrino events could be more than one type, these 12 events marked by † in Table 2 may be really associated with GRBs. Therefore we do a new combined fitting which includes these 12 events and 13 higher-energy events as shown in Fig. 3, and get the slope and the intercept

[TABLE]

with the correlation coefficient $r=0.989$. Therefore the LV scale is

[TABLE]

and Eq. (11) indicates that GRB neutrinos might be emitted about five minutes before GRB photons at the source. The new combined fitting result is well consistent with the 60 TeV to 2 PeV regularity equations (6) and (7). Since the lower limit of these neutrino energies is several hundred GeVs, such a consistency among the gap over 4 orders of magnitude in energy scale, from a few hundred GeV to several PeV, can be a new support for the energy-dependent speed variation of ultrahigh-energy cosmic neutrinos.

V Neutrinos emitted before GRB photons

The combined fitting provides us with the probability to reveal the intrinsic time difference of GRB neutrinos, since it includes neutrino events observed close to the GRB trigger time. As shown in Eq. (11), the error range of $\Delta t_{\rm in}$ is much smaller than that in Eq. (7), and no longer covers the zero point. It indicates clearly that GRB neutrinos might be emitted before the GRB photons. Actually, the scenario of “early neutrino” is supported by the evidence from astronomical observations. In the first observation of supernova neutrinos Hirata:1987hu ; Bionta:1987qt in 1987, more than 10 neutrino events were observed several hours before the optical lights of the associated Supernova 1987A. In our discussion, $\Delta t_{\rm in}$ is only related to the intrinsic mechanism of the GRB. It is natural to understand the negative $\Delta t_{\rm in}$ under the framework of the GRB fireball model Eichler:1989ve ; Waxman:1995vg ; Waxman:1997ti ; Piran:1999bk . As a relativistically expanding plasma of electrons, photons and protons, the fireball becomes cooler and cooler during the expansion. At ultrahigh temperatures, neither neutrinos nor photons can freely escape from the intense plasma due to frequent collisions and reactions. With reduction of temperature, neutrinos and photons run out and propagate outward. Since the decoupling temperature of neutrinos is much higher than that of photons, neutrinos leak out of the dense fireball before the optical depth of photons become less than 1. Therefore, it is natural that GRB neutrinos are emitted before GRB photons. The negative $\Delta t_{\rm in}$ revealed by the combined analysis on neutrino events of different energy scales is thus supported by a rational explanation.

The regularity presented by Eqs. (4) or (5) reveals not only an energy-dependent speed variation, but also the $CPT$ reversal symmetry violation. Neutrinos and antineutrinos might have different signs of LV time correlation. For 60 TeV to 2 PeV GRB neutrinos, both “time delay” and “time advance” events exist Huang:2018 . Although the observed time difference $\Delta t_{\rm obs}$ of northern hemisphere events in Table 2 could be positive or negative, these 19 events are all time delay events. To select neutrino events, the IceCube Collaboration introduced a time probability distribution function Aartsen:2014aqy ; Aartsen:2016qcr ; Aartsen:2017wea which only includes neutrino events satisfying $-100~{}{\rm s}<\Delta t_{\rm obs}<300~{}{\rm s}$. However, if considering a $\Delta t_{\rm in}$ of $-$280 s at the source and assuming $z=2.15$, the arrival time of the time advance events on the Earth is at least 882 s earlier than the trigger time of the associated GRB, i.e.,

[TABLE]

Therefore, we suggest analyzing neutrino events detected a few minutes before GRBs as a method to test whether the $CPT$ violation of ultrahigh-energy neutrino does exist or not.

VI Discussion

In a recent study Aartsen:2017ibm , the IceCube Collaboration reported some limits on the Lorentz invariance violation by searching the anomalous neutrino oscillations of high-energy atmospheric neutrinos under a Lorentz violation framework of neutrino oscillations. According to the IceCube analysis, the off-diagonal term of dimension-five LV coefficients $\mathring{a}_{\mu\tau}^{(5)}$ was restricted to $\sim 10^{-32}\ {\rm GeV}^{-1}$, which seems to go beyond our result in Eqs. (10) and (12). However, the Lorentz violation scale $E_{\rm LV}$ or its reciprocal $1/E_{\rm LV}$ in our work represents actually the diagonal terms $\mathring{a}_{\mu\mu}^{(5)}$ (or other flavor corresponding parameters) in an oscillation-independent framework Zhang:2018otj . The off-diagonal coefficients mainly represent the flavor transition characters, and can thus be constrained by the measurements of LV neutrino oscillations. But the diagonal terms have no effect on the neutrino flavor transition, as shown by the oscillation formula in the Methods of Ref. Aartsen:2017ibm . The IceCube analysis shows that the LV is dominated by the large diagonal component, which just corresponds to the oscillation-independent parameters in our work. Therefore, our results are not in conflict with the recent limits on the oscillation-dependent parameters Aartsen:2017ibm .

Our analyses include both shower and track events of IceCube neutrinos. The energy uncertainties of these two kinds of events are different. For shower events, the energy is estimated with a good precision. Only a part of the energy in neutral current processes might be missed. For track events, the energy collected by the IceCube detector may be only a part of the muon energy, since the produced vertex of muon tracks may be located outside the instrumental volume. The muon energy is only a part of the total energy of the neutrino too. Therefore, the energies of track events are reported as a lower bound. In our analyses, the positive error of the energy is set as 30% for shower events and 50% for track events, the same as the approaches in Ref. Huang:2018 . Actually, the true energy of neutrinos may be even much higher than the deposited energy of track events. If we extend the error range to a much larger value, the time window needs to be extended too, and then more GRBs with longer observed time differences should be taken into account. In consideration of the narrow time window adopted by the IceCube Collaboration when finding the correlated GRBs of the near-TeV events, we choose the 50% positive energy error for track events as a reasonable assumption.

The statistical significance of the near-TeV neutrino events has been analyzed in detail by the IceCube Collaboration, in the form of the $S/B$ value Aartsen:2016qcr ; Aartsen:2017wea . Here we use the list of GRBs with the highest significance and choose the possible correlations between IceCube neutrinos and GRBs. As discussed above, because of the large amount of atmospheric $\mu$ and $\nu_{\mu}$ particles, these near-TeV events are still consistent with the atmospheric neutrino backgrounds. Hence the 12 northern hemisphere track events selected by us should be regarded as referential evidence to some extent. If these near-TeV neutrinos are generated from cosmic sources and really associated with GRBs, they would be excellent supports for the energy-dependent speed variation regularity. Even if some of these near-TeV neutrinos are just atmospheric background, such a regularity still cannot be ignored, since the higher-energy neutrinos at the scale of a few tens of TeV to PeV are certainly outstanding against the backgrounds Amelino-Camelia:2016ohi ; Huang:2018 .

VII Summary

In summary, by analyzing near-TeV IceCube neutrino events Aartsen:2014aqy ; Aartsen:2016qcr ; Aartsen:2017wea that are likely associated with GRBs, we find that another 12 northern hemisphere track events are consistent with the energy-dependent speed variation regularity found from 60 TeV to 2 PeV GRB neutrinos Amelino-Camelia:2016ohi ; Huang:2018 . Such a consistency over 4 orders of magnitude in energy provides a strong support of the revealed regularity. We also suggest that GRB neutrinos are emitted several minutes before GRB photons at the source. These findings are of fundamental importance concerning the dynamics of GRBs and properties of neutrinos and antineutrinos. Because of the continuous running of the IceCube Neutrino Observatory and many other detectors, these findings are expected to be testable in the foreseeable future.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11475006) and the Huabao Student Research Collaborative Innovation Fund of Peking University.