TeV-PeV Neutrino Oscillation of Low-luminosity Gamma-ray Bursts
aa r X i v : . [ a s t r o - ph . H E ] A p r TeV-PeV Neutrino Oscillation of Low-luminosity Gamma-rayBursts
D. Xiao , and Z. G. Dai , School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China;[email protected] Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry ofEducation, China
ABSTRACT
There is a sign that long-duration gamma-ray bursts (GRBs) originate fromthe core collapse of massive stars. During a jet puncturing through the progenitorenvelope, high energy neutrinos can be produced by the reverse shock formed atthe jet head. It is suggested that low-luminosity GRBs (LL-GRBs) are possiblecandidates of this high energy neutrino precursor up to ∼ PeV. Before leavingthe progenitor, these high energy neutrinos must oscillate from one flavor toanother with matter effect in the envelope. Under the assumption of a power-lawstellar envelope density profile ρ ∝ r − α with an index α , we study the propertiesof TeV − PeV neutrino oscillation. We find that adiabatic conversion is violatedfor these neutrinos so we do certain calibration of level crossing effect. Theresonance condition is reached for different energies at different radii. We noticethat the effective mixing angles in matter for PeV neutrinos are close to zero so thetransition probabilities from one flavor to another are almost invariant for PeVneutrinos. We plot all the transition probabilities versus energy of TeV − PeVneutrinos from the birth place to the surface of the progenitor. With an initialflavor ratio φ ν e : φ ν µ : φ ν τ = 1 : 2 : 0, we plot how the flavor ratio evolveswith energy and distance when neutrinos are still in the envelope, and furtherget the ratio when they reach the Earth. For PeV neutrinos, the ratio is always φ ν e : φ ν µ : φ ν τ ≃ .
30 : 0 .
37 : 0 .
33 on Earth. In addition, we discuss thedependence of the flavor ratio on energy and α and get a pretty good result.This dependence may provide a promising probe of the progenitor structure. Subject headings: gamma-ray bursts: general — neutrinos 2 –
1. Introduction
The idea that GRBs can serve as the sources of high energy neutrinos has long been dis-cussed (Waxman & Bahcall 1997, 2000; Dai & Lu 2001; Li et al. 2002; Dermer et al. 2003;Razzaque 2013; Vietri 1995; M´esz´aros & Rees 2000; Guetta et al. 2004; Murase & Nagataki2006; Rees & M´esz´aros 2005; Murase 2008; Wang & Dai 2009; Gao et al. 2012; M´esz´aros & Waxman2001; Enberg et al. 2009; Murase & Ioka 2013; Pruet 2003; Razzaque et al. 2003; Ando & Beacom2005; Horiuchi & Ando 2008). Before breaking out, a relativistic jet punctures throughthe stellar envelope and transports energy to electrons and protons via shock acceleration(Zhang & M´esz´aros 2004; Piran 2005; M´esz´aros 2006; Woosley 1993; MacFadyen & Woosley1999). Accelerated electrons dominate radiation by synchrotron or inverse Compton mech-anism while accelerated protons produce neutrinos by proton-proton collision and photo-pion process (Waxman & Bahcall 1997, 1999; Rachen & M´esz´aros 1998; Alvarez-Muniz et al.2000; Bahcall & Waxman 2001; Guetta & Granot 2003; Murase et al. 2006; Becker 2008).This neutrino signal is prior to the main burst, as a precursor. However, it has been arguedthat this high energy neutrino precursor of energy ranging from TeV up to PeV cannot beproduced for a typical GRB (Levinson & Bromberg 2008; Katz et al. 2010; Murase & Ioka2013). The reason is that the reverse shock occurring at the interface of the jet head wouldbe radiation mediated, resulting in an inefficient shock acceleration. Nevertheless, a matteris different for LL-GRBs (Murase & Ioka 2013; Xiao & Dai 2014). Due to its low power, theThomson optical depth is low even inside a star so that efficient shock acceleration would beexpected. We assume the same LL-GRB as our previous work (Xiao & Dai 2014), in whichwe have shown that our LL-GRB is responsible for TeV − PeV neutrinos. Further in thispaper we focus on the oscillation properties of these high energy neutrinos.Neutrino oscillation in matter has been studied for a long while. The resonant con-version of neutrinos from one flavor to another with matter effect was first discussed insolar neutrino problem (Chen 1985). While propagating in a medium, ν e interacts via neu-tral current (NC) and charged current (CC), whereas ν µ and ν τ interacts only via NC. Thismechanism is called the Mikheyev-Smirnov-Wolfenstein (MSW) effect (Mikheyev & Smirnov1985; Wolfenstein 1978). As we know, solar neutrinos are relative low in energy ( ≤ π ′ s and µ ′ s which would modify the flavorratio produced by GRBs and concluded that the flavor ratio on Earth φ ν e : φ ν µ : φ ν τ is1 : 1 : 1 at low energy to 1 : 1 . . . − R and discussed its variation with source properties and neutrino oscillation param-eters. Razzaque & Smirnov (2010) also presented a detailed and comprehensive study offlavor conversion of neutrinos from hidden sources (jets) but their results differ from theresults of Mena et al. (2007). Sahu & Zhang (2010) showed that the resonant oscillationcould take place within the inner high density region of the choked jet progenitor and thefinal flavor ratio detected on Earth is further modified to either 1 : 1 .
095 : 1 .
095 for thelarge mixing angle solution to the solar neutrino data, or 1 : 1 . . ≤ − PeV neutrinos created ininternal shocks at different places in the star, estimating the flavor ratios on Earth.The main difference of our paper with previous works is at the starting point that we takea LL-GRB as the source of high energy neutrinos and hence we focus on higher energy fromTeV to PeV. We find that the mixing angles in matter for PeV neutrinos are close to zero sothe transition probabilities from one flavor to another are almost invariant, which are differentwith MeV − TeV neutrinos. Thus we get a constant ratio of φ ν e : φ ν µ : φ ν τ ≃ .
30 : 0 .
37 : 0 . α and neutrino energy, providing a promising way to probe the GRB progenitorstructure through a neutrino precursor signal in the future.This paper is organised as follows. We present all results in section 2. Subsection 2.1is about the density profile of the envelope and subsection 2.2 is about adiabatic violationfor neutrinos above TeV. In subsection 2.3 we discuss the three neutrino mixing both inthe envelope and in vacuum, and then the dependence on envelope density profile index isexhibited in subsection 2.4. We finish with discussions and conclusions in section 3.
2. Neutrino Mixing2.1. Density Profile of the Envelope
In this subsection, we take α = 2 as our premise, and discuss the dependence on α laterin subsection 2.4.We assume a power-law envelope density profile ρ ( r ) = Ar − α , where A = (3 − α ) M He / (4 πR − α )and 2 ≤ α < M He and R being the mass and radius of the helium envelope. For a 4 –helium core of mass M He = 2 M ⊙ and radius R = 4 × cm, the ambient envelope den-sity can be expressed as ρ ( r ) = 7 . × r − g cm − . The number density of electrons inthe envelope is N e ( r ) = ρ ( r )4 m p Y e , where the number of electrons per nucleon Y e needs to beobtained.The Saha’s equation readslog N r +1 N r = log 2 u r +1 ( T ) u r ( T ) + 52 log T − T χ r − log P e − . , (1)where N r , u r , and χ r stand for the number density, partition function and ionization energyof r th ionization ions, respectively. T is the temperature and P e ≡ N e kT is the electronpressure. For our pure helium envelope, we can get N (He ) N (He + ) = 1 . × − , N (He + ) N (He) = 0 . , (2)if we adopt typical values as T = 15000K , χ = 24 . , χ = 54 . Y e = N (He + ) N (He)+ N (He + ) ≃ . V eff = √ G F N e , (3)where G F is the Fermi coupling constant. Before we start to consider the neutrino oscillation, we need to check whether theadiabatic approximation is valid. The adiabatic parameter γ is defined as γ ≡ δm E sin 2 θ tan 2 θ | d ln N e dr | res , (4)where δm is the mass square difference between the neutrino mass eigenstates, E is theneutrino energy and θ is the mixing angle. The subscript “res” represents the place atwhich resonance happens. We can easily see that γ is in proportion to 1 /E and the adia-batic approximation requirement γ ≫ ≤ γ ≫ γ and 5 – γ are needed because there are at most two level crossing for neutrinos in the three fla-vor case (Dighe & Smirnov 2000; Yasuda 2014). The vacuum oscillation parameters weadopt are δm = 7 . × − eV , δm = 2 . × − eV , sin θ = 0 . , sin θ =2 . × − , sin θ = 3 . × − , CP violation phase δ = 1 . π and normal mass hierarchyis assumed (Fogli et al. 2012).For the reason above, we are obliged to do certain calibration of level crossing effect.The jumping probability is approximately computed by the WKB method (Yasuda 2014;Kuo 1989): P = exp[ − π γF ] − exp[ − π γ F sin θ ]1 − exp[ − π γ F sin θ ] , (5)where F is a factor depending on the density profile. Then in Figure 2 we plot P H and P L versus neutrino energy, representing the jumping probability from energy eigenstate ν m to ν m and from ν m to ν m respectively. The evolution equation for neutrinos in matter is given by i d Ψ dt = [ U H U † + V eff ]Ψ , (6)where H = E diag( − δm , , δm ) and Ψ T ≡ ( ν e , ν µ , ν τ ) is the flavor eigenstate (Fraija2014). U is the three neutrino mixing matrix, U = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c , (7)where s ij ≡ sin θ ij , c ij ≡ cos θ ij .Neutrino mixing angles in matter can be expressed as (Fraija 2014; Yasuda 2014)sin 2 θ ,m = sin 2 θ p (cos 2 θ − EV eff /δm ) + (sin 2 θ ) , sin 2 θ ,m = sin 2 θ p (cos 2 θ − EV eff c /δm ) + (sin 2 θ ) , (8) 6 –The effective mixing angles θ ,m , θ ,m become maximum π/ θ =2 EV eff /δm and cos 2 θ = 2 EV eff c /δm are fulfilled respectively. We can see that θ ,m , θ ,m are functions of neutrino energy and radius since V eff = V eff ( r ). We plot θ ,m , θ ,m versus energy at radius r = 1 . × cm in Figure 3(a) and find that θ ,m reaches max-imum for ∼ r is treated as the born site of high en-ergy neutrinos. The reason is that the Thomson optical depth is τ T = nσ T l = ρ j m p σ T r Γ j ≃ . L j erg s − ) ( θ . ) ( M He M ⊙ ) ( r × cm ) − and at r = r we have τ T ≃
1, thus ensuringthe efficient shock acceleration (Xiao & Dai 2014). We also plot the effective mixing anglesversus energy at progenitor surface R = 4 × cm in Figure 3(b) and how they evolve withpropagation distance for 1TeV neutrinos in Figure 3(c) respectively. In addition, we kindlyfind that effective mixing angles tend to be constant zero for PeV neutrinos, which is themain reason for constant flavor ratio and will be shown later.The transition probability from one flavor to another after level crossing calibration canbe expressed as (Yasuda 2014) P ( ν α → ν β ) = (cid:0) | U β ,m | | U β ,m | | U β ,m | (cid:1) − P L P L P L − P L
00 0 1 × − P H P H P H − P H | U α ,m | | U α ,m | | U α ,m | , (9)where α, β → e, µ, τ and U αi,m , U βj,m are mixing matrix elements in matter.We plot the nine mutual transition probabilities between three neutrino flavors as func-tions of neutrino energy at different radii in Figure 4(a)- 4(b) and the evolution withpropagation distance for given energies in Figure 4(c)- 4(d). Given an initial flavor ratio φ ν e : φ ν µ : φ ν τ = 1 : 2 : 0, we can plot how the flavor ratio changes with energy and distancewhen neutrinos are still in the envelope in Figure 5(a)- 5(d). The deviation from 1 : 2 : 0 atborn site r is due to level crossing effect for different energies. We can see the trend thatthe flavor ratio changes more gently for neutrinos with higher energy. For PeV neutrinos,the flavor ratio is almost a constant value φ ν e : φ ν µ : φ ν τ ≃ .
21 : 0 .
77 : 0 .
02 in the envelope.
Neutrinos go through vacuum oscillation after leaving the progenitor surface and hasbeen well understood. We can express the transition probability P αβ as the first-order 7 –expansion of the small parameter sin θ (Xing & Zhou 2006): P ee = 1 −
12 sin θ ,P eµ = P µe = 12 sin θ cos θ + 14 sin 4 θ sin 2 θ sin θ cos δ ,P eτ = P τe = 12 sin θ sin θ −
14 sin 4 θ sin 2 θ sin θ cos δ ,P µµ = 1 −
12 sin θ −
12 sin θ cos θ −
12 sin 4 θ sin 2 θ cos θ sin θ cos δ ,P µτ = P τµ = 12 sin θ −
18 sin θ sin θ + 18 sin 4 θ sin 4 θ sin θ cos δ ,P ττ = 1 −
12 sin θ −
12 sin θ sin θ + 12 sin 4 θ sin 2 θ sin θ sin θ cos δ . (10)The flavor ratio on Earth is φ ν e φ ν µ φ ν τ Earth = P ee P eµ P eτ P µe P µµ P µτ P τe P τµ P ττ φ ν e φ ν µ φ ν τ source . (11)We plot the flavor ratio versus neutrino energy on just leaving the progenitor and on Earthin Figure 6(a)- 6(b). We can see clearly that the flavor ratio varies with energy in the rangeof less than 100TeV, while the flavor ratio keeps invariant φ ν e : φ ν µ : φ ν τ ≃ .
30 : 0 .
37 : 0 . α It is reasonable to argue that the final neutrino flavor ratio depends on the densityprofile of the progenitor envelope. Apparently, with the same assumed envelope mass andradius, different values of the power law index lead to different ambient envelope densities,thus effective potentials are different. This will have an impact on the resonance conditions,effective mixing angles in matter and transition probabilities. In this subsection, we wouldlike to investigate how large this impact could be. Here we adopt the same helium progenitorbut with different power law index α = 2 . , .
7. Respectively, we can write them as ρ ( r ) =2 . × r − . g cm − and ρ ( r ) = 3 . × r − . g cm − and all calculations have beenrepeated for these two cases.We present our result in Figure 7. For simplicity, we only show the flavor ratio at thesurface of the progenitor (Figure 7(a)) and on Earth ( Figure 7(b)). It is clear that α has 8 –an impact on the flavor ratio. At a given radius, the resonance conditions are shifted tohigher energies for larger α . The lower limit of neutrino energy for constant flavor ratio ishighest for α = 2 .
7, which is several PeV, compared with subPeV for α = 2 . ∼ α = 2. The reason is that the effective potential of neutrinos is lower for a steeperenvelope density profile at the same radius, so higher neutrino energies are required to reachresonance conditions. Furthermore, the flavor ratio is evidently different before it reachesconstant for the three cases, so that we can use the observed ratio of neutrino energy fromTeV to several hundred TeVs to probe the stellar structure, provided that we can observeprecursor neutrinos of a GRB with km scale detectors like IceCube in the future.
3. Discussions and Conclusions
High-energy neutrinos can be produced while the jet is still propagating in the envelopeand LL-GRBs with typical parameters are responsible for TeV − PeV neutrinos. Theseneutrinos will oscillate with matter effect in the envelope and go through vacuum oscillationafter leaving the progenitor till they arrive at the Earth. We investigate the three-neutrinomixing properties with matter effect and then get an expected flavor ratio on Earth, givenan initial ratio φ ν e : φ ν µ : φ ν τ = 1 : 2 : 0.We notice that adiabatic conversion is violated because level crossing effect is non-negligible for such high energy neutrinos. After calibrating this effect, we get the neutrinomixing angles in matter and nine transition probabilities. We find that the effective mixingangles tend to be zero for neutrinos on the high energy end ( ∼ PeV), resulting in constanttransition probability and constant flavor ratio. For PeV neutrinos, we always get φ ν e : φ ν µ : φ ν τ ≃ .
30 : 0 .
37 : 0 .
33 on Earth.From our expectations, the final neutrino ratio will depend on the density profile param-eter α . We take α = 2 , . , . T . Though the number density of electrons varies fordifferent T , the effective mixing angles for PeV neutrinos are always zero and constant theflavor ratio φ ν e : φ ν µ : φ ν τ ≃ .
30 : 0 .
37 : 0 .
33 on Earth is still in expectation. However,this constant value may differ from 0 .
30 : 0 .
37 : 0 .
33 due to the uncertainties of vacuumoscillation parameters and neutrino mass hierarchy.The IceCube experiment has recently reported the observation of 37 high-energy ( ≥ φ ν e : φ ν µ : φ ν τ = 1 : 1 : 1 at Earth is disfavoredat 92% C.L. with the recently released 3-yr data. On one hand, neutrino flavor ratio in thedetector may be changed by the Earth matter effect (Varela et al. 2014). On the other hand,this does not conflict with our conclusion. We just recommend to do an analysis of observedshower-to-track ratio in different energy bins when doing data reduction since we know thatthe flavor ratio depends strongly on neutrino energy and an overall 1 : 1 : 1 ratio does notmake any sense. Nevertheless, we expect constant flavor ratio for PeV neutrinos but we haveobserved only three PeV events now. So our result is to be verified with a larger dataset ofTeV − PeV neutrinos of IceCube in the future. If this constant value appears in next fewdecades, when we would have observed tens of PeV neutrino events, we can constrain thestructure of LL-GRB progenitors and vacuum oscillation parameters by exactly measuringthis value. If there is no sign of such a constant ratio, the most probable reason is that thereexist other dominate PeV neutrino sources. The hypothesis that all observed neutrinos areproduced in the jet propagation process of LL-GRBs may be not so complete because theymay also origin from other cosmic-ray sources like AGNs or they can be produced in otherstages of a GRB event such as by internal shocks and external shocks. Especially, we hopethat one day we could observe the neutrino precursor of a GRB event, this neutrino-GRBcorrelation is crucial for our understanding about the structure of the progenitor envelopeand the jet propagation dynamics. For a complete comparison with the observation, theflavor ratio in TeV range of diffuse neutrino background produced by GRBs needs to bedone and is beyond the scope of this paper.We thank an anonymous referee for his/her helpful suggestions. This work is supportedby the National Basic Research Program of China (973 Program, grant 2014CB845800) andthe National Natural Science Foundation of China (grant 11033002). 10 –
REFERENCES
Ando, S., Beacom, J. F. 2005, Phys. Rev. Lett., 95, 061103Alvarez-Muniz, J., Halzen, F., & Hooper, D. W. 2000, Phys. Rev. D, 62, 093015Bahcall, J., & Waxan, E. 2001, Phys. Rev. D, 64, 023002Becker, J. K. 2008, Phys. Rep., 458, 173Chen, H. H. 1985, Phys. Rev. Lett., 55, 1534Dai, Z. G., & Lu, T. 2001, ApJ, 551, 249Dermer, C. D., & Atoyan, A. 2003, Phys. Rev. Lett., 91, 071102Dighe, A. S., & Smirnov, A. Y. 2000, Phys. Rev. D, 62, 033007Enberg, R., Reno, M. H., & Sarcevic, I. 2009, Phys. Rev. D, 79, 053006Fogli, G. L., Lisi, E., Marrone, A., Montanino, D., Palazzo, A., & Rotunno, A. M. 2012,Phys. Rev. D, 86, 013012Fraija, N. 2014, MNRAS, 437, 2187Gao, S., Asano, K., & M´esz´aros, P. 2012, JCAP, 1211, 058Guetta, D., & Granot, J. 2003, Phys. Rev. Lett., 90, 201103Guetta, D., Hooper, D., Alvarez-Mu˜niz, J., Halzen, F., & Reuveni, E. 2004, AstroparticlePhysics, 20, 429Horiuchi, S., & Ando, S. 2008, Phys. Rev. D, 77, 063007IceCube Collaboration, 2014, Phys. Rev. Lett., 113, 101101Kashti, T., & Waxman, E. 2005, Phys. Rev. Lett., 95, 181101Katz, B., Budnik, R., & Waxman, E. 2010, ApJ, 716, 781Kuo, T. K. 1989, Phys. Rev. D, 39, 1930Levinson, A., & Bromberg, O. 2008, Phys. Rev. Lett., 100, 131101Li, Z., Dai, Z. G., & Lu, T. 2002, A&A, 396, 303Lund, T., & Kneller J. P. 2013, Phys. Rev. D, 88, 3008 11 –MacFadyen, A., & Woosley, S. E. 1999, ApJ, 524, 262Mena, O., Mocioiu, I., & Razzaque, S. 2007, Phys. Rev. D, 75, 063003Mena, O., Palomares-Ruiz, S., & Vincent, A. C. 2014, Phys. Rev. Lett., 113, 1103M´esz´aros, P., & Rees, M. J. 2000, ApJ, 541, L5-L8M´esz´aros, P., & Waxman, E. 2001, Phys. Rev. Lett., 87, 171102M´esz´aros, P. 2006, Rept. Prog. Phys., 69, 2259Mikheyev, S. P., & Smirnov, A. Y. 1985, Yad. Fiz., 42, 1441Murase, K., Ioka, K., Nagataki, S., &Nakamura, T. 2006, ApJ, 651, L5Murase, K., & Nagataki, S. 2006, Phys. Rev. D, 73, 063002Murase, K. 2008, Phys. Rev. D, 78, 101302Murase, K., & Ioka, K. 2013, Phys. Rev. Lett., 111, 121102Osorio Oliveros, A. F., Sahu, S., & Sanabria, J. C. 2013, The European Physical Journal C,73, 2574Piran, T. 2005, Rev. Mod. Phys., 76, 1143Pruet, J. 2003, ApJ, 591, 1104-1109Rachen, J. P., & M´esz´aros, P. 1998, Phys. Rev. D, 58, 123005Razzaque, S., M´esz´aros, P., & Waxman, E. 2003, Phys. Rev. Lett., 90, 241103Razzaque, S., & Smirnov, A. Y. 2010, Journal of High Energy Physics, 3, 31Razzaque, S. 2013, Phys. Rev. D, 88, 103003Rees, M. J., & M´esz´aros, P. 2005, ApJ, 628, 847Sahu, S., & Zhang, B. 2010, Research in Astronomy and Astrophysics, 10, 943Scholberg, K. 2012, Annual Review of Nuclear and Particle Science, 62, 81Varela, K., Sahu, S., Osorio Oliveros, A. F., & Sanabria, J. C. 2014 arXiv:1411.7992.Vietri, M. 1995, ApJ, 453, 883 12 –Wang, X. Y., & Dai, Z. G. 2009, ApJ, 691, 67Waxman, E., & Bahcall, J. 1997, Phys. Rev. Lett., 78, 2292Waxman, E., & Bahcall, J. 1998, Phys. Rev. D, 59, 023002Waxman, E., & Bahcall, J. 2000, ApJ, 541, 707Wolfenstein, L. 1978, Phys. Rev. D, 17, 2369Woosley, S. E. 1993, ApJ, 405, 273Xiao, D., & Dai, Z. G. 2014, ApJ, 790, 59Xing, Z. Z., & Zhou, S. 2006, Phys. Rev. D, 74, 3010Yasuda, O. 2014, Phys. Rev. D, 89, 3023Zhang, B., & M´esz´aros, P. 2004, Int. J. Mod. Phys. A 19, 2385
This preprint was prepared with the AAS L A TEX macros v5.2.
13 – Γ Γ H eV L Γ Fig. 1.— Adiabatic parameters versus neutrino energy of our LL-GRB progenitor. The redline represents adiabatic parameter for level crossing between 1st and 3rd energy eigenstates,while the blue line is that for level crossing between 1st and 2nd energy eigenstates. 14 – P L P H H eV L J u m p i n g P r o b a b ili t y P Fig. 2.— Jumping probabilities versus neutrino energy according to adiabatic parameters.The red line represents the jumping probability between 1st and 3rd energy eigenstates, whilethe blue line is that for the jumping probability between 1st and 2nd energy eigenstates. 15 – H eV L Θ m (a) Effective mixing angles versus neutrino energyat born site r = 1 × cm H eV L Θ m (b) Effective mixing angles versus neutrino energyat progenitor surface R = 4 × cm ´ ´ ´ ´ H cm L Θ m (c) Effective mixing angles evolve versus radius for1TeV neutrinos Fig. 3.— Effective mixing angles in matter for neutrinos with different energies at differentradii. In all three subfigures above, the red line represents θ ,m and the blue line is θ ,m .All angles are measured in radians. 16 – H eV L T r a i n s i t i o n P r o b a b ili t y P (a) Transition probabilities versus neutrino energyat born site r = 1 × cm H eV L T r a i n s i t i o n P r o b a b ili t y P (b) Transition probabilities versus neutrino energyat progenitor surface R = 4 × cm ´ ´ ´ ´ H cm L T r a i n s i t i o n P r o b a b ili t y P (c) Transition probabilities evolve versus radius for1TeV neutrinos ´ ´ ´ ´ H cm L T r a i n s i t i o n P r o b a b ili t y P (d) Transition probabilities evolve versus radius for1PeV neutrinos Fig. 4.— Transition probabilities in matter for neutrinos with different energies at differentradii. Nine different colors represent nine mutual transition probabilities and are listed below: P ee → red, P eµ → orange, P eτ → green, P µe → blue, P µµ → brown, P µτ → purple, P τe → black, P τµ → yellow, P ττ → cyan. These apply for all four subfigures above. 17 – ´ ´ ´ ´ H cm L F l avo r R a t i o Φ (a) Flavor ratio evolves versus radius for 1TeV neu-trinos ´ ´ ´ ´ H cm L F l avo r R a t i o Φ (b) Flavor ratio evolves versus radius for 10TeVneutrinos ´ ´ ´ ´ H cm L F l avo r R a t i o Φ (c) Flavor ratio evolves versus radius for 100TeVneutrinos ´ ´ ´ ´ H cm L F l avo r R a t i o Φ (d) Flavor ratio evolves versus radius for 1PeVneutrinos Fig. 5.— Flavor ratio in matter evolves versus radius for neutrinos with four differentenergies. In all four subfigures above, the red line represents the fraction of ν e , the greenline is ν µ and the blue line is ν τ . 18 – H eV L F l avo r R a t i o Φ (a) Flavor ratio versus energy when neutrinos arejust leaving the progenitor H eV L F l avo r R a t i o Φ (b) Flavor ratio versus energy on Earth Fig. 6.— Neutrino flavor ratio versus energy before and after long distance vacuum oscilla-tion. Same as Figure 5, the red line represents the fraction of ν e , the green line is ν µ and theblue line is ν τ . H eV L F l avo r R a t i o Φ (a) Comparison of flavor ratios versus energy whenneutrinos are just leaving the progenitors for threedifferent envelope power law indexes H eV L F l avo r R a t i o Φ (b) Comparison of flavor ratios versus energy onEarth for three different progenitor envelope powerlaw indexes Fig. 7.— The dependence of flavor ratios versus energy before and after long distance vacuumoscillation on progenitor envelope power law index. The red lines represent the fractions of ν e , the green lines are ν µ and the blue lines are ν τ . Besides, solid lines are responsible for α = 2, dotted lines are for α = 2 . α = 2 ..