Effects of conversions for high energy neutrinos originating from cosmological gamma-ray burst fireballs H. Athar Departamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Spain; E-mail: [email protected] (February 7, 2008) 9 9 9 We study neutrino conversions in the recently envisaged source of 1 high energy (E∼>106 GeV) neutrinos, that is, in the vicinity of cosmo- n logical gamma-ray burst fireballs (GRB). We consider mainly the possi- a J bility of neutrino conversions due to an interplay of neutrino transition 9 magnetic moment, µ, and the violation of equivalence principle (VEP), 2 parameterized by ∆f, in a reasonable strength of magnetic field in the 1 vicinity of the GRB. We point out that for ∆f 10−25(δm2/1eV2), v ∼ 0 a resonant spin-flavour precession between ν¯µ and ντ may occur in the 5 vicinity of GRB for µ 10−12µ (µ is Bohr magneton), thus enhanc- B B 4 ∼ ing the expected high energy ν flux from GRBs. 1 τ 0 PACS numbers: 95.85.Ry, 14.60.Pq, 13.15.+g, 04.80.Cc 9 9 / h p - p I. INTRODUCTION e h v: Recently, cosmologicalfireballs aresuggested asa possible productionsite for gamma-ray Xi bursts as well as the high energy (E∼>106 GeV) neutrinos [1]. Although, the origin of these r cosmological gamma-ray burst fireballs (GRB) is not yet well-understood, the observations a [2] suggest that generically a very compact source of linear scale 107 cm through internal ∼ or/and external shock propagation produces these gamma-ray bursts as well as the (burst of) high energy neutrinos. Typically, this compact source is hypothesized to be formed possibly due to the merging of binary neutron stars or due to collapse of a supermassive star. The main source of high energy tau neutrinos in GRBs is the production and decay of ± ± D . The production of D may be through pγ and/or through pp collisions. In [3], the ν S S e and ν flux is estimated in pp collisions, whereas in [1], the ν and ν flux is estimated in µ e µ pγ collisions for GRBs. In pp collisions, the flux of ν may be obtained through the main τ process of p+p D++X. The D+ decays into τ+ lepton and ν with a branching ratio of → S S τ 3%. This τ+ lepton further decays into ν . The cross-section for D+ production, which ∼ τ S is main source of ν ’s, is 1-2 orders of magnitude lower than that of D+ and/or D− which τ 1 subsequently produces ν and ν . The branching ratio for ν and/or ν production is higher e µ e µ ± upto an order of magnitude than for ν production (through D ). These imply that the ν τ S τ flux in pp collisions is suppressed up to 3-4 orders of magnitude relative to corresponding ν and/or ν fluxes. In pγ collisions, the main process for the production of ν may be e µ τ p+γ D++Λ0+D¯0 with similar relevant branching ratios and corresponding cross-section → S values. Herethecorrespondingmainsourceforν andν productionisp+γ ∆+ π++n. e µ → → Therefore,inpγ collisions, theν fluxisalsosuppressedupto3-4ordersofmagnituderelative τ to ν and/or ν flux. Thus, in both type of collisions, including the relevant kinematics, the e µ ν flux is estimated to be rather small relative to ν and/or ν fluxes from GRBs, typically τ e µ being, (ν +ν¯ )/(ν +ν¯ )<(10−4 10−3) [4]. τ τ µ µ ∼ − In this brief report, we consider the possibility of obtaining higher ν neutrino flux, that τ is, (ν +ν¯ )/(ν +ν¯ ) 10−4, from GRBs through neutrino conversions as compared to no τ τ µ µ ≫ conversion situation [5]. Inparticular, we obtaintherangeof neutrino mixing parameters re- sulting from an interplay of possible violation of the equivalence principle (VEP) parameter- izedby∆f andthemagneticfieldinthevicinityofGRBsyielding(ν +ν¯ )/(ν +ν¯ ) 10−4. τ τ µ µ ≫ The possibility of VEP arises from the realization that different flavours of neutrinos may couple differently to gravity [6]. Thepresent studyisparticularlyusefulasthenewice/waterCˇerenkov lightneutrinotele- scopes namely AMANDA and Baikal (also the NESTOR and ANTARES) will have energy, angle and flavour resolution for high energy neutrinos originating at cosmological distances [7]. Recently, there are several discussions concerning the signatures of a possible neutrino burst from GRBs correlated in time and angle [8]. In particular, there is a suggestion of measuring ν flux from cosmologically distant sources through a double bang event [9] or τ through a small pile up of upgoing µ-like events near (104 105) GeV [10]. − The plan of this brief report is as follows. In Sect. II, we briefly describe the mat- ter density and magnetic field in the vicinity of GRBs. In Sect. III, we discuss in some detail, the range of neutrino mixing parameters that may give rise to relatively large pre- cession/conversion probabilities resulting from an interplay of VEP and the magnetic field in the vicinity of GRBs for vanishing gravity and vacuum mixing angles. In the same Sect., we briefly discuss the relevant neutrino mixing parameter range for non-vanishing gravity and vacuum mixing angles with vanishing neutrino magnetic moment. In Sect. IV, we give estimates of separable but contained double bang event rates induced by high energy ν ’s τ originatingfromGRBswithout/withconversions forillustrative purposes andfinally inSect. V, we summarize our results. 2 II. THE MATTER DENSITY AND THE MAGNETIC FIELD IN THE VICINITY OF GRB According to [1], the high energy neutrino production may take place in the vicinity of r Γ2c∆t. Here Γ is the Lorentz factor (typically Γ 300) and ∆t is the observed GRB p ∼ ∼ variability time scale (typically ∆t 1 ms). Thus, the fireball matter density is ρ 10−13 ∼ ∼ g cm−3 in the vicinity of r [1]. In these models, the typical distance traversed by neutrinos p may be taken as, ∆r<(10−4 1) pc, in the vicinity of GRB, where 1 pc 3 1018 cm. ∼ − ∼ × These imply that average effective width of matter traversed by neutrinos originating from GRBis l ρ ∆r 104 g cm−2. In the presence of matter, the relevant effective width GRB ≃ × ∼ of matter needed for appreciable spin-flavour conversions is lm √2πmN/GF 2 109 g ∼ ∼ × cm−2. Thus, lGRB lm, and hence no matter effects are expected due to coherent forward ≪ scattering of neutrinos off the background for high energy neutrinos originating from GRBs. Taking the observed duration of the typical gamma-ray burst as, ∆t<1 ms, we obtain ∼ the mass of the source as, M <∆t/G , where G is the gravitational constant. Let us GRB∼ N N mention here that for the relatively shorter observed duration of gamma-ray burst from a typical GRB, ∆t 0.2 ms, implying M 40M (where M 2 1033 g is solar GRB ⊙ ⊙ ∼ ∼ ∼ × mass). We use M 2 102M in our estimates. GRB ⊙ ∼ × The magnetic field in the vicinity of a GRB is obtained by considering the equipartition arguments [1]. We use the following profile of magnetic field, B , for r > r [11] GRB p r 2 p B (r) B , (1) GRB 0 ≃ r (cid:18) (cid:19) where, B L1/2c−1/2(r Γ)−1 with L being the total wind luminosity (typically L 1051 0 p ∼ ∼ erg s−1). III. NEUTRINO CONVERSIONS IN GRB Consider a system of two mixed neutrinos ν¯ and ν for simplicity. The difference of µ τ diagonal elements of the 2 2 effective Hamiltonian describing the dynamics of the mixed × system of these oscillating neutrinos in the basis ψT = (ν¯ ,ν ) for vanishing vacuum and µ τ gravity mixing angles is [12] ∆H = V δ, (2) G − whereas, each of the off diagonal elements is µB (µ is neutrino magnetic moment). In Eq. (2), δ = δm2/2E, where δm2 = m2(ν ) m2(ν¯ ) > 0 and E being the neutrino energy. τ µ − Here V is the effective potential felt by the neutrinos at a distance r from a gravitational G source of mass M due to VEP and is given by [6] 3 V = ∆fφ(r)E, (3) G where ∆f = f f is a measure of the degree of VEP and φ(r) = G Mr−1 is the 3 2 N − gravitational potential in the Keplerian approximation. In Eq. (3), f G and f G are 3 N 2 N respectively the gravitational couplings of ν and ν¯ , such that f = f . There are three τ µ 2 3 6 relevant φ(r)’s that need to be considered [14]. The effect of φ(r) due to supercluster named Great Attractor with φ (r) in the vicinity of GRB; φ(r) due to GRB itself, which is, SC φ (r), in the vicinity of GRB and the galactic gravitational potential, which is, φ (r). GRB G Therefore, we use, φ(r) φ (r)+φ (r)+φ (r). However, φ (r) φ (r), φ (r) SC GRB G G SC GRB ≡ ≪ in the vicinity of GRB. Thus, φ(r) φ (r)+φ (r). If the neutrino production region SC GRB ≃ r is <1013 cm then at r , we have φ (r) > φ (r). At r 6 1013 cm, φ (r) p ∼ p GRB SC GRB ∼ ∼ × ∼ φ (r) and for r>1014 cm, φ (r) < φ (r). If the supercluster is a fake object then SC ∼ GRB SC φ(r) φ (r). Here we assume the smallness of the effect of φ(r) due to an active galactic GRB ≃ nucleus (AGN), if any, nearby to GRB. The possibility of vanishing gravity and vacuum mixing angle in Eq. (2) allows us to identify the range of ∆f relevant for the neutrino magnetic moment effects only. Latter in this Sect., we briefly comment on the ranges of relevant neutrino mixing parameters for non-vanishing gravity and vacuum mixing angles with vanishing neutrino magnetic moment. The case of ν¯ ν can be studied by replacing ν¯ with ν¯ along with corresponding e τ µ e → changes in V and δm2. Let us here mention that initial flux of ν¯ may be smaller than G e that of ν by a factor of less than 2 [1], thus also possibly enhancing the expected ν flux µ τ from GRBs through ν¯ ν . However, we have checked that observationally this possibility e τ → leads to quite similar results in terms of event rates and are therefore not discussed here further. We now intend to study in some detail, the various possibilities arising from relative comparison between δ and V in Eq. (2). G Let us first ignore the effects of VEP (∆f = 0). For constant B, the spin-flavour precession probability P(ν¯ ν ) is obtained using Eq. (2) as µ τ → (2µB)2 ∆r P(ν¯ ν ) = sin2 (2µB)2 +δ2 . (4) µ → τ "(2µB)2 +δ2# (cid:18)q · 2 (cid:19) We take µ 10−12µ or less, where µ is Bohr magneton, which is less than the stringent B B ∼ astrophysical upper bound on µ based on cooling of red giants [13]. We here consider the transitionmagneticmoment,allowingthepossibilityofsimultaneouslychangingtheneutrino flavour as well as the helicity . Thus, the precessed ν is an active neutrino and interacts τ weakly. If µ is of Dirac type, then the precession leads to disappearance/nonobservation of ν as the precessed ν is now a sterile one. In Eq. (4), ∆r is the width of the region τ τ with B. Note that if δ 2µB, then, for E 2 106 GeV and using Eq. (1), we obtain ≪ ∼ × δm2 5 10−8 eV2. We take, δm2 10−9 eV2, as an example and consequently we obtain ≪ × ∼ from Eq. (4) an energy independent large (P > 1/2) spin-flavour precession probability for 4 µ 10−13µ with 10−4<∆r/pc<1 . Let us mention here that the typical relevant energy B ∼ ∼ ∼ span for high energy neutrinos originating from the considered class of GRBs is an order of magnitude, that is, 2 106<E/GeV<2 107 (see Sect. IV). Thus, for µ of the order ∼ ∼ × × of 10−13µ , the ν flux may be higher than the expected one from GRBs in the absence of B τ spin-flavour precessions, that is, (ν + ν¯ )/(ν + ν¯ ) 10−3 due to neutrino spin-flavour τ τ µ µ ≫ precession effects. Since the neutrino spin-flavour precession effect is essentially determined by the product µB so one may rescale µ and B to obtain the same results. For δ 2µB ≃ and δ 2µB, we obtain from Eq. (4), an energy dependent P such that P < 1/2. ≫ With non-vanishing ∆f (∆f = 0), a resonant character in neutrino spin-flavour pre- 6 cession can be obtained for a range of values of relevant neutrino mixing parameters 1 Two conditions are essential to obtain a resonant character in neutrino spin-flavour precession: the level crossing and the adiabaticity condition. The level crossing condition is obtained by taking ∆H = 0 and is given by: ∆f 2 2 δm | | eV . (5) ∼ 10−25! The other essential condition, namely, the adiabaticity in the resonance reads [12] 2(2µB)2 κ = . (6) dV /dr G | | Note that here κ depends explicitly on E through V unlike the case of ordinary neutrino G spin-flip induced by the matter effects. A resonant character in neutrino spin-flavour pre- cession is obtained if κ>1 such that Eq. (5) is satisfied. We notice that, B /B <1 for ∼ ad GRB∼ µ 10−12µ . Here B is obtained by setting κ 1 in Eq. (6). The general expression B ad ∼ ∼ for relevant neutrino spin-flavour conversion probability is given by [15] 1 1 P(ν¯ ν ) = P cos2θ , (7) µ τ LZ B → 2 − 2 − (cid:18) (cid:19) where P = exp( πκ) and tan2θ = (2µB)/∆H is being evaluated at the high energy LZ −2 B neutrino production cite in the vicinity of GRB. In Fig. 2, we plot P(ν¯ ν ), given by Eq. µ τ → (7), as a function of neutrino energy E (GeV) for different ∆f as well as δm2 values. Note that the resonant spin-flavour precession probability is > 1/2 for a relatively large range of δm2 (and ∆f values). The expected spectrum F(ν +ν¯ ) of the high energy tau neutrinos τ τ originating from GRBs due to spin-flavour conversions is calculated as [15] 1From above discussion, it follows that E dependent/independent spin-flavour precession may also be obtained for nonzero ∆f, however, given the current status of the high energy neutrino detection, for simplicity, we ignore these possibilities which tend to overlap with this case for a certain range of relevant neutrino mixing parameters; for details of these possibilities in the context of AGN, see [5]. 5 0 0 F(ν +ν¯ ) = [1 P(ν¯ ν )]F (ν +ν¯ )+P(ν¯ ν )F (ν +ν¯ ), (8) τ τ µ τ τ τ µ τ µ µ − → → where F0’s are the neutrino flux spectrums originating from GRBs. In Fig. 3, we plot the expectedν spectrumobtainedbyneutrinospin-flavourprecessions andconversions(induced τ by an interplay of VEP and the magnetic field in GRBs). We use the GRB spectrum for (ν +ν¯ ), i.e., F0(ν +ν¯ ) from [1] and F0(ν +ν¯ ) from [4] and multiply these by respective µ µ µ µ τ τ P(ν¯ ν ) given by Eq. (8) to obtain F(ν +ν¯ ) due to resonant spin-flavour precession µ τ τ τ → (lower curve in Fig. 2). The upper curve in Fig. 2 is obtained by multiplying P given by Eq. (4) for small δm2. Let us now consider briefly the effects of nonvanishing gravity mixing angle θ and non- G vanishing vacuum mixing angle θ for vanishing neutrino magnetic moment. If θ = 0 and G θ = 0, then taking the distance between a typical GRB and our galaxy as, L 1000 6 ∼ Mpc, the vacuum flavour oscillations may occur between ν and ν for δm2 10−3eV2 µ τ ∼ with maximal vacuum flavour mixing between ν and ν . These values of neutrino mixings µ τ have been suggested as a possible explanation of recent superkamiokande data concerning the deficit of atmospheric muon neutrinos [16]. The corresponding expression for flavour oscillations in vacuum is δm2 2 2 P(ν ν ) = sin 2θsin L . (9) µ τ → 4E ! Note that here the resulting P is 1/2 due to the fact that 4E/δm2 L. For θ = 0 G ∼ ≪ 6 and θ = 0, in the case of massless or degenerate neutrinos, the corresponding vacuum flavour oscillation analog for ν ν is obtained through θ θ and δm2 V . For e → τ → G 4E → G maximal θ , the sensitivity of ∆f may be estimated by equating the argument of second sin G factor equal to π/2 in the corresponding expression for P [14]. This implies ∆f 10−41 ∼ with φ(r) φ (r). Note that this value of ∆f is of the same order of magnitude as SC ≃ that expected for neutrinos originating from AGNs [14]. In contrast to neutrino spin-flavour precession effects given by Eq. (4)resulting inP > 1/2, the vacuum flavour oscillations give P 1/2, thus, allowing the possibility of isolating the mechanism of oscillation since the ∼ high energy neutrino telescopes may attempt to measure (ν +ν¯ )/(ν +ν¯ ). Further, with τ τ µ µ the improved information on either ∆f and/or µ, one may be able to distinguish between the situations of resonant and nonresonant spin-flavour precession induced by an interplay of VEP and µ in B . Concerning possibilities of resonant flavour conversion, a resonant GRB or/and nonresonant flavour conversion between ν and ν in the vicinity of a GRB is also e τ possible due to an interplay of vanishing/nonvanishing vacuum and gravity mixing angles [14]. For instance, with θ 0, a resonant flavour conversion between ν and ν may be e τ → obtained if sin22θ 0.25. Here the relevant level crossing may occur at r 0.1 pc with G ≫ ∼ δm2 10−6 eV2. ∼ 6 IV. SIGNATURES OF HIGH ENERGY ν IN NEUTRINO TELESCOPES τ Thekm2 scalehighenergyneutrinotelescopesmaybeabletoobtainfirstexamplesofhigh energy ν , through double bangs, originating from GRBs correlated in time and direction τ with corresponding gamma-ray burst [9]. The first bang occurs due to charged current interaction of high energy tau neutrinos near/inside the neutrino telescope producing the tau lepton and the second bang occurs due to hadronic decay of this tau lepton. Following [17], we present in Table I, the expected contained but separable double bang event rates for downgoing ν in 1 km2 size water Cˇerenkov high energy neutrino telescopes for illustrative τ purposes. To calculate the event rates, we use Martin Roberts Stirling (MRS 96 R ) parton 1 distributions [18] and present event rates in units of yr−1 sr−1. We have checked that other recent parton distributions give quite similar event rates and are therefore not depicted here. From Table I, we notice that the event rates for neutrino spin-flavour precession are up to 4 orders of magnitude higher than that for typical intrinsic (no oscillations) tau neutrino ∼ flux. The condition of containedness is obtained by requiring that the separation between the two bangs is less than the typical km size of the neutrino telescope. It is obtained by ∼ equating the range of tau neutrino induced tau leptons with the size of detector implying E<2 107 GeV. The condition of separableness is obtained by demanding the separation ∼ × between the two bangs is larger than the typical spread of the bangs such that the amplitude of the second bang is essentially 2 times the first bang. This leads to E>2 106 GeV [9]. ∼ × Thus, the two bangs may be separated by a µ-like track within these energy limits. The upgoing tau neutrinos at these energies may lead to a small pile up of upgoing µ-like events near (104 105) GeV with flat zenith angle dependence [10]. − The possibility of measuring the contained but separable double bang events may enable onetodistinguishbetweenthehighenergytauneutrinosandelectronand/ormuonneutrinos originating from cosmologically distant GRBs while providing useful information about the relevant energy interval at the same time. The chance of having double bang events induced by electron and/or muon neutrinos is negligibly small for the relevant energies [9]. V. RESULTS AND DISCUSSION We have studied in some detail the effects of neutrino spin-flavour conversions due to an interplay of the effect of possible VEP and the magnetic field in GRBs and have obtained the relevant range of neutrino mixing parameters for appreciable neutrino conversion prob- abilities. We have also briefly commented on the corresponding range of neutrino mixing parameters for vanishing neutrino magnetic moment. The matter density in the vicinity of GRB is quite small (upto 4-5 orders of magnitude) 7 toinduceanyresonantflavour/spin-flavour neutrino conversion duetonormalmattereffects. We have pointed out that a resonant character in the neutrino spin-flavour conversions may nevertheless be obtained due to possible VEP. The corresponding degree of VEP may be (10−35 10−25) depending on δm2 value. ∼ − The double bang event rate for intrinsic (no oscillations/conversions) high energy tau neutrinos originating from GRBs turns out to be small as compared to that due to preces- sion/conversion effects up to a factor of 10−4. Thus, the high energy neutrino telescopes ∼ may provide useful upper bounds on intrinsic properties of neutrinos such as mass, mixing and magnetic moment, etc.. The relevant tau neutrino energy range for detection in 1 km2 neutrino telescopes may be 2 106<E/GeV<2 107 through characteristic contained but ∼ ∼ × × separable double bang events. Observationally, the high energy ν burst from a GRB may possibly be correlated to the τ corresponding gamma-ray burst/highest energy cosmic rays (if both have common origin) in time and in direction thus raising the possibility of its detection even if there is rela- tively large background flux of high energy tau neutrinos from AGNs also from oscillations. If the range of neutrino mixing parameters pointed out in this study is realized terres- tially/extraterrestially then a relatively large ν flux from GRBs is expected as compared τ to no oscillation/conversion scenario. Since the high energy neutrino telescopes may measure the ratio of the sum of tau and antitau neutrinos and the muon and antimuon neutrinos, so in principle, any change in the expected relatively small (upto 4 orders of magnitude) ratio may be attributed to spin- flavour precession/conversion effects as the ν ν channel may alternatively be a possible µ s → explanation of recent superkamiokande results concerning the deficit of atmospheric muon neutrinos. The flavour/spin-flavour conversions may occur possibly through several mechanisms. We have discussed in some detail mainly the spin-flavour precession/conversion situation induced by a nonzero neutrino magnetic moment and by a relatively small violation of equivalence principle as an example to point out the possibility of obtaining higher tau neutrino fluxes as compared to no oscillations/conversions scenarios from gamma-ray burst fireballs. Acknowledgments. TheauthorthanksAlexeiSmirnovandEnriqueZasforusefuldiscus- sionandAgencia Espan˜ola deCooperaci´onInternacional(AECI), Xunta de Galicia(XUGA- 20602B98) and CICYT (AEN96-1773) for financial support. [1] E. Waxman and J. Bahcall, Phys. Rev. Lett. 78, 2292 (1997); Phys. Rev. D 59, 023002 (1999). 8 [2] For a recent review, see, for instance, T. Piran in Unsolved problems in astrophysics, eds. J. Bahcall and P. Ostriker, Princeton University Press 1997, p. 343 and references cited therein. [3] B. Paczyn´ski and G. Xu, Ap. J 427, 708 (1994). [4] H. Athar, R. A. V´azquez, E. Zas (in preparation). [5] Fora recent discussion ofneutrino spin-flip inAGN,see, AtharHusain, hep-ph/9811221. [6] M. Gasperini, Phys. Rev. D 38, 2635 (1988); D 39, 3606 (1989); For an independent similar possibility of testing VEP by neutrinos, see, A. Halprin and C. N. Leung, Phys. Rev. Lett. 67, 1833 (1991). [7] See, for instance, L. Moscoso, in Fifth School on Non-accelerator Particle Astrophysics, June/July 1998, ICTP, Trieste, Italy (to appear in its proceedings) and F. Halzen, astro-ph/9810368. [8] T. J. Weiler et al., hep-ph/9411432; F. Halzen and G. Jaczko, Phys. Rev. D 54, 2779 (1996). [9] J. G. Learned and S. Pakvasa, Astropart. Phys. 3, 267 (1995). [10] F. Halzen and D. Saltzberg, Phys. Rev. Lett. 81, 4305 (1998). [11] See, for instance, P. M´esz´aros, P. Laguna and M. J. Rees, Ap. J 415, 181 (1993). [12] E. Kh. Akhmedov, S. T. Petcov and A. Yu. Smirnov, Phys. Rev. D 48, 2167 (1993). [13] G. G. Raffelt, Phys. Rev. Lett. 64, 2856 (1990). [14] For details in the context of AGNs, see, H. Minakata and A. Yu. Smirnov, Phys. Rev. D 54, 3698 (1996). [15] G. Raffelt, in Summer School in High Energy Physics and Cosmology, June/July 1998, ICTP, Trieste, Italy (to appear in its proceedings). [16] Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998); Phys. Lett. B 433, 9 (1998); 436, 33 (1998). [17] R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Astropart. Phys. 5, 81 (1996); Phys. Rev. D 58, 093009 (1998); H. Athar, G. Parente and E. Zas (to be submitted). [18] A. D. Martins, R. G. Roberts and W. J. Stirling, Phys. Lett. B 387, 419 (1996). 9 TABLE I. Event rate (yr−1sr−1) for high energy tau neutrino induced contained but separable double bangs in various energy bins using MRS 96 parton distributions. Energy Interval Rate (yr−1sr−1) no osc spin-flavour precession vac osc 2 106∼<E/GeV∼<5 106 10−5 1 10−1 0.5 10−1 × × × × 5 106∼<E/GeV∼<7 106 2 10−6 2 10−2 10−2 × × × × 7 106∼<E/GeV∼<1 107 2 10−6 2 10−2 10−2 × × × × 1 107∼<E/GeV∼<2 107 2 10−6 2 10−2 10−2 × × × × 10