Neutrino Flavor Detection at Neutrino Telescopes and Its Uses

Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 In ternational Journal o f Mo dern Ph ysics A c  W orld Scientific Publishing Company NEUTRINO FLA V OR DETECTION A T NEUTRINO TELESCOPES AND ITS USES SANDIP P AKV ASA Dep artment of Physics and Astr onomy, Unive rsity of Hawaii Honolulu, HI 96822 1. In tro duction W e a ssume that sources of high energy astrophysical neutrinos exis t with detectable fluxes. The existence of ga mma ra y sources with energies of upto 100 s of T eV provides some evidence that this ass umption may be correct 1 , assuming tha t the gamma r ays are co ming from π 0 ’s. W e also need to assume that in the not too dis- tant future, lar ge enough detectors, well instrumented and with go o d ang ular and energy reso lution will b e op era ting (hope fully of several KM3 size) 2 , 3 , 4 . W e assume also that neutrino signals will b e seen, and furthermore that the detectors will have the ability to distinguish fla vors(at the momen t this is only as sured for H 2 O ˆ c de- tectors). If all these o ptimistic a ssumptions turn out to b e v alid, there ar e a num b er of uses that thes e detectors can b e put to 5 . I would lik e to discuss s ome o f them. 2. Astroph ysical neutrino fla v or conten t Since mo st o f the s ources are tenous a nd emit neutrinos fro m π / K decays via the π − µ − e chain, we exp ect that the ratio o f ν e to ν µ is 1:2 ; furthermore, es tima tes of ν τ pro duction even at v ery high energ ies yield very small fraction of ν τ s 6 . Hence, for practical purp o ses the flav or ratio pro duced in such ”conven tio nal” sources (either via p-p or γ -p collisions) is ν e : ν µ : ν τ = 1 : 2 : 0. The mixture of ν ′ s and ¯ ν ′ s is exp ected to be 1 :1 with some exceptions. There ar e als o sourc e s in which muons lose energ y by mag netic fields or other means(called damp ed muon sourc e s ); in this case the flavor mix b ecomes ν e : ν µ : ν τ = 0 : 1 : 0, this ca n be an ener g y dep endent effect, making the flav o r mix ener gy dependent 7 . There ma y b e sour ces, in which the dominant co mpo nent is from neutron decays, resembling a ” beta b eam” 8 , with the resultant mix being : ν e : ν µ : ν τ = 1 : 0 : 0 . If the density is high enough fo r pions to interact before decaying, then heavy fla vors dominate and the fla vor mix bec omes: ν e : ν µ : ν τ = 1 : 1 : 0. This is the cas e,for example in the so- called slow-jet sup e rnov as 9 . There ar e also the neutrinos emitted as a by pro duct in the GZK 10 reaction, prop erly called the BZ(Ber ezinsky-Zatse pin) 11 neutrinos. The reaction is of course 1 Nov ember 12 , 20 18 10:58 NEUTRINO FLA VOR DETECTION A T NEUTRINO TELESCOP ES AND ITS USES flav or1 2 Pakvasa p + γ → ∆ + → n + π + . In this case the flav or mix dep ends on the ene r gy 12 . Below ab out 100 PeV, it is a pure ”b eta b eam” with ν e : ν µ : ν τ = 1 : 0 : 0 ; and a bove 10 0 PeV it is conv ent iona l ν e : ν µ : ν τ = 1 : 2 : 0. 3. Effect of Oscillatio ns The current kno wledge of neutrino ma sses and mixing 13 can b e summarized a s follows 14 . The mixing matrix elements are given to a v er y go o d appr oximation b y the so -called tr i-bi-maximal matrix 15 . The b ound on the element | U e 3 | comes from the CHOOZ exp eriment 16 and is given by | U e 3 | < 0 . 17. The mass sp ectrum has t wo p ossibilities: no rmal o r inverted. The mass differences are given by | δ m 2 32 | ∼ 2 . 4 . 10 − 3 eV 2 (with the + sign corres p onding to normal hierarch y and - sig n to the inv er ted one) and δ m 2 21 ∼ +7 . 6 . 1 0 − 5 eV 2 . Since δ m 2 L/ 4 E for the distances to GRB’s and A GN’s (even for energies up to and b eyond PeV) is very larg e ( > 10 7 ) the oscillations have alwa ys av era ged o ut and the conv ersion(or surviv al) pr obability is given by P αβ = X i | U αi | 2 | U β i | 2 (1) Assuming no s ignificant matter effects en-route, it is ea sy to show that the tri-bi- maximal mixing matrix leads to a simple propa gation matrix P , whic h, for any v alue of the solar mixing a ngle, conv erts a flux ratio of ν e : ν µ : ν τ = 1 : 2 : 0 in to one of 1 : 1 : 1. Hence the flav or mix e x pec ted at arriv al is simply an equal mixture o f ν e , ν µ and ν τ as was obser ved long a go 6 , 17 . Similar ly , for the other cas es neutrino mixing modifies the flavor mixes a s: Initial After Mixing Damped Muo n (0:1:0) (4:7:7) Beta Beam (1:0:0) (5:2:2) Prompt (1:1:0) (14:11:1 1) If the universal fla vor mix is co nfirmed by future observ ations , our current knowl- edge of neutrino masse s a nd mixing is reinforced a nd conv entional wis dom ab out the b eam dump na ture of the pro duction proces s is confirmed as well. How ever, it would b e muc h mo r e exciting to find deviations from it, and lear n so mething new. How can this c ome ab out? I give b elow a shopping list of v ariety of wa ys in which this could co me to pa ss, a nd wha t ca n b e learne d in each case. 4. Discriminating Fla v ors W e define tw o ra tios to distinguish v ar ious flavor mixes as: f e (= e/ ( e + µ + τ ) and R (= µ/ ( e + τ ). Then we hav e the follo wing for the v arious p o ssible flavor mixes exp ected at ea rth fr o m v ario us source t yp es: Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 Neutrino Flavor Detection at Neutrino T elesco p es and Its Uses (Pap er’s Title) 3 Source Type f e R Pionic 0.33 0.5 Damped- m uon 0.22 0.64 Beta-Beam 0.55 0.2 9 Prompt 0.39 0.4 4 It has b een shown that R and/o r f e can b e determined up to an accura cy of ab out 0.08 or so in an ice-cub e type detector 18 . Hence, pionic , damped m uon and Beta-b eam t yp e os sources can be distinguished but proba bly not the prompt. There have b een many sugges tions that small dev iations from the canonical fla- vor mixes ca n b e us e d for a v ariety of purp o s es, such as deter mine s mall deviations from T ri-bi-maximal mixing(i.e. mea sure θ 13 and δ ), small mixing with sterile neu- trinos etc 19 . How ever, there are several r e asons why this is rather impractical. In addition to the limits on the precision with which f e and/or R can be measured, there a r e inherent uncertainties in the source flavor mixes themselves. F or exa mple, in the π /K ca s e the fla vor mix is not ex p ected to b e e x actly ν e : ν µ : ν τ = 1 : 2 : 0 but rather more like 1:1 .85:0 20 . The main reas on for this effect is the muon po lar- ization in the π − µ decay which gives rise to a ν µ of lower ener gy than ν e , with additional subtle effects in K- decays. Similarly , the fla vor mixes in the damped m uon case and Beta-b eam case ar e also not exp ected to give rise to the simple pur e flav or mixes . These e ffects com bined with the 8 % uncertaint y in the experimental determination o f the flav or mix re nders extremely difficult an y attempt to measure small effects in mixing 21 . 5. Deviation from Canonical Mix Let us consider ho w the conv entional mix may undergo ma jor deviations which are detectable. The p oss ibilit y that the ma ss differences betw een neutr ino mass eigenstates are zero in v acuum (a nd b ecome no n-zero only in the presence of matter) has be e n rais e d 22 . If this is true, then the final flav or mix should b e the same a s initial, namely: 1 : 2 : 0 . How e ver, very recently , ana lysis o f low energ y atmospher ic neutrino data by Sup er-Ka miok ande has r uled out a wide v ar iety of mo dels for such behavior 23 . Neutrino decay is a nother impor tant p oss ible w ay for the fla vor mix to devia te significantly from the democ r atic mix 24 . W e now know that neutrinos hav e non- zero ma sses and non-trivia l mixing, ba sed o n the evidence for neutrino mixing and oscillations from the data o n a tmo spheric, solar a nd r e a ctor neutr ino s. Once neutrinos have masses and mixing , then in general, the hea vier ne utr inos are e x pec ted to deca y into the lighter ones via flav or c hanging pro ce s ses 25 . The only remaining questio ns a re (a) whether the lifetimes are short enough to b e phe- nomenologica lly in teres ting (or are they to o long ?) and (b) wha t are the do minant decay mo des. Since we are interested in decay mo des which are lik ely to hav e rates (or lead to lifetimes) whic h ar e phenomenologic a lly interesting, we can rule out sev- eral classes of decay mo des immediately . F or example, the very strong constraints 25 Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 4 Pakvasa on radia tiv e decay mo des and on three b o dy modes such as ν → 3 ν render them unin teres ting. The only deca y mo de s which can ha ve interestingly fast decay rates are t wo bo dy mo des such as ν i → ν j + x w he r e x is a v ery ligh t or ma ssless particle, e.g. a Ma joro n. In general, the Ma joron is a mixture o f the Gelmini- Ro ncadelli 26 and Chik asige- Mohapatra- Peccei 27 t yp e Ma jorons. E xplicit models of this kind which can give rise to fast neutrino decays hav e b een discussed 28 . T he mo dels with ∆ L = 2 are unconstrained by µ and τ deca ys which cannot be eng endered by such couplings. Both(∆ L = 2 and ∆ L = 0 ) kinds of mo dels with couplings of ν µ and ν e are co nstrained by the limits o n multi-bo dy π , K decays, and on µ − e univ ersa lit y violation in π and K decays 29 , but thes e bo unds allow fast neutrino decays. Direct limits on such decay mo des are rather weak. Current b ounds on suc h decay mo des are as fo llows. F or the mass eigenstate ν 1 , the limit is a bo ut τ 1 ≥ 10 5 sec/eV (2) based on observ a tion o f ¯ ν e s from SN1987A 30 (assuming CPT in v a r iance). F or ν 2 , strong limits can b e deduced fro m the non-observ a tio n of solar an ti-neutrinos in KamLAND 31 . A more general but similar b o und is obta ine d from a n analys is of solar neutrino da ta 32 . This b ound is given by: τ 2 ≥ 10 − 4 sec/eV (3) F o r ν 3 , one can derive a bound from the a tmospheric neutrino observ atio ns o f up- coming neutrinos 33 : τ 3 ≥ 10 − 10 sec/eV (4) The s trongest lifetime limit is th us to o w eak to e limina te the p ossibility of as- trophysical neutrino decay by a factor abo ut 10 7 × ( L/ 100 Mp c) × (10 T eV/E). It was noted that the dis app earance o f all states except ν 1 would pr epare a b eam that could in principle b e used to measure elements of the neutrino mixing matrix 34 , namely the ratios | U e 1 | 2 : | U µ 1 | 2 : | U τ 1 | 2 . The p ossibility o f measuring neutrino lifetimes over long baselines was mentioned in Ref. 35 , a nd some predictions for de- cay in four- neutrino mo dels were given in Ref. 37 . The particular v alues and small uncertainties on the neutrino mixing parameters allow for the first time very dis- tinctiv e signatur es o f the effects o f neutrino deca y on the detected flavor ratios. The exp ected increa se in neutrino lifetime sensitivity (and corresp onding ano ma- lous neutrino co uplings) by s everal order s of magnitude makes for a very interesting test of physics b eyond the Standard Mo del; a dis cov ery would mea n ph ysics muc h more exo tic than neutrino ma ss and mixing alo ne. Because of its unique signature, neutrino decay ca nno t b e mimic ked by either differen t neutrino flavor r atios a t the source or o ther non-standard neutr ino interactions. A c hara cteristic feature of decay is its strong ener gy dependence: exp( − Lm/ E τ ), where τ is the r est-frame lifetime. F or simplicity , we will consider the ca se that decays are a lwa ys complete, i.e., tha t these exp o nent ial fa ctors v anish. The simplest Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 Neutrino Flavor Detection at Neutrino T elesco p es and Its Uses (Pap er’s Title) 5 case (and the most g eneric ex pec ta tion) is a normal hiera rch y in which b oth ν 3 and ν 2 decay , leaving only the lig h test stable eigenstate ν 1 . In this case the flav or r atio is 34 | U e 1 | 2 : | U µ 1 | 2 : | U τ 1 | 2 . Th us, if | U e 3 | = 0 we hav e φ ν e : φ ν µ : φ ν τ ≃ 4 : 1 : 1 , (5) where we used the propagation matrix derived from the tri- bi-maximal mixing. Note that this is an extreme deviation of the flav or ratio fro m that in the absence of decays. It is difficult to imagine other mechanisms tha t would lead to such a high ratio of ν e to ν µ . In the case of inv erted hier arch y , ν 3 is the lightest and hence s table state, and so 24 we hav e instea d φ ν e : φ ν µ : φ ν τ = | U e 3 | 2 : | U µ 3 | 2 : | U τ 3 | 2 = 0 : 1 : 1 . (6) If | U e 3 | = 0 and θ atm = 45 0 , each mass eig enstate ha s equal ν µ and ν τ comp onents. Therefore, dec ay cannot break the equality betw een the φ ν µ and φ ν τ fluxes a nd thus the φ ν e : φ ν µ ratio contains all the useful information. When | U e 3 | is not zero, and the hierarch y is no rmal, it is poss ible to obtain information on the v alues of | U e 3 | as w ell as the CPV pha se δ 36 . The fla vor ratio e/µ v aries fro m 4 to 10 (as | U e 3 | g o es from 0 to 0.2) for cos δ = +1 but from 4 to 2.5 fo r cos δ = − 1 . The ratio τ /µ v ar ies from 1 to 4 (cos δ = +1) or 1 to 0.25 (cos δ = − 1 ) for the same range of U e 3 . If the decays are not complete a nd if the daughter do es not ca rry the full energy of the par ent neutrino; the r esulting flav or mix is somew ha t different but in an y case it is still quite distinct from the simple 1 : 1 : 1 mix 24 . There is a very re c en t exhaustive study of the v arious po ssibilities 38 . If the path of neutrinos tak es them thru reg io ns with significant magnetic fields and the neutrino magnetic moments are larg e enough, the flavor mix can be affected 39 . The main effect of the passage thru magnetic field is the conversion of a given helicity into a n equal mixture of b oth helicity states. This is also true in passage thru ra ndom magnetic fields 40 . It ha s b een shown recently that the pres- ence of a mag ne tic field of a few(10 or more) Gauss at the s ource can mak e the neutrinos decohere as they traverse cosmic distances 41 . If the neutr inos are Dirac particles, and a ll magnetic momen ts ar e co mparable, then the e ffect of the spin-flip is to simply reduce the o verall flux of all flav or s by ha lf, the other half b ecoming the s terile Dirac pa r tners. If the neutrinos ar e Ma jorana particles, the fla vor comp osition r emains 1 : 1 : 1 when it starts from 1 : 1 : 1 , a nd the abso lute flux remains unchanged. Other neutrino prop erties ca n also affect the neutrino flav or mix and modify it from the canonical 1 : 1 : 1 . If neutrino s ha ve fla vor(and equiv alence principle) violating couplings to gravit y(FVG) ; then there can be r e s onance effects which make for o ne wa y transitions(ana lo gues of MSW trans itions) e.g. ν µ → ν τ but not vice versa 42 , 43 . In ca se of FV G fo r example, this can g ive rise to a n anisotr o pic deviation of the ν µ /ν τ ratio from 1, b ecoming less than 1 for even ts coming from the direction towards the Grea t Attractor, while re ma ining 1 in other directions 42 . Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 6 Pakvasa If such striking effects are not seen, then the current b ounds on such vio lations ca n be impr oved by six to seven or ders o f magnitude. Complete quantu m deco her ence w ould g ive rise to a flavor mix given b y 1 : 1 : 1, which is identical to the ca s e of av erag ed o ut oscilla tions as we sa w a bove. The distinction is that co mplete decohere nce alwa ys lea ds to this result; w her eas av eraged out oscillations lead to this r esult only in the s pecia l case of the initial flav or mix b eing 1 : 2 : 0 . T o find evidence for decoherence, therefore, requires a source which ha s a different flavor mix . O ne p os sible prac tical exa mple is the “b eta” b eam source with a n initial flavor mix of 1 : 0 : 0 . In this case decoher e nce leads to the universal 1 : 1 : 1 mix where a s the averaged out o scillations lead to 2 . 5 : 1 : 1 44 . The tw o cases can be ea s ily distinguished from each other. Violations o f Lorentz in v ariance and/or CP T inv aria nce can change the final flav or mix from the canonical universal mix of 1 : 1 : 1 significantly . With a sp ecific choice o f the change in dispers ion rela tion due to Lorentz Inv ariance Violation, the effects can be dr amatic. F or example, the final flav or mix at sufficien tly high energies can become 7 : 2 : 0 44 . If e ach of the three neutrino mass eigensta tes is actually a doublet with very small mass differenc e (smaller than 10 − 6 eV ), then there a re no current exp eriments that co uld hav e detected this. Such a p os sibilit y w as r aised long ago 45 . It turns out that the only wa y to detect such small mass differences (1 0 − 12 eV 2 > δ m 2 > 10 − 18 eV 2 ) is b y measuring flav or mixes of the high energy neutrinos from cosmic sources. Relic super nov a neutrino signals and A GN neutrinos are se ns itiv e to mass difference squared down to 10 − 20 eV 2 46 . Let ( ν + 1 , ν + 2 , ν + 3 ; ν − 1 , ν − 2 , ν − 3 ) denote the six ma ss eigenstates where ν + and ν − are a nearly degenerate pair. A 6 x6 mixing ma trix rotates the mass basis into the flav or basis. F or pseudo- Dirac neutrinos, Kobay ashi and Lim 47 hav e giv en an elegant pro of that the 6x6 matrix V K L takes the very simple form (to lowest o rder in δ m 2 /m 2 : V K L =  U 0 0 U R  ·  V 1 iV 1 V 2 − iV 2  , (7) where the 3 × 3 matrix U is just the usual mixing (MNSP)matrix determined by the atmos pher ic and solar o bserv a tions, the 3 × 3 matrix U R is a n unknown uni- tary matrix and V 1 and V 2 are the diagonal matrices V 1 = diag (1 , 1 , 1) / √ 2, and V 2 =diag( e − iφ 1 , e − iφ 2 , e − iφ 3 ) / √ 2, with the φ i being arbitrary phases. The flav or ratios deviate from 1 : 1 : 1 when one o r tw o of the pseudo -Dirac oscillation mo de s is accessible. In the ultimate limit where L/E is so la rge that all three oscillating factors hav e av era ged to 1 2 , the flavor ratios return to 1 : 1 : 1, with only a net suppres sion of the measurable flux, by a factor of 1 / 2. As a bo nu s, if such s ma ll pseudo-Dirac mass differences exist, it w ould enable us to measure cosmological parameter s such as the red shift in neutr inos(rather than in photons) 35 , 46 . Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 Neutrino Flavor Detection at Neutrino T elesco p es and Its Uses (Pap er’s Title) 7 6. Exp erimental Fl a vor Iden tification It is o b vio us fro m the ab ov e discussion that flavor identification is cr ucial for the purp ose at ha nd. In a water(or ice) cerenko v detector flav o rs can b e identified as follows. The ν µ flux can b e measured by the µ ′ s pr o duced by the charged curr ent inter- actions and the resulting µ tracks in the detector which are long a t these energies. ν ′ e s pro duce show e r s by both CC and NC in teractio ns. The total ra te for showers includes thos e pro duced by NC interactions of ν ′ µ s and ν ′ τ s as well and those hav e to be (and ca n b e) subtra cted off to get the rea l flux of ν ′ e s . Double-bang a nd lollip op even ts are signatures unique to ν ′ τ s , made p o ssible by the fact that tau lepto ns decay befo re they lose a significant fraction of their energy . A double-bang even t consis ts of a hadr onic shower initiated by a charged-c ur rent in tera ction of the ν τ follow ed by a second energetic shower from the decay of the r esulting τ 6 . A lollip op event consists of the second of the double-bang sho wers along with the re constructed tau lepton track (the fir st bang may b e detected or not). In principle, with a sufficient num b er of even ts, a fair ly go o d estimate of the flav o r r atio ν e : ν µ : ν τ can b e re constructed (with cav eats ab out uncertainties) as has bee n discus s ed recently . Deviations of the flav or ratios from 1 : 1 : 1 due to p ossible decays are s o extr e me that they s hould b e readily ident ifiable 18 . High energy neutrino telescop es, such as Icec ube 48 , will not hav e p erfect ability to sepa rately measure the neutrino flux in each flav or. How ever, the situation is salv agea ble. In the limit of ν µ − ν τ symmetry the fluxes for ν µ and ν τ are a lwa ys in the ratio 1 : 1 , with or without decay . This is useful since the ν τ flux is the har dest to mea sure. Even when the tau events are not all identifiable, the rela tive num b er of show er even ts to tra ck ev ents can b e re la ted to the most interesting qua nt ity for testing decay scenar ios, i.e., the ν e to ν µ ratio. The pr e cision of the upco ming exp eriments should b e go o d enough to test the extreme fla vor ratios pro duced by decays. If electromagne tic and ha dronic show ers can b e sepa rated, then the pr ecision will b e even b etter 18 .Comparing, for exa mple, the sta ndard flavor ratios of 1 : 1 : 1 to the p ossible 4 : 1 : 1 (or 0 : 1 : 1 for in verted hiera rch y)generated by decay , the higher(low er) elec tr on neutrino flux will result in a substa ntial increase(decr ease) in the relative num b er o f show e r even ts.The measurement will be limited o nly by the energy resolutio n of the detector and the ability to reduce the atmospheric neutrino background(which dro ps rapidly with energy and s hould be negligibly small a t and ab ov e the PeV scale). 7. Discussion and Conclus ions The flux r atios we discus s are energ y -indep e nden t to the extent that the following assumptions are v a lid: (a)the ra tios at pro duction ar e energ y- indep e nden t, (b) all oscillations a re averaged out, and (c) that a ll p oss ible decays are complete. In the standard sc e nario with only osc illa tions, the final flux r atios ar e φ ν e : φ ν µ : φ ν τ = 1 : 1 : 1. In the cases with decay , w e have found rather different p ossible flux ratios, for Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 8 Pakvasa example 4 : 1 : 1 in the normal hierarchy and 0 : 1 : 1 in the inv erted hierarch y . These deviations from 1 : 1 : 1 are so e xtreme that they should b e readily measura ble. If w e are very fortunate, w e may b e able to observe a reaso nable num b er of even ts fro m se veral sources (of known distance) and/o r over a sufficient range in energy . Then the res ulting depe ndence of the flux r atio ( ν e /ν µ ) on L/E as it evolves from say 4 (or 0) to 1 can b e clear evidence o f decay and fur ther ca n pin down the actual lifetime instead of just plac ing a b ound 49 . T o summarize, we sug g est that if future meas urements of the flav or mix a t e a rth of high ene r gy as trophysical neutrinos find it to b e φ ν e /φ ν µ /φ ν τ = α/ 1 / 1; (8) then (i) α ≈ 1 (the most bor ing case) c o nfirms o ur kno wledge of the MNSP 13 matrix and o ur prejudice ab out the pro duction mechanism; (ii) α ≈ 1 / 2 indicates that the s ource emits pure ν ′ µ s and the mixing is conven- tional; (iii) α ≈ 3 from a unique direction, e.g. the Cygnus reg ion, would b e evidence in fa vor of a pure ¯ ν e pro duction as ha s b een sug g ested recently 8 ; (iv) α > 1 indicates that neutrinos are de c aying with normal hierar ch y ; and (v) α ≪ 1 would mean that neutrino decays are o ccurring with inv erted hier- arch y; (vi) V a lues of α which cov er a broader range and deviation of the µ/τ ratio from 1 can yield v aluable information ab out U e 3 and co s δ . Deviations of α which are less ex tr eme(betw een 0 .7 a nd 1.5) can also prob e very small pseudo-Dirac δ m 2 (smaller than 10 − 12 eV 2 ). Incident ally , in the la s t three cases , the results have absolutely no dep endence on the initial flav or mix, and so a re co mpletely free of any dep endence on the pr o duction mo del. So either one le arns ab out the pro duction mechanism and the initial flavor mix, as in the first three cases , o r one lear ns only ab out the neutrino prop er ties , as in the last three cases. T o summarize, the measurement of neutrino flav o r mix at neutrino telescop es is absolutely essential to uncov er new and interesting physics of neutrinos. In any case, it should b e evident that the construction of very la rge neutrino telescop es is a “no lose” prop osition. 8. Ac kno wledgm e n ts This talk is bas ed on work done in colla bo ration with J o hn Bea com, Nicole Bell, Dan Ho op er, John Lear ned, W er ner Ro dejohann and T om W eiler . I thank them all for a most enjoy able collab ora tion. I would like to thank the or ganizers of CTP 2 009 for the opp ortunity to pr esent this talk as w ell a s their ho spitality and for providing a most stim ulating atmosphere during the meeting. This work was suppo rted in part b y U.S.D.O.E. under grant DE-FG02-04ER41 291. Nov ember 12, 20 18 10:58 NE UTRINO FLA V OR DETECTION A T NEU- TRINO TELESCOP ES AND ITS USES flav or1 Neutrino Flavor Detection at Neutrino T elesco p es and Its Uses (Pap er’s Title) 9 References 1. The H ES S Collaboration, F. Aharonian et al., astro-ph /06118 13 . 2. Descriptions of severa l KM3 detectors can be found at h ttp ://icecube.wisc.edu/ and at http://ww w.km3net.org/home.php 3. The Auger detector is described at http://www .auger.org/ ; for the planned JEM- EUSO detector see http://jemeuso.rik en.jp/ or http://aquila.lbl.EUSO/. 4. Examples are A N IT A:hep-ph/0503304, and SALSA :astro-ph/0412128. 5. S. 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