Exact formula of 3 flavor ν oscillation probability and its application to high nrgy astrophysical ν Osamu Yasuda Tokyo Mtropolitan nivrsity 1-16 16-5 at Miami5
Contnts 1. Introduction 1.1 Status of ν oscillation study 1. Exact oscillation probability in mattr 1.3 High nrgy astrophysical ν. Anothr proof of Kimura-Takamura- Yokomakura formula & its gnralization 3. Flavor ratio of high nrgy astrophysical ν 4. Summary
1. Introduction 1.1 Status of ν oscillation study N ν =3 : ν atm +ν solar +ν ractor θ Mixing matrix 1 = μ1 τ1 1 1 3 / π 6, μ τ θ 3 θ ε 13.1 Δm 5 = 8 1 V Δm =.5 1 3 μ3 τ3 / 3 π 4 / V c1 s1/ s / 1 s1 c1/ c / Mixing angls & mass squard diffrncs 1 1 1 ε / / n θ 13 :only uppr bound is known n δ :undtrmind n Both hirarchis ar allowd 3 1 1 3
1. Exact oscillation probability in mattr with constant dnsity 1 = μ1 τ1 μ τ 3 μ3 τ3 E p + j m j oscillation in vacuum mattr ffct Evaluation of is th hardst part
Kimura, Takamura, Yokomakura Bilinar quantity in mattr can b xprssd as linar in bilinar quantity in vacuum: simpl known functions nthir drivation is complicatd & confind only for N ν =3 in constant mattr Anothr proof & gnralization is givn hr nthr has bn no xampl which was drivd for th first tim using KTY a nw rsult using KTY is prsntd
1.3. High nrgy astrophysical ν Flux of high nrgy cosmic ν from Activ Galactic Nucli or Gamma Ray Burst tc. E j(e) V cm - sc -1 sr -1 1 5 1 4 1 3 1 1 1 1-1 1-1 -3 S/N ratio is xpctd to b larg du to littl background of atmosphric ν 1-4 1 9 1 1 1 11 1 1 1 13 1 14 1 15 1 16 1 17 1 18 1 19 1 1 1 1 1 3 1 4 1 5 Nutrino Enrgy (V)
n Prcis normalization of flux is not known Th ratio of diffrnt flavors is important quantity to obsrv n Initial flux: Just lik in ν atm, th sourc of ν is π dcay ): 1:: ): F (ν F (ν F μ (ν τ n Obsrvd flux on Earth: Du to ν oscillations θ 13 <<1, π/4-θ 3 <<1 ) π + μ + + π μ + F(ν ): 1:1:1 F(ν μ ν μ + + ν + ν μ ν μ + ν + ν μ ): F(ν τ )
A fw scnarios to prdict dviation from 1:1:1 hav bn proposd Standard flux + ν dcay α:1:1 (α=1.4~6) Standard flux + psudo-dirac ν α:1:1 (α=/3~14/9) Elctromagntic nrgy losss of π & μ α:1:1 (α=1/1.8~1) Bacom-Bll-Hoopr- Pakvasa-Wilr 3 Bacom -Bll- Hoopr-Larnd- Pakvasa-Wilr 4 Kashti-Waxman 5 Hr I will considr th possibility of standard flux + ν magntic transitions + magntic fild
. Anothr proof of Kimura-Takamura Takamura- Yokomakura formula : Mixing matrix in vacuum : Mixing matrix in mattr E p + j m j Basic stratgy is to tak α,β componnt of both sids and to us th following idntity:
W gt a linar quation for W can solv th quation and rproduc KTY s rsult:
n It can b gnralizd to th cas with adiabatically varying mass matrix in L= limit: Only in th limit, KTY mthod works
n Gnralization of Kimura- Takamura-Yokomakura s rsult for th cas with magntic transitions & magntic fild μ αβ : magntic transitions In principl can b obtaind from this
3. Flavor ratio of high nrgy astrophysical ν In standard N ν =3, whn L oscillation probability in vacuum P αβ F(ν F(ν F(ν μ τ = j ) = F ) = F ) = F αj (ν (ν (ν )(P )(P )(P βj μ τ + P + P + P αj μ μμ μτ ) = F ) = F ) = F (ν c s s (ν (ν 1 1 )(1-P 1 / / c c )(1-P )(1-P s 1 1 1 τ τμ ττ / / + P μ + P + P 1/ 1/ μτ ) = 1 μμ ) = 1 ) = 1 Larnd- Pakvasa 95 F(ν ) = F (ν )P + F (ν )P = F (ν )(P + P α α μ μα α μα ) P + P = (P + P ) + P = 1- P + P = α μα α μα μα τα μα 1
CHOOZ+ν atm : θ 13 <<1 ν atm : π/4 -θ 3 <<1 Dviation from 1:1:1 is small Athar- Jzabk-OY N ν =3 no osc Fτ Fμ μ F τ μ τ
For simplicity w considr th Majorana cas with CP invarianc Majorana ν: μ αα =, μ αβ = -μ βα =pur imaginary no mattr ffct: A= KM lik CP phas δ dcoupls ( θ 13 <<1) no CP phas from th chargd lpton sctor: β, γ = NB: β, γ (Majorana CP phass) don t appar in th q.
In this cas analysis of 6x6 matrix is rducd to that of 3x3: W considr th situation in which B(t=) B(t=L)= occurs adiabatically
Assuming adiabatic approximation ( B(t=) > B(t=L)=), oscillation probability in th limit L can b analytically xprssd : F(ν α ) = F (ν ) X ~ (X ~ - X αα μμ ττ j j j j ) F(ν X = μμ ττ X j j ) :F(ν ) :F(ν ) = μ τ F(ν ) = F(ν μ α: 1: 1 τ )
Th ratio of ν + ν can b analytically xprssd :
Th ffct of th magntic transitions bcoms largst whn μ μτ >> μ μ, μ τ, Δ E jk / B In this cas th ratio 3/ nfortunatly, for th abov condition to b satisfid, (ν nrgy) (magntic fild at production point) has to b unralistically larg: (E ν /1TV)( B /1G) >1 assuming μ μτ 1-1 μ B But if this condition is satisfid, thn nontrivial nrgy dpndnc of th ratio should b obsrvd
4. Summary Simpl proof and gnralization of Kimura- Takamura-Yokomakura formula is givn Gnralization to th cas with magntic transitions Gnralization to th cas with nonconstant adiabatically varying mattr ffct and/or magntic fild Flavor ratio of high nrgy astrophysical ν is xprssd analytically taking into account of N ν =3 mixing matrix undr crtain assumptions (Majorana ν & CP invarianc)
Furthr situations to b considrd in futur: Possibility of non-adiabatic transition Effct of phass from th chargd lpton sctor: β, γ