Neutrinos: Three-Flavor Effects in Sparse and Dense Matter

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1 Neutrinos: Three-Flavor Effects in Sparse and Dense Matter Tommy Ohlsson Royal Institute of Technology (KTH) & Royal Swedish Academy of Sciences (KVA) Stockholm, Sweden Neutrinos and the Early Universe, ECT*, Trento, Italy October 5, 2004 Tommy Ohlsson - ECT*, October 4-8, 2004 p. 1/29

2 Outline Introduction Three-flavor and matter effects Solar day-night effect (sparse matter) Effective neutrino mixing and oscillations in dense matter Summary & Conclusions In collaboration with: Mattias Blennow and Håkan Snellman as well as Evgeny Akhmedov, Robert Johansson, Manfred Lindner, and Thomas Schwetz Based on: hep-ph/ and hep-ph/ as well as hep-ph/ Tommy Ohlsson - ECT*, October 4-8, 2004 p. 2/29

3 Introduction Neutrinos were introduced in 1930 by W. Pauli, n p + e + ν e. The first neutrino was detected in 1956 by C.L. Cowan and F. Reines (the electron antineutrino). Neutrinos come in three flavors (ν e, ν µ, and ν τ ). Initially, neutrinos were believed to be massless. Neutrino oscillations were first considered in 1957 by B. Pontecorvo. Neutrino oscillations indicate non-zero neutrino masses (and vice versa). Tommy Ohlsson - ECT*, October 4-8, 2004 p. 3/29

4 Two-flavor neutrino oscillations Two flavors: 2 parameters Three flavors: 6 parameters The two-flavor formula (probability for the transition ν α ν β ): P αβ (L/E) = δ αβ (2δ αβ 1) sin 2 2θ sin 2 m2 L, α, β = e, µ, τ. 4E sin 2 2θ: oscillation amplitude m 2 : oscillation frequency L baseline length E neutrino energy For example, if α = β = e or α = e, β = µ, then P ee = 1 sin 2 2θ sin 2 m2 L 4E, P eµ = sin 2 2θ sin 2 m2 L 4E. Note! P ee = 1 P eµ = 1 P µe = P µµ Solar (or reactor) neutrinos: θ := θ 12 and m 2 := m Atmospheric neutrinos: θ := θ 23 and m 2 := m But: Three-flavor effects 10 % Today: Three flavors a must! Tommy Ohlsson - ECT*, October 4-8, 2004 p. 4/29

5 General definitions The leptonic mixing matrix: U = O 23 U δ O 13 U δ O 12 c 12 c 13 s 12 c 13 s 13 e iδ CP = s 12 c 23 c 12 s 13 s 23 e iδ CP c 12 c 23 s 12 s 13 s 23 e iδ CP c 13 s 23, s 12 s 23 c 12 s 13 c 23 e iδ CP c 12 s 23 s 12 s 13 c 23 e iδ CP c 13 c 23 where s ij = sin θ ij and c ij = cos θ ij. Here θ ij is the mixing angle in the ij-plane and δ CP is the CP-violating phase. 3 mixing angles and 1 CP-violating phase. The mass squared differences: m 2 ij = m 2 i m 2 j 3 neutrino flavors: 2 mass squared differences, since m m m2 31 = 0. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 5/29

6 Phenomenological facts Parameter Best-fit value Range (3σ) m ev 2 ( ) 10 5 ev 2 m ev 2 ( ) 10 3 ev 2 θ θ θ δ CP - 0 2π (M. Maltoni et al., hep-ph/ ; hep-ph/ ) Bilarge leptonic mixing, i.e., θ 12 and θ 23 are large and θ 13 is small. Note! Leptons (MNS): 2 large mixing angles and 1 small mixing angle Quarks (CKM): 3 small mixing angles Tommy Ohlsson - ECT*, October 4-8, 2004 p. 6/29

7 Three-flavor effects, matter effects, and notation If any of the elements U αa of the leptonic mixing matrix U or any of the mass squared differences equals zero (e.g. θ 13 0 or α 0), then one has effective two-flavor scenarios. Three-flavor effects: P 3fe αβ P 3ν αβ P 2ν αβ If V 0, then one has vacuum scenarios. Matter effects: Sparse matter : Dense matter : P me αβ P matter αβ 2EV/ m 2 1 2EV/ m 2 1 P vacuum αβ Tommy Ohlsson - ECT*, October 4-8, 2004 p. 7/29

8 Effective two-flavor formulas Amplitudes and frequencies: P αβ sin 2 2θ m 2 θ 13 = 0 P ee sin 2 2θ 12 m 2 21 P eµ = P µe c 2 23 sin2 2θ 12 m 2 21 P eτ = P τe s 2 23 sin2 2θ 12 m 2 21 α = 0 P eµ s 2 23 sin2 2θ 13 m 2 31 = m2 32 P µτ c 4 13 sin2 2θ 23 m 2 31 = m2 32 θ 13 0: E.g. oscillations of anti-electron neutrinos from reactor experiments (KamLAND) α 0: E.g. oscillations of atmospheric neutrinos (Super-Kamiokande) Tommy Ohlsson - ECT*, October 4-8, 2004 p. 8/29

9 Neutrino evolution The Schrödinger equation for the neutrino vector of state in the flavor basis: with the effective Hamiltonian i d ν(t) = H ν(t) dt H 1 2E U diag(0, m2 21, m 2 31)U + diag(v, 0,0). The potential V (x) is given by ( ) ρ(x) V (x) g/cm 3 Y e (x) }{{} 1/2 ev, where ρ(x) is the matter density. Earth s crust: ρ crust 3 g/cm 3 Earth s mantle: ρ mantle 4.5 g/cm 3 Antineutrinos: U U and V V Tommy Ohlsson - ECT*, October 4-8, 2004 p. 9/29

10 Neutrino oscillation probabilities In general: 3 neutrino flavors 18 neutrino and antineutrino oscillation probabilities I. Antineutrinos: Pᾱ β = P αβ (δ CP δ CP, V V ) 9 neutrino oscillation probabilities II. Unitarity (conservation of probability): α P αβ = β P αβ = 1 5 of the 9 neutrino oscillation probabilities can be expressed in the other 4 III. µ τ symmetry: Pαβ P αβ (s 2 23 c2 23, sin2θ 23 sin 2θ 23 ) P eτ = P eµ, P τµ = P µτ, P ττ = P µµ Out of these 3 conditions, only 2 are independent (unitarity) 2 independent neutrino oscillation probabilities We will use P eµ and P µτ! (E. Akhmedov, R. Johansson, M. Lindner, T. Ohlsson, and T. Schwetz, hep-ph/ ) Useful abbreviations: m2 31 L 4E, A 2EV m 2 31 = V L 2. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 10/29

11 Series expansions of neutrino oscillation probabilities Neutrino oscillation probabilities [P αβ P(ν α ν β )]: P αβ = P αβ ( m 2 21, m 2 31, θ 12, θ 13, θ 23, δ CP ;E, L, V (x)), α, β = e, µ, τ, where E is the neutrino energy, L is the baseline length, and V (x) is the matter induced effective potential, x [0, L]. Different types of expansions: α expansion: Single series expansion up to first order in α s 13 expansion: Single series expansion up to first order in s 13 Double expansion: Series expansion up to second order in both α and s 13 Expansion parameters: α m2 21 m and θ s Tommy Ohlsson - ECT*, October 4-8, 2004 p. 11/29

12 Double expansion Neutrino oscillation probabilities in constant matter density: P ee = 1 α 2 sin 2 2θ 12 sin 2 A A 2 4 s 2 13 sin 2 (A 1) (A 1) 2, P eµ = α 2 sin 2 2θ 12 c 2 23 sin 2 A A s 2 13 s 2 sin 2 (A 1) 23 (A 1) α s 13 sin 2θ 12 sin2θ 23 cos( δ CP ) sin A A sin(a 1) A 1, P µτ = sin 2 2θ 23 sin 2 α c 2 12 sin 2 2θ 23 sin 2 + α 2 c 4 12 sin 2 2θ 23 2 cos 2 1 ( 2A α2 sin 2 2θ 12 sin 2 sin A 2θ 23 sin cos(a 1) ) A 2 sin ( sin(a 1) A 1 s2 13 sin2 2θ 23 sin cos A A ) A 1 2 sin α s 13 sin 2θ 12 sin2θ 23 sin δ CP sin sin A 2 A 1 α s 13 sin 2θ 12 sin2θ 23 cos 2θ 23 cos δ CP sin A sin(a 1) A 1 ( Asin sin A A ) cos(a 1). Tommy Ohlsson - ECT*, October 4-8, 2004 p. 12/29

13 Double expansion II In vacuum (i.e., A 0): P vac ee = 1 α 2 sin 2 2θ s 2 13 sin2, P vac eµ = α 2 sin 2 2θ 12 c s 2 13 s 2 23 sin α s 13 sin2θ 12 sin2θ 23 cos( δ CP ) sin, Pµτ vac = sin 2 2θ 23 sin 2 α c 2 12 sin 2 2θ 23 sin2 + α 2 sin 2 2θ 23 (c cos 2 1 ) 2 sin2 2θ 12 sin 2 2 s 2 13 sin 2 2θ 23 sin α s 13 sin2θ 12 sin2θ 23 (sin δ CP sin cos 2θ 23 cos δ CP cos ) sin. Other works: A. Cervera et al., hep-ph/ M. Freund, hep-ph/ M. Freund, P. Huber, and M. Lindner, hep-ph/ V. Barger, D. Marfatia, and K. Whisnant, hep-ph/ Tommy Ohlsson - ECT*, October 4-8, 2004 p. 13/29

Solar day-night effect SUN Neutrino production N EARTH D Day at detector Night at detector D day rate of solar neutrinos N night rate of solar neutrinos The Sun in

14 Solar day-night effect SUN Neutrino production N EARTH D Day at detector Night at detector D day rate of solar neutrinos N night rate of solar neutrinos The Sun in neutrino light. The Sun seen in ES detection of neutrinos. Is it brighter at night than at day? Are the rates D and N different? Tommy Ohlsson - ECT*, October 4-8, 2004 p. 14/29

15 Solar day-night effect II Flux of solar ν e depleted due to neutrino oscillations. What is the influence of the Earth on neutrino oscillations? The day-night asymmetry: A n d = 2 N D N + D c2 13, where N and D are night and day neutrino rates, respectively. The asymmetry is a measure of the effect. We study this effect in a three neutrino flavor framework. In the future: Determination of the mixing angle θ 13 as an alternative to super-beams, future reactor experiments, and neutrino factories. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 15/29

16 Analytical approach Approximation: Earth is a sphere of constant density. For the neutrino energies used, the third mass eigenstate is hardly affected by the presence of matter in the Sun and Earth. The difference between the night and day survival probabilities is: (P D c 4 13 ) P N P D 2c 6 13D 3ν EV E m 2 sin2 (2θ 12 ) sin 2 ( ) m 2 4E L c 6 13, (M. Blennow, T. Ohlsson, and H. Snellman, hep-ph/ ) where V E is the effective ν e potential arising from interactions with matter and D 3ν = R 0 dr f(r) cos[2ˆθ 12 (r)]. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 16/29

17 Numerical calculation (Super-Kamiokande) The day-night asymmetry numerically calculated with our analytical approach (for ES detection at the Super-Kamiokande experiment): θ 13 = 0 θ 13 = 9.2 o θ 13 = 12 o 0 log( m 2 /ev 2 ) % CL 95% CL 99% CL 99.73% CL sin 2 θ 12 (M. Blennow, T. Ohlsson, and H. Snellman, hep-ph/ ) Tommy Ohlsson - ECT*, October 4-8, 2004 p. 17/29

18 Numerical calculation (SNO) The day-night asymmetry numerically calculated with our analytical approach (for CC detection at the SNO experiment): θ 13 = 0 θ 13 = 9.2 o θ 13 = 12 o log( m 2 /ev 2 ) % CL 95% CL 99% CL 99.73% CL sin 2 θ 12 (M. Blennow, T. Ohlsson, and H. Snellman, hep-ph/ ) Tommy Ohlsson - ECT*, October 4-8, 2004 p. 18/29

19 Another analytical approach An approximate analytic expression for the day-night effect valid for an arbitrary matter density profile of the Earth. The neutrino potential in the Earth V is small (V/2δ 0.05). It can be considered as a perturbation. Use the technique from hep-ph/ The night-day probability difference is: P N P D = (Pe2 P e1 )(P 2e P (0) 2e ) = c2 13 cos 2ˆθ 12 (P2e P (0) 2e ), P 2e P (0) 2e = c4 13 sin 2 2θ L Constant matter density (i.e., V = const.) 0 dx V (x) sin[2δ(l x)]. P N P D = c 6 13 cos 2ˆθ 12 sin 2 2θ 12 V δ 2ω 2 sin2 ωl. (E. Akhmedov, M. Tórtola, and J. Valle, hep-ph/ ) Tommy Ohlsson - ECT*, October 4-8, 2004 p. 19/29

20 What can one learn about θ 13 from A n d? SK 1σ region σ = 25% σ SK σ = 10% σ SK 0.04 σ = 25% σ SK σ = 10% σ SK A ND 0.02 A ND cos 2 θ cos 2 θ cos 2 θ cos 2 θ 13 (E. Akhmedov, M. Tórtola, and J. Valle, hep-ph/ ) At present: Experimental errors too large In the future: UNO and Hyper-Kamiokande Constant matter density: M. Blennow, T. Ohlsson, and H. Snellman Arbitrary matter density: E. Akhmedov, M. Tórtola, and J. Valle Tommy Ohlsson - ECT*, October 4-8, 2004 p. 20/29

21 Neutrino oscillations in sparse matter Two-flavor neutrino oscillations in low-density medium: V (x) m2 2E, Small parameter: i.e., potential kinetic energy ǫ(x) 2EV (x) m 2 1 perturbation theory in ǫ(x) Useful for solar and supernova neutrinos. (A. Ioannisian and A. Smirnov, hep-ph/ ; A. Ioannisian, N. Kazarian, A. Smirnov, and D. Wyler, hep-ph/ ) (Cf. talk by A. Smirnov) Neutrino oscillations in dense matter: 2EV m 2 1 Tommy Ohlsson - ECT*, October 4-8, 2004 p. 21/29

22 Mixing and oscillations in dense matter The Hamiltonian in matter: V 0 0 H = m2 31 2E U 0 α 0 U }{{}}{{} kinetic term H k interaction term H m When 2EV m 2 31 ( dense matter): H k perturbation to H m. H m has two degenerate eigenvectors degenerate perturbation theory degenerate sector of H m spanned by ν µ = (0, 1, 0) T and ν τ = (0, 0, 1) T H d = ( ) a b b d H = m2 4E ( cos 2θ sin2θ sin2θ cos 2θ ) Here a, b, and d depend on the fundamental neutrino parameters as well as E and V. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 22/29

23 Effective two-flavor neutrino parameters The effective mixing angle and mass squared difference in dense matter: sin 2 (2θ ) = 4 b 2 (a d) b 2 = 4 {s 23c 23 [c 2 13 α(c2 12 s2 12 s2 13 )] αs 12c 12 s 13 c δ cos 2θ 23 } 2 + α 2 s 2 δ s2 12 c2 12 s2 13 [c 2 13 α(c2 12 s2 13 s2 12 )]2 + 4αc 2, 12 s2 12 s2 13 m 2 = 2E (a d) b 2 = m 2 31 c αc2 13 (c2 12 s2 13 s2 12 ) + α2 (1 c 2 13 s2 12 )2 = m 2 31 [c 2 13 α(c2 12 s2 13 s2 12 )]2 + 4αc 2 12 s2 12 s2 13. (M. Blennow and T. Ohlsson, hep-ph/ ) Note! θ 13 = 0 or α = 0 θ θ 23 and m 2 m 2 32 For reasonable values of the fundamental neutrino parameters: θ θ 23 Tommy Ohlsson - ECT*, October 4-8, 2004 p. 23/29

24 The ratio m 2 / m 2 32 as a function of θ m 2 / m α = α = α = α = 0 α = 0.02 α = 0.04 α = θ 13 [ o ] The sign of the correction to the approximation m 2 = m 2 32 depends on θ 13 and α. If m 2 and m 2 32 measured, then valuable information on the mass hierarchy (sign of α) and θ 13. Note! For α 0 There is a value of θ 13 : m 2 = m 2 32 Tommy Ohlsson - ECT*, October 4-8, 2004 p. 24/29

25 Effective neutrino parameters in matter Antineutrinos Neutrinos Angle [ o ] ~ θ ~ 13 θ 23 ~ θ ~ 12 θ oo Mass Squared Difference [ev 2 ] e-2 1e-2 1e-4 EV [ev 2 ] 1e-4 EV [ev 2 ] 1e-6 ~ 2 m 21 ~ 2 m 32 ~ 2 m oo 1e e-6 1e-4 1e-2 EV [ev 2 ] 1e-6 1e-4 1e-2 EV [ev 2 ] Three different regions for EV : 2EV m 2 21 (left): {fundamental neutrino parameters} {vacuum neutrino parameter} 2EV m 2 21 (middle): resonance phenomena 2EV m 2 21 (right): two-flavor mixing between ν µ and ν τ (ν e eigenstate to the Hamiltonian) Tommy Ohlsson - ECT*, October 4-8, 2004 p. 25/29

26 The relative accuracy of the approximation EV c 13 m2 31 2EV 1 4 sin2 2θ 13 2αs 2 13 c2 13 s α2 c 2 13 s2 12 (c c2 13 s2 12 ) Neutrino oscillation channel 1 P µµ : LBL exps. and ν-factories 15 < 0.25 % < < 0.5 % < < 1 % 15 < 0.25 % < < 0.5 % < < 1 % θ 13 [ o ] θ 13 [ o ] log(ve / 1 ev 2 ) log(ve / 1 ev 2 ) L/E = 7000 km/50 GeV L/E = 3000 km/50 GeV (M. Blennow and T. Ohlsson, hep-ph/ ) Tommy Ohlsson - ECT*, October 4-8, 2004 p. 26/29

27 Open questions in neutrino physics Are neutrinos Dirac or Majorana particles? Are there sterile neutrinos? MiniBooNE will most probably have the answer. What is the sign of m 31? Normal or inverted mass hierarchy? What is the value of θ 13? Upper limit from the CHOOZ experiment exists. Is there CP violation in the leptonic sector? What is the value of δ CP? So far, we do not know anything about δ CP. What is the absolute neutrino mass scale? What are the symmetries? θ 23 = π/4 (maximal atmospheric mixing)? θ 12 + θ 13 = π/4 or θ 12 + θ C = π/4? Here θ C is the Cabibbo angle. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 27/29

28 Summary & Conclusions A complete handbook of neutrino oscillation formulas have been developed. (hep-ph/ ) The series expansions for the neutrino oscillation probabilities can be used to better understand neutrino oscillation physics. The solar day-night effect has been studied analytically with three neutrino flavors. (hep-ph/ ) The solar day-night effect can maybe (in the future) be used to determine the mixing angle θ 13. Effective neutrino mixing and oscillations have been studied in the approximation EV (dense matter). (hep-ph/ ) This could provide valuable information on the mass hierarchy and the mixing angle θ 13. Tommy Ohlsson - ECT*, October 4-8, 2004 p. 28/29

29 References E. Akhmedov, R. Johansson, M. Lindner, T. Ohlsson, and T. Schwetz, JHEP 04, 078 (2004), hep-ph/ M. Blennow, T. Ohlsson, and H. Snellman, Phys. Rev. D 69, (2004), hep-ph/ E. Akhmedov, M. Tórtola, and J. Valle, JHEP 05, 057 (2004), hep-ph/ M. Blennow and T. Ohlsson, hep-ph/ Tommy Ohlsson - ECT*, October 4-8, 2004 p. 29/29

Damping signatures in future neutrino oscillation experiments

Damping signatures in future neutrino oscillation experiments Based on JHEP 06(2005)049 In collaboration with Tommy Ohlsson and Walter Winter Mattias Blennow Division of Mathematical Physics Department