Neutrino Mixing and Oscillations Carlo Giunti INFN, Sez. di Torino, - PowerPoint PPT Presentation

� � � � � � � � � � � � � � � � � � � Dirac and Majorana Degrees of Freedom CPT CPT p, − h ) p, − h ) ν ( � p, h ) ν ( � ¯ ν ( � p, h ) ν ( � � � � � � � � � � � � � � � � � � � � � 180 ◦ Rotation Boost Boost Boost Boost � � � ����� � ����� � � � � � � � ν ( − � p, − h ) ν ( − � ν ( − � p, − h ) ν ( − � ¯ p, h ) p, h ) CPT CPT ν ( � p, h ) and ¯ ν ( − � p, h ) ν ( � p, h ) and ν ( − � p, h ) ν ( − � p, − h ) and ¯ ν ( � p, − h ) ν ( − � p, − h ) and ν ( � p, − h ) have different interactions have same interactions ⇓ ⇓ four degrees of freedom two degrees of freedom C. Giunti , Neutrino Mixing and Oscillations − 14

Majorana Mass ∇ ) χ L + m σ 2 χ ∗ σ · � ( ∂ 0 − � Two-Component Majorana Equation: L = 0 � � � 0 � � � Four Components σ · � i ( ∂ 0 − � ∇ ) 0 − iσ 2 χ ∗ − ( m 0 0 m ) = 0 L χ L σ · � ∇ ) 0 i ( ∂ 0 + � 0 � �� � (chiral representation) � �� � ν L ν c L iγ µ ∂ µ ν L + mν c Four-Component Majorana Equation: L = 0 Lagrangian: L L = 1 2 [ − iν L γ µ ( ∂ µ ν L ) + i ( ∂ µ ν L ) γ µ ν L − m ( ν c L ν L + ν L ν c )] L � �� � − ν T L C † ν L + ν L C ν L T � �� � T , ν c ν c L = − ν T L C † L = C ν L − νL C T νLT Euler-Lagrange ∂ ( ∂ µ ν L ) − ∂ L L ∂ L L ∂ν L = 0 ⇒ 1 T − m C T ν L 2( iγ µ ∂ µ ν L + iγ µ ∂ µ ν L + m C ν L T ∂ µ ) = 0 Equations | {z } −C ν LT � � L M L = − 1 L ν L + ν L ν c ν c Majorana Mass Term: 2 m L C. Giunti , Neutrino Mixing and Oscillations − 15

Majorana Neutrino ⇐ ⇒ No Conserved Lepton Number L e , L µ , L τ , L = L e + L µ + L τ ❩❩❩ ✚ ❩❩❩ ✚ ✚✚✚ ✚✚✚ ν c = ν L = − 1 L = +1 ← − − → ❩ ❩ Noether − − − − − − → Conserved Lepton Number Global Gauge Invariance ← − − − − − − Theorem Dirac mass term Majorana mass term � � L D = − m D ( ν L ν R + ν R ν L ) L M = − m M ν L ν c L + ν c L ν L invariant under not invariant under L → e − i Λ ν c ν L → e i Λ ν L ν R → e i Λ ν R ν L → e i Λ ν L ν c L ν L → e − i Λ ν L ν R → e − i Λ ν R ν L → e − i Λ ν L ν c L → e i Λ ν c L Majorana Neutrino = Truly Neutral Fermion C. Giunti , Neutrino Mixing and Oscillations − 16

the chiral fields ν L and ν R (if it exists!) are the building blocks of the neutrino Lagrangian ONLY ν L = ⇒ Majorana Mass Term � � � � L = − 1 2 m L ν ν = − 1 L ) = − 1 L M ( ν L + ν c L ν L + ν L ν c ν L + ν c ν c 2 m L 2 m L L L = 1 L C † ν L − ν L C ν L 2 m L ( ν T T ) � �� � L = C ν LT , ν † L C ν ∗ ν c ν c L = − ν T L C † L ⇒ ν L AND ν R = Dirac Mass Term L D = − m D ν ν = − m D ( ν L + ν R ) ( ν L + ν R ) = − m D ( ν L ν R + ν R ν L ) C. Giunti , Neutrino Mixing and Oscillations − 17

SURPRISE! ν L AND ν R = ⇒ Dirac–Majorana Mass Term   L D+M = L M L + L M R + L D  m L m D      M = � � m D m R  m L m D  ν L = − 1  + H . c .  ν c ν R   L 2 ν c m D m R  ν L R  N L = = 1 L C † M N L + H . c . ν c 2 N T R     diagonalization  ν 1 L  m 1 0 U T M U =   N L = U n L , n L = ⇒ ⇓ ν 2 L 0 m 2 fields with definite mass � � L D+M = 1 kL C † ν kL + h . c . = − 1 m k ν T m k ν k ν k 2 2 k =1 , 2 k =1 , 2 ν k = ν kL + ν c Massive neutrinos are Majorana! kL C. Giunti , Neutrino Mixing and Oscillations − 18

0 1 0 1 @ m L m D @ ν L “ ” L D+M = − 1 A + H . c . = 1 L C † M N L + H . c . 2 N T ν c ν R A L 2 ν c m D m R R m L , m R can be chosen real ≥ 0 by rephasing the fields ν L , ν R simplest case: real m D = ⇒ U = O ρ (CP invariance) ` m L m D ` cos ϑ “ ” | ρ k | 2 = 1 , “ ” ρ 1 0 ρ 1 cos ϑ ρ 2 sin ϑ sin ϑ ´ ´ O = M = , , ρ = , U = m D m R − sin ϑ cos ϑ 0 ρ 2 − ρ 1 sin ϑ ρ 2 cos ϑ “ m ′ 2 m D 2 , 1 = 1 » – q O T M O = 0 ” ( m L − m R ) 2 + 4 m 2 ⇒ tan 2 ϑ = m ′ = 1 , m L + m R ± m ′ D 0 m R − m L 2 2 1 negative if m 2 m ′ D > m L m R ” “ m ′ „ « „ « ρ 2 1 m ′ ρ 2 U T MU = ρ T O T M O ρ = “ ” “ ” 0 0 1 = ± 1 ρ 1 0 ρ 1 0 = ⇒ m k = ρ 2 k m ′ 1 = 1 m ′ 2 m ′ k 0 ρ 2 0 ρ 2 ρ 2 ρ 2 0 0 2 =1 2 2      cos ϑ sin ϑ  i cos ϑ sin ϑ ρ 2  ρ 2  ⇒ U = 1 = − 1 = ⇒ U = 1 = 1 = − sin ϑ cos ϑ − i sin ϑ cos ϑ C. Giunti , Neutrino Mixing and Oscillations − 19

→ η k γ 0 ν k ( t, − � CP η k = i ρ 2 − − k = ± i ν k ( t,� x ) x ) CP parity of ν k [Wolfenstein, Phys. Lett. B107 (1981) 77] important in neutrinoless double- β decay [Bilenky, Nedelcheva, Petcov, Nucl. Phys. B247 (1984) 61] [Kayser, Phys. Rev. D30 (1984) 1023]   → η k γ 0 ν c CP − − k ( t, − � ν k ( t,� x ) x ) the product of the CP parities of in general k γ 0 ν k ( t, − � CP  particle and antiparticle is − 1 ν c → − η ∗ − − k ( t,� x ) x ) T ) ( | η k | 2 = 1 , ψ c = C ψ ⇒ η k = − η ∗ Majorana Constraint ν c k = ν k = k = ⇒ η k = ± i imaginary CP parity! C. Giunti , Neutrino Mixing and Oscillations − 20

� ν L � CP transformation of N L = is determined by CP invariance of Lagrangian ν c R L D+M = − 1 L M N L − 1 2 N L M ∗ N c ( M T = M ) 2 N c L 9 → ξ γ 0 N c CP N L − − = → 1 L + 1 L ξ † M ∗ ξ † N L L CP L D+M 2 N L ξ M ξ N c 2 N c ⇒ − − = → − ξ † γ 0 N L CP N c − − ; L 8 → i γ 0 N c CP N L − − < L M real ⇒ CP invariance ⇔ ξ M ξ = − M ⇒ ξ = ( i 0 0 i ) = i I ⇒ → i γ 0 N L CP N c − − : L n L = U † N L U = O ρ N L = U n L ρ kj = ρ k δ kj O T O = I L = U ∗ n c L = U T N c N c n c ρ 2 k = ± 1 L L n L = U † N L → i U † γ 0 N c L = i U † U ∗ γ 0 n c CP − − � �� � L η � � ∗ = i � � ∗ = i ρ 2 η = i U † U ∗ = i U T U ρ O T O ρ η k = iρ 2 k = ± i C. Giunti , Neutrino Mixing and Oscillations − 21

= − g 2 ν L γ µ ℓ L W µ − g 2 ℓ L γ µ ν L W † CP invariance of L CC √ √ µ ? I T → i γ 0 C ν LT → i γ 0 C ℓ L CP CP − − − − ν L ℓ L CP → − W µ † W µ − − L C † γ 0 L C † γ 0 CP CP → − i ν T → − i ℓ T − − − − ν L ℓ L → − g 2 ℓ L γ µ † ν L W µ † − g 2 ν L γ µ † ℓ L W µ CP L CC − − √ √ I � γ † � � � γ µ † = γ 0 † ,� γ 0 , − � = γ = γ µ → − g ℓ L γ µ ν L W µ † − g CP L CC ν L γ µ ℓ L W µ − − √ √ I 2 2 CP invariance OK! CP parity of charged lepton is also imaginary! C. Giunti , Neutrino Mixing and Oscillations − 22

Maximal Mixing » – q 2 m D 2 , 1 = 1 ( m L − m R ) 2 + 4 m 2 m ′ tan 2 ϑ = m L + m R ± D m R − m L 2 m ′ m L = m R = ⇒ ϑ = π/ 4 , 2 , 1 = m L ± | m D | 8 ρ 2 ν 1 L = − i 2 ( ν L − ν c m 1 = | m D | − m L , 1 = − 1 , R ) < √ | m D | > m L ≥ 0 ⇒ ρ 2 1 2 ( ν L + ν c m 2 = | m D | + m L , 2 = +1 , ν 2 L = R ) : √ 8 ν 1 = ν 1 L + ν c 2 [( ν L + ν R ) − ( ν c L + ν c 1 L = − i < R )] √ Majorana Neutrino Fields: ν 2 = ν 2 L + ν c 1 2 [( ν L + ν R ) + ( ν c L + ν c 2 L = R )] : √ C. Giunti , Neutrino Mixing and Oscillations − 23

⇒ m L = m R = 0 = Dirac Neutrino Field ν 1 and ν 2 have the same mass m 1 = m 2 = | m D | and opposite CP parities. The two Majorana fields ν 1 and ν 2 can be combined to give one Dirac field ν 1 √ ν = 2 ( iν 1 + ν 2 ) = ν L + ν R Viceversa, one Dirac field ν can always be splitted in two Majorana fields „ « „ ν + ν c « − i ν − ν c ν = 1 i + 1 1 2 [( ν − ν c ) + ( ν + ν c )] = √ √ √ √ √ = ( iν 1 + ν 2 ) 2 2 2 2 2  ν 1 = − i  2 ( ν − ν c ) √  Majorana Neutrino Fields ( ν 1 = ν c 1 , ν 2 = ν c 2 ): 1  ( ν + ν c )  √ ν 2 = 2 In general: one Dirac field ≡ two Majorana fields with same mass and opposite CP parities C. Giunti , Neutrino Mixing and Oscillations − 24

CP parity of Dirac = 2 Majorana neutrino field → − i γ 0 ν 1 ( t, − � → i γ 0 ν 2 ( t, − � CP CP ν 1 ( t,� x ) − − x ) ν 2 ( t,� x ) − − x ) 1 → i γ 0 1 CP √ √ ν = 2 ( iν 1 + ν 2 ) − − 2 ( − iν 1 + ν 2 ) 1 L = − i ν 1 = ν 1 L + ν c 2 [( ν L + ν R ) − ( ν c L + ν c R )] √ 1 ν 2 = ν 2 L + ν c 2 [( ν L + ν R ) + ( ν c L + ν c 2 L = R )] √ → i γ 0 ( ν c CP L + ν c R ) = i γ 0 ν c − − ν Dirac neutrino field has definite CP parity = i C. Giunti , Neutrino Mixing and Oscillations − 25

Pseudo-Dirac Neutrinos 2 , 1 ≃ m L + m R m ′ ρ 2 ρ 2 m L , m R ≪ | m D | ⇒ ± | m D | ⇒ 1 = − 1 , = = 2 = +1 2 m 1 ≃ | m D | − m L + m R m 2 ≃ | m D | + m L + m R ∆ m 2 ≃ | m D | ( m L + m R ) , = ⇒ 2 2 2 m D tan 2 ϑ = ≫ 1 = ⇒ ϑ ≃ π/ 4 practically maximal mixing! m R − m L ν 1 L ≃ − i 2 ( ν L − ν c 1 R ) ν L ≃ 2 ( iν 1 L + ν 2 L ) √ √ ⇐ ⇒ 1 2 ( ν L + ν c ν c 1 ν 2 L ≃ R ) R ≃ 2 ( − iν 1 L + ν 2 L ) √ √ 0 1 0 1 0 1 @ i 1 @ 1 1 @ i 0 1 1 A = U ≃ √ √ A A 2 2 − i − 1 1 1 0 1 active ( ν L ) – sterile ( ν R ) oscillations! C. Giunti , Neutrino Mixing and Oscillations − 26

See-Saw Mechanism [Yanagida, 1979] [Gell-Mann, Ramond, Slansky, 1979] [Witten, Phys. Lett. B91 (1980) 81] [Mohapatra, Senjanovic, Phys. Rev. Lett. 44 (1980) 912] » – q 2 m D 2 , 1 = 1 ( m L − m R ) 2 + 4 m 2 m ′ tan 2 ϑ = m L + m R ± D m R − m L 2 1 ≃ − ( m D ) 2 tan 2 ϑ = 2 m D m ′ m ′ m L = 0 , | m D | ≪ m R = ⇒ , , 2 ≃ m R m R m R ν m 1 ≃ ( m D ) 2 ρ 2 1 ≪ | m D | 1 = − 1 m R ν ρ 2 m 2 ≃ m R 2 = +1 2 tan ϑ ≃ m D ν 2 L ≃ ν c ≪ 1 ⇒ ν 1 L ≃ − ν L , = R m R Example: | m D | ∼ M EW ∼ 10 2 GeV , m R ∼ M GUT ∼ 10 15 GeV m 1 ∼ 10 − 2 eV ⇒ =    0 m D  See-Saw Mass Matrix: M = Why m L = 0 ? m D m R C. Giunti , Neutrino Mixing and Oscillations − 27

   ν L � I 3 =1 / 2 L M ∼ ν T L σ 2 Φ) C − 1 (Φ T σ 2 L L ) Symmetry  ( L T ν T − − − − − → L L = L ν L L ν L Breaking ℓ L non-renormalizable I 3 =1 doublet triplet Effective Lagrangian [Weinberg, Phys. Rev. Lett. 43 (1979) 1566, Phys. Rev. D22 (1980) 1694] [Weldon, Zee, Nucl. Phys. B173 (1980) 269] minimum dimension lepton-number violating operator invariant under SU(2) L × U(1) Y g L σ 2 Φ) C − 1 (Φ T σ 2 L L ) + H . c . M ( L T “ ” “ ” φ + Symmetry 0 Φ ≡ − − − − − → √ φ 0 v/ 2 Breaking gv 2 L C − 1 ν L + H . c . ∼ − m 2 L M = 1 M ( ν L ) c ν L + H . c . M ν T D 2 m L ∼ m 2 Plausible Cut-Off: M � M P ∼ 10 19 GeV D See-Saw Type M C. Giunti , Neutrino Mixing and Oscillations − 28

General Considerations on Fermion Masses In Standard Model fermion masses are generated through Yukawa couplings � y ℓ L H,ℓ = − αβ L αL Φ ℓ βR + H . c . α,β = e,µ,τ the coefficients y α,β are parameters of the model ⇓ explanation of parameters must come from new physics Beyond the SM ⇓ all fermion masses give info on new physics BSM C. Giunti , Neutrino Mixing and Oscillations − 29

u s d b t � � � � 1 2 3 e � � 4 � 3 � 2 � 1 0 1 2 3 4 5 6 7 8 9 10 11 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 ⇒ more info? smallness of ν masses is additional mystery = m [ eV ℄   ⋆ See-Saw Mechanism known natural explanations of smallness of ν masses:  ⋆ Effective Lagrangian   ⋆ Majorana ν masses!    see-saw type relation m light ∼ m 2 D both imply ⋆  M    New high energy scale M ⋆ general features of SU(2) L × U(1) Y invariant models with additional scalars and fermions (unless special symmetries forbid all Majorana mass terms) C. Giunti , Neutrino Mixing and Oscillations − 30

neutrino masses provide a window on New Physics Beyond the Standard Model most accessible window on NPBSM at low energy the lepton-number violating dimension 5 operator ( L T L )(Φ T Φ) → m L ν T L ν L is the operator beyond the Standard Model with minimum dimension (quarks are Dirac!) Y ( q U Y ( q D Y ( L L ) = − 1 , Y ( ℓ R ) = − 2 , R ) = − 2 / 3 Y (Φ) = 1 , Y ( Q L ) = 1 / 3 , R ) = 4 / 3 , next: lepton and barion number violating dimension 6 operators ∼ qqqℓ ( ∆ L = ∆ B ) T q U T ℓ R “ ” “ ” “ ” “ ” “ ” “ ” q D Q T Q T q U Q T Q T L L L , L Q L , L Q L L L L , R R R T q U T ℓ R T q U T ℓ R “ ” “ ” “ ” “ ” p → e + π 0 , q D q U q U q D ⇒ , = etc. R R R R R R C. Giunti , Neutrino Mixing and Oscillations − 31

Majorana mass term for ν R respects SU(2) L × U(1) Y Standard Model Symmetry! � � R = − 1 L M R ν R + ν R ν c ν c 2 m R Majorana mass term for ν R breaks Lepton number conservation!   − Lepton number can be explicitly broken       − Lepton number is spontaneously broken locally, with a mas- Three possibilities: sive vector boson coupled to the lepton number current     − Lepton number is spontaneously broken globally and a    massless Goldstone boson appears in the theory (Majoron) C. Giunti , Neutrino Mixing and Oscillations − 32

Singlet Majoron Model [Chikashige, Mohapatra, Peccei, Phys. Lett. B98 (1981) 265, Phys. Rev. Lett. 45 (1980) 1926] � � L L Φ ν R + ν R Φ † L L L Φ = − y d − − − − → − m D ( ν L ν R + ν R ν L ) � Φ �� =0 � � � � R ν R + η † ν R ν c − 1 η ν c ν c R ν R + ν R ν c L η = − y s − − − − → 2 m R R R � η �� =0 � � � ν L � L mass = − 1 η = 2 − 1 / 2 ( � η � + ρ + i χ ) 0 m D 2 ( ν c L ν R ) + H . c . ν c m D m R R ⇒ See-Saw: m 1 ≃ m 2 ≫ m R m D = D m R scale of L violation EW scale ρ = massive scalar χ = massless pseudoscalar Goldstone boson = Majoron " # « 2 Majoron weakly coupled L χ − ν = iy s ν 2 γ 5 ν 2 − m D „ m D ν 2 γ 5 ν 1 + ν 1 γ 5 ν 2 ν 1 γ 5 ν 1 ˆ ´ √ χ + m R m R 2 to light neutrino Majoron weakly coupled weak long-range force m 2 χ − f = ± y s G F L eff D χ fγ 5 f 16 π 2 m f to matter through with spin-dependent m R potential ∼ 10 − 65 cm 2 /r 3 W − ν loop and Z − χ mixing C. Giunti , Neutrino Mixing and Oscillations − 33

Three-Neutrino Mixing [Bilenky & Petcov, Rev. Mod. Phys. 59 (1987) 671] Dirac neutrino mass term generated SM with ν eR , ν µR , ν τR = ⇒ by standard Higgs mechanism � L D = − M D = complex 3 × 3 matrix ν αR M D αβ ν βL + H . c . ( α, β = e, µ, τ ) α,β M D can be diagonalized by the biunitary transformation V † M D U = M V † = V − 1 , U † = U − 1 , M kj = m k δ kj , real m k ≥ 0 POSSIBLE? C. Giunti , Neutrino Mixing and Oscillations − 34

Proof that M D can be diagonalized by a biunitary transformation consider M D ( M D ) † : Hermitian = ⇒ can be diagonalized by the unitary transformation V † M D ( M D ) † V = M 2 , V † = V − 1 , M 2 kj = m 2 real m 2 k δ kj , k choosing an appropriate matrix U , it is always possible to write � M D = V M U † V † M D U = M m 2 ⇒ with M kj = k δ kj = m k δ kj = only problem: is U unitary? U † = M − 1 V † M D , U = ( M D ) † V M − 1 ( M † = M ) magically U is unitary! U † U = M − 1 V † M D ( M D ) † V M − 1 = 1 UU † = ( M D ) † V M − 2 V † M D = ( M D ) † V V † (( M D ) † ) − 1 ( M D ) − 1 V V † M D = 1 C. Giunti , Neutrino Mixing and Oscillations − 35

3 � L D = − diagonalized Dirac mass term: m k ν k ν k k =1  3 �    ν αL = U αk ν kL   k =1 mixing: ( α = e, µ, τ ) 3 �    ν αR = V αk ν kR   k =1 no right-handed fields in weak interaction Lagrangian ⇓ right-handed singlets are sterile and not mixed with active neutrinos 3 � � � † = 2 j CC weak charged current: ℓ αL γ ρ ν αL = 2 ℓ αL γ ρ U αk ν kL ρ α = e,µ,τ α = e,µ,τ k =1 U = unitary 3 × 3 mixing matrix we assumed for simplicity that the mass matrix of charged leptons is diagonal otherwise U = U ( ℓ ) † U ( ν ) C. Giunti , Neutrino Mixing and Oscillations − 36

Physical Parameters in N × N Mixing Matrix   N ( N − 1) Mixing Angles N × N Unitary Mixing Matrix ⇒ N 2 parameters 2  N ( N +1) Phases 2 � � † = 2 j CC Weak Charged Current: ℓ αL γ ρ ν αL = 2 ℓ αL γ ρ U αk ν kL ρ α α,k Lagrangian is invariant under global phase transformations of Dirac fields  � † → 2 j CC ℓ αL e − iθ α γ ρ U αk e iφ k ν kL   ρ       α,k ℓ α → e iθ α ℓ α � ℓ αL e − i ( θ e − φ 1 ) e − i ( θ α − θ e ) γ ρ U αk e i ( φ k − φ 1 ) ν kL ⇒ = = 2   ν k → e iφ k ν k   ↑ ↑ ↑ α,k    1 N − 1 N − 1 number of independent phases that can be eliminated: 2 N − 1 (not 2 N !) number of physical phases: N ( N + 1) − (2 N − 1) = ( N − 1) ( N − 2) 2 2 ⇒ conservation of L remains global phase freedom of lepton fields = C. Giunti , Neutrino Mixing and Oscillations − 37

N × N Unitary Mixing Matrix: N ( N − 1) Mixing Angles and ( N − 1) ( N − 2) Phases 2 2 N = 3 ⇒ 3 Mixing Angles and 1 Physical Phase (as in the quark sector) standard parameterization (convenient) ( c ij ≡ cos ϑ ij , s ij ≡ sin ϑ ij )       0 s 13 e − iδ 13 1 0 0 c 13 c 12 s 12 0             U = R 23 W 13 R 12 = 0 c 23 s 23 0 1 0 − s 12 c 12 0       − s 13 e iδ 13 0 0 − s 23 c 23 c 13 0 0 1   s 13 e − iδ 13 c 12 c 13 s 12 c 13     − s 12 c 23 − c 12 s 23 s 13 e iδ 13 c 12 c 23 − s 12 s 23 s 13 e iδ 13 = s 23 c 13   s 12 s 23 − c 12 c 23 s 13 e iδ 13 − c 12 s 23 − s 12 c 23 s 13 e iδ 13 c 23 c 13 phase δ 13 associated with s 13 ⇒ CP violation is small if ϑ 13 is small in other parameterizations phase can be associated with s 12 or s 23 ⇓ CP violation is small if any mixing angle is small if any element of U is zero the phase can be rotated away ⇒ no CP violation C. Giunti , Neutrino Mixing and Oscillations − 38

µ ± → e ± + γ Dirac mass term allows L e , L µ , L τ violating processes like � µ ± → e ± + e + + e − W W µ − → e − + γ � � � � e k � U U ek �k (A) � U ∗ µk U ek = 0 ⇒ GIM Mechanism W W � � k � � 2 � � � � � � Γ = G F m 5 � � � e � � e 3 α m k k k � � µ U ∗ µk U ek � � 192 π 3 32 π � m W � (B) (C) k � �� � BR Suppression factor: m k � 10 − 11 for m k � 1 eV m W ( BR ) exp � 10 − 11 ( BR ) the � 10 − 25 14 orders of magnitude smaller! C. Giunti , Neutrino Mixing and Oscillations − 39

NUMBER OF MASSIVE NEUTRINOS? Z → ν ¯ ⇒ ν ν e ν µ ν τ active flavor neutrinos N � N ≥ 3 ⇒ mixing ν αL = U αk ν kL α = e, µ, τ no upper limit! k =1 · · · Mass Basis: ν 1 ν 2 ν 3 ν 4 ν 5 Flavor Basis: ν e ν µ ν τ ν s 1 ν s 2 · · · ACTIVE STERILE STERILE NEUTRINOS singlets of SM = ⇒ no interactions! active → sterile transitions are possible if ν 4 , . . . are light (no see-saw) ⇓ disappearance of active neutrinos C. Giunti , Neutrino Mixing and Oscillations − 40

Dirac-Majorana mass term active ν αL ( α = e, µ, τ ) + sterile ν sR ( s = s 1 , s 2 , . . . , s N ) � L D = − ν sR M D sα ν αL + H . c . s,α � L = − 1 L M αL M L ν c L D+M = L M L + L D + L M αβ ν βL + H . c . 2 R α,β � R = − 1 L M ν sR M R ss ′ ν c s ′ R + H . c . 2 s,s ′ M D , M L , M R are complex matrices M L , M R are symmetric C. Giunti , Neutrino Mixing and Oscillations − 41

αL = C ν αLT , ν c αL = − ν T αL C † ν c example:  � � αL C † M L  αL M L ν T ν c αβ ν βL = − αβ ν βL      α,β α,β   �   βL ( C † ) T M L  ν T  = αβ ν αL     α,β  M L αβ = M L   � βα  C T = −C � = − βL C † M L ν T αβ ν αL ⇒ = �   α,β  M L is symmetric!  �    βL M L ν c = αβ ν αL      α,β   �    αL M L ν c  α ⇆ β � = βα ν βL   α,β C. Giunti , Neutrino Mixing and Oscillations − 42

L D+M = L M L + L D + L M R � � � = − 1 sα ν αL − 1 αL M L ν sR M D ν sR M R ss ′ ν c ν c αβ ν βL − s ′ R + H . c . 2 2 s,s ′ s,α α,β write Lagrangian in compact form for mass diagonalization       ν c ν eL s 1 R      ν L .      ν c . column matrix of left-handed fields: N L ≡ ν L ≡ R ≡ ν µL   .     ν c R ν c ν τL s N R L D+M = − 1 L M D+M N L + H . c . = 1 L C † M D+M N L + H . c . 2 N T 2 N c    M L ( M D ) T M D+M ≡  (3 + N ) × (3 + N ) symmetric mass matrix: M D M R U T M D+M U = M , m k ≥ 0 , diagonalization: N L = U n L , M kj = m k δ kj , U † = U − 1 POSSIBLE? C. Giunti , Neutrino Mixing and Oscillations − 43

Proof that M D+M = ( M D+M ) T can be diagonalized by U T M D+M U = M an arbitrary complex matrix can be diagonalized by the biunitary transformation V † M D+M W = M , V † = V − 1 , W † = W − 1 m k ≥ 0 , M kj = m k δ kj ,   M D+M = V M W †    M D+M ( M D+M ) † = V M 2 V † ⇒ = = M D+M ( M D+M ) † = ( W † ) T M 2 W T    ( M D+M ) T = ( W † ) T M V T V M 2 V † = ( W † ) T M 2 W T W T V M 2 = M 2 W T V ⇒ W T V = D , D kj = e 2 iλ k δ kj M D+M = V M W † = ( W † ) T W T V M W † = ( W † ) T D M W † = ( W † ) T D 1 / 2 M D 1 / 2 W † = ( D 1 / 2 W † ) T M ( D 1 / 2 W † ) = ( U † ) T M U † ⇓ U T M D+M U = M C. Giunti , Neutrino Mixing and Oscillations − 44

  ν eL       ν µL left-handed     ν 1 L       ν τL components  ν L .      = U † N L  ≡ . n L ≡ N L ≡ = U n L     . ν c    ν c of fields with  s 1 R  R   ν (3+ N ) L . .   definite mass .   ν c L D+M = − 1 L M D+M N L + H . c . 2 N c s N R 3+ N � = − 1 L M n L + H . c . = − 1 2 n c m k ν c kL ν kL + H . c . 2 k =1   ν 1   .   L = U † N L + U T N c .  = n L + n c n ≡ fields with definite mass are Majorana:   . L  ν 3+ N 3+ N � L D+M = − 1 2 n M n = − 1 m k ν k ν k 2 k =1 C. Giunti , Neutrino Mixing and Oscillations − 45

3+ N � ν αL = U αk ν kL ( α = e, µ, τ ) k =1 mixing relations: 3+ N � ν c sR = U sk ν kL ( s = s 1 , . . . , s N ) k =1 Sterile neutrino fields ν sR are connected to Active neutrino fields ν αL trough the Massive neutrino fields ν kL ⇓ Active ⇆ Sterile oscillations are possible! ⇓ disappearance of active neutrinos C. Giunti , Neutrino Mixing and Oscillations − 46

Physical Parameters in N × N Mixing Matrix for Majorana Neutrinos N ( N − 1) angles N × N Unitary Mixing Matrix ⇒ N 2 parameters 2 N ( N +1) phases 2 not rephasable � ↓ † = 2 j CC Weak Charged Current: ℓ αL γ ρ U αk ν kL ρ ↑ α,k rephasable Lagrangian is not invariant under global phase transformations ν k → e iφ k ν k kT C − 1 ν kL → e 2 iφ k ν T kT C − 1 ν kL Majorana mass term: ν T Lepton number is not conserved! only N phases in the mixing matrix can be eliminated rephasing the charged lepton fields � † → 2 j CC ℓ αL e − iθ α γ ρ U αk ν kL ρ ↑ α,k N C. Giunti , Neutrino Mixing and Oscillations − 47

� � − N = N ( N − 1) N ( N + 1) same number as number of physical phases: mixing angles 2 2 N ( N − 1) = ( N − 1) ( N − 2) + N − 1 � �� � 2 2 � �� � “Majorana phases” “Dirac phases”   1 0 ··· 0 0 e iλ 21 ··· 0 U αk = U (D) αk e iλ k 1 , U = U (D) D ( λ ) ,   λ 11 = 0 = ⇒ D ( λ ) = . . . ... . . . . . . overall phase ··· e iλN 1 0 0 C. Giunti , Neutrino Mixing and Oscillations − 48

Three Light Majorana Neutrinos ( ⇐ See-Saw) N = 3 = ⇒ 3 Mixing Angles 1 Dirac Phase 2 Majorana Phases standard parameterization (convenient) ( c ij ≡ cos ϑ ij , s ij ≡ sin ϑ ij ) U = R 23 W 13 R 12 D ( λ ) 0 1 0 1 0 1 0 1 0 s 13 e − iδ 13 1 0 0 c 13 c 12 s 12 0 1 0 0 B C B C B C B C 0 e iλ 21 = 0 c 23 s 23 0 1 0 − s 12 c 12 0 0 B C B C B C B C @ A @ A @ A @ A − s 13 e iδ 13 0 e iλ 31 0 − s 23 c 23 c 13 0 0 1 0 0 0 1 0 1 s 13 e − iδ 13 c 12 c 13 s 12 c 13 1 0 0 B C B C − s 12 c 23 − c 12 s 23 s 13 e iδ 13 c 12 c 23 − s 12 s 23 s 13 e iδ 13 0 e iλ 21 = s 23 c 13 0 B C B C @ A @ A s 12 s 23 − c 12 c 23 s 13 e iδ 13 − c 12 s 23 − s 12 c 23 s 13 e iδ 13 e iλ 31 c 23 c 13 0 0 Majorana phases are relevant only in processes involving Lepton number violation ββ 0 ν , ν α ⇆ ¯ ν β , . . . these processes are suppressed by smallness of neutrino masses because of helicity mismatch in the limit of negligible neutrino massess Dirac = Majorana! C. Giunti , Neutrino Mixing and Oscillations − 49

CP invariance → i γ 0 N c → i γ 0 N L CP CP CP invariance of L CC N c ⇒ − − ⇒ − − N L I L L L D+M = − 1 L M D+M N L − 1 2 N L M D+M ∗ N c ( M D+M T = M D+M ) 2 N c L → − 1 L − 1 L M D+M ∗ N L 2 N L M D+M N c CP L D+M 2 N c − − M D+M = M D+M ∗ CP invariance ⇐ ⇒ real! n L = U † N L U = O D N L = U n L D kj = D k δ kj L = U ∗ n c L = U T N c O T O = I N c n c D 2 k = ± 1 L L n L = U † N L → i U † γ 0 N c L = i U † U ∗ γ 0 n c CP − − η k = CP parity of ν k � �� � L η � � ∗ = i � � ∗ = i D 2 η = i U † U ∗ = i U T U D O T O D η k = iD 2 k = ± i important: relative CP parities η kj ≡ η k /η j = D 2 k /D 2 j = ± 1 C. Giunti , Neutrino Mixing and Oscillations − 50

standard parameterization of CP-invariant Majorana mixing matrix U = R 23 R 13 R 12 D ( λ ) 0 1 0 1 0 1 0 1 1 0 0 c 13 0 s 13 c 12 s 12 0 1 0 0 B C B C B C B C 0 e iλ 21 = 0 c 23 s 23 0 1 0 − s 12 c 12 0 0 B C B C B C B C @ A @ A @ A @ A e iλ 31 0 − s 23 c 23 − s 13 0 c 13 0 0 1 0 0 0 1 0 1 c 12 c 13 s 12 c 13 s 13 1 0 0 B C B C 0 e iλ 21 = − s 12 c 23 − c 12 s 23 s 13 c 12 c 23 − s 12 s 23 s 13 s 23 c 13 0 B C B C @ A @ A e iλ 31 s 12 s 23 − c 12 c 23 s 13 − c 12 s 23 − s 12 c 23 s 13 c 23 c 13 0 0 equal or opposite λ kj = 0 , π η kj = e 2 iλ kj = ± 1 ⇐ ⇒ 2 CP parities if λ kj = π e iλ kj = i ⇒ ⇒ = = complex U ! 2 C. Giunti , Neutrino Mixing and Oscillations − 51

d u W � e U ek Neutrinoless Double- β Decay ( ββ 0 ν ): ∆ L = 2 m � k k N ( A, Z ) → N ( A, Z + 2) + e − + e − Γ ββ 0 ν ∝ |� m �| 2 U ek � e � � � � effective � W � � U 2 d u |� m �| = ek m k � � Majorana � � mass k  76 Ge → 76 Se + e − + e −  100 Mo → 100 Ru + e − + e − examples: 130 Te → 130 Xe + e − + e −  136 Xe → 136 Ba + e − + e − d u W � e Two-Neutrino Double- β Decay ( ∆ L = 0 ) � � e � � e � N ( A, Z ) → N ( A, Z ) + e − + e − + ¯ e ν e + ¯ ν e W d u second order weak interaction process C. Giunti , Neutrino Mixing and Oscillations − 52 1 1

Im h m i 2 2 i� 31 j U j e m e 3 3 h m i 2 2 i� 21 j U j e m e 2 2 2 j U j m e 1 1 Re h m i � � � � � � � U 2 |� m �| = ek m k � � � � k complex U ek ⇒ possible cancellations among m 1 , m 2 , m 3 contributions! � � � | U e 1 | 2 m 1 + | U e 2 | 2 e 2 iλ 21 m 2 + | U e 3 | 2 e 2 iλ 31 m 3 � |� m �| = λ kj = 0 , π e 2 iλ kj = η kj = ± 1 ⇒ ⇒ conserved CP = = 2 e 2 iλ kj = − 1 opposite CP parities of ν k and ν j = ⇒ = ⇒ maximal cancellation! C. Giunti , Neutrino Mixing and Oscillations − 53 1

EXAMPLE: 2 MASSIVE NEUTRINOS � � � | U e 1 | 2 m 1 + | U e 2 | 2 e 2 iλ 21 m 2 � |� m �| = � � λ 21 = π � | U 2 � | U 2 ⇒ |� m �| = e 1 | m 1 − e 2 | m 2 = 2 ↑ conserved CP cancellation opposite CP parities if m 1 ≃ m 2 and | U 2 e 1 | ≃ | U 2 e 2 | ≃ 1 / 2 = ⇒ |� m �| can be extremely small! Dirac neutrino: perfect cancellation   equal mass   1 Dirac neutrino ≡ 2 Majorana neutrinos with maximal mixing    opposite CP parities   m 1 = m 2   | U e 1 | 2 = | U e 2 | 2 = 1 / 2 = ⇒ |� m �| = 0    λ 21 = π/ 2 C. Giunti , Neutrino Mixing and Oscillations − 54

See-Saw Mechanism   ( M D ) T  0 M L = 0 = ⇒ M D+M =  M D M R eigenvalues of M R ≫ eigenvalues of M D M D+M is block-diagonalized ⇒ =    M light 0 W † ≃ W − 1 W T M D+M W ≃  0 M heavy corrections ∼ ( M R ) − 1 M D   † ) − 1 M D 2( M D ) † ( M R ) †− 1  ( M D ) † ( M R ( M R ) W = 1 − 1  ( M R ) − 1 M D ( M D ) † ( M R ) †− 1 2 − 2( M R ) − 1 M D M light ≃ − ( M D ) T ( M R ) − 1 M D M heavy ≃ M R C. Giunti , Neutrino Mixing and Oscillations − 55

M light ≃ − ( M D ) T ( M R ) − 1 M D M R = M I ⇒ M = high energy scale = QUADRATIC SEE-SAW M light ≃ − ( M D ) T M D m k ∼ ( m f k ) 2 = ⇒ M M 1 ) 2 : ( m f 2 ) 2 : ( m f m 1 : m 2 : m 3 ∼ ( m f 3 ) 2 M R = M ⇒ M D = scale of M D M D = LINEAR SEE-SAW M D M light ≃ −M D m k ∼ M D M m f M M D ⇒ = k m 1 : m 2 : m 3 ∼ m f 1 : m f 2 : m f 3 C. Giunti , Neutrino Mixing and Oscillations − 56

Summary of Part 1: Neutrino Masses and Mixing in the “Standard Model” neutrino are massless by construction implementation of “two-component theory” “Standard Model” can be naturally extended to include neutrino masses add ν eR , ν µR , ν τR surprise: Majorana Masses known natural explanations of smallness of ν masses See-Saw Mechanism, Effective Lagrangian ⇓ Majorana ν Masses, New High Energy Scale ⇓ Neutrino Masses are powerful window on New Physics Beyond Standard Model C. Giunti , Neutrino Mixing and Oscillations − 57

Part 2: Neutrino Oscillations in Vacuum and in Matter C. Giunti , Neutrino Mixing and Oscillations − 58

Detectable Neutrinos are Extremely Relativistic Only neutrinos with energy larger than some fraction of MeV are detectable! Charged-Current Processes: Threshold ☼ ν e + 37 Cl → 37 Ar + e − E th = 0 . 81 MeV ν + A → B + C ☼ ν e + 71 Ga → 71 Ge + e − E th = 0 . 233 MeV ⇓ s = 2 Em A + m 2 A ≥ ( m B + m C ) 2 ν e + p → n + e + ♁ ¯ E th = 1 . 8 MeV ⇓ ♁ ν µ + n → p + µ − E th = 110 MeV E th = ( m B + m C ) 2 − m A m 2 ♁ ν µ + e − → ν e + µ − µ 2 m A 2 E th ≃ 2 m e = 10 . 9 GeV Elastic Scattering Processes: Cross Section ∝ Energy ☼ ν + e − → ν + e − σ 0 ∼ 10 − 44 cm 2 σ ( E ) ∼ σ 0 E/m e Background ⇒ E th ≃ 5 MeV (SK, SNO) Laboratory and Astrophysical Limits = ⇒ m ν � 1 eV C. Giunti , Neutrino Mixing and Oscillations − 59

π + → µ + + ν µ π − → µ − + ¯ Easy Example of Neutrino Production: ν µ E 2 k = p 2 k + m 2 two-body decay = ⇒ fixed kinematics k  � � 2 � � 1 − m 2 1 + m 2  k = m 2 − m 2 + m 4  µ µ  p 2 π k k   m 2 m 2 4 m 2 4 2 π π π π at rest: � � 2 � �  1 − m 2 1 − m 2 k = m 2 + m 2 + m 4   µ µ π E 2 k k   m 2 m 2 4 m 2 4 2 π π π � � 1 − m 2 0 th order: m k = 0 ⇒ p k = E k = E = m π µ ≃ 30 MeV m 2 2 π � � 1 − m 2 E k ≃ E + ξ m 2 p k ≃ E − (1 − ξ ) m 2 ξ = 1 1 st order: µ k k ≃ 0 . 2 m 2 2 E 2 E 2 π � � general! C. Giunti , Neutrino Mixing and Oscillations − 60

Neutrino Oscillations in Vacuum: Plane Wave Model � � Neutrino Production: j CC = 2 ν αL γ ρ ℓ αL ν αL = U αk ν kL Fields ρ α = e,µ,τ k � X X U ∗ U αk U ∗ U αk U ∗ | ν α � = αk | ν k � States � 0 | ν αL | ν β � = βj � 0 | ν kL | ν j � ∝ βk = δ αβ | {z } k,j k k ∝ δ kj � αk e − iE k t + ip k x | ν k � | ν k ( x, t ) � = e − iE k t + ip k x | ν k � U ∗ ⇒ | ν α ( x, t ) � = = k ↑ �� � � � | ν k � = U βk | ν β � U ∗ αk e − iE k t + ip k x U βk | ν α ( x, t ) � = | ν β � β = e,µ,τ β = e,µ,τ k � �� � A να → νβ ( x,t ) Transition Probability � � 2 � � � � � � 2 = � � P ν α → ν β ( x, t ) = |� ν β | ν α ( x, t ) �| 2 = � A ν α → ν β ( x, t ) U ∗ αk e − iE k t + ip k x U βk � � � � k C. Giunti , Neutrino Mixing and Oscillations − 61

ultrarelativistic neutrinos = ⇒ t ≃ x = L source-detector distance E k t − p k x ≃ ( E k − p k ) L = E 2 k − p 2 m 2 L ≃ m 2 k k k L = 2 E L E k + p k E k + p k � � 2 � � � � � k L/ 2 E U βk αk e − im 2 U ∗ P ν α → ν β ( L, E ) = � � � � k � | U αk | 2 | U βk | 2 = ⇐ constant term k � � ∆ m 2 � kj L U ∗ αk U βk U αj U ∗ − i ⇐ oscillating term + 2Re βj exp 2 E k>j � coherence ∆ m 2 kj ≡ m 2 k − m 2 j C. Giunti , Neutrino Mixing and Oscillations − 62

NEUTRINOS AND ANTINEUTRINOS ν CP = γ 0 C ν T = −C ν ∗ antineutrinos are described by CP-conjugated fields: ⇒ C = Particle ⇆ Antiparticle ⇒ P = Left-Handed ⇆ Righ-Handed � � CP → ν CP U ∗ αk ν CP Fields: ν αL = U αk ν kL − − αL = kL k k � � CP U ∗ States: | ν α � = αk | ν k � − − → | ¯ ν α � = U αk | ¯ ν k � k k U ∗ NEUTRINOS U ANTINEUTRINOS ⇆ � � � � ∆ m 2 kj L | U αk | 2 | U βk | 2 + 2Re U ∗ αk U βk U αj U ∗ P ν α → ν β ( L, E ) = βj exp − i 2 E k k>j � � ∆ m 2 � � kj L | U αk | 2 | U βk | 2 + 2Re U αk U ∗ βk U ∗ P ¯ ν β ( L, E ) = αj U βj exp − i ν α → ¯ 2 E k k>j C. Giunti , Neutrino Mixing and Oscillations − 63

CPT Symmetry CPT P ν α → ν β − − − → P ¯ ν β → ¯ ν α A CPT = P ν α → ν β − P ¯ CPT Asymmetries: ν β → ¯ ν α αβ A CPT ⇒ Local Quantum Field Theory = = 0 CPT Symmetry αβ � � � � ∆ m 2 kj L | U αk | 2 | U βk | 2 + 2Re U ∗ αk U βk U αj U ∗ indeed, P ν α → ν β ( L, E ) = βj exp − i 2 E k k>j U ∗ is invariant under CPT: U ⇆ α ⇆ β P ν α → ν β = P ¯ ν β → ¯ ν α in particular P ν α → ν α = P ¯ (solar ν e , reactor ¯ ν e , accelerator ν µ ) ν α → ¯ ν α C. Giunti , Neutrino Mixing and Oscillations − 64

CP Symmetry CP − − → P ν α → ν β P ¯ ν α → ¯ ν β A CP A CP αβ = − A CP CP Asymmetries: αβ = P ν α → ν β − P ¯ CPT ⇒ ν α → ¯ ν β βα ∆ m 2 ∆ m 2 ! ! kj L kj L X X A CP U ∗ αk U βk U αj U ∗ U αk U ∗ βk U ∗ − i − 2Re − i αβ ( L, E ) = 2Re βj exp αj U βj exp 2 E 2 E k>j k>j � � � ∆ m 2 kj L A CP αβ ( L, E ) = 4 J αβ ; kj sin 2 E k>j � � Jarlskog rephasing ( U αk → e iλ α U αk e iη k ) invariants: U ∗ αk U βk U αj U ∗ J αβ ; kj = Im βj violation of CP symmetry depends only on Dirac phases (three neutrinos: J αβ ; kj = ± c 12 s 12 c 23 s 23 c 2 13 s 13 sin δ 13 ) � � A CP = 0 = ⇒ observation of CP violation needs measurement of oscillations αβ C. Giunti , Neutrino Mixing and Oscillations − 65

T Symmetry T P ν α → ν β − → P ν β → ν α A T T Asymmetries: αβ = P ν α → ν β − P ν β → ν α 0 = A CPT ⇒ = P ν α → ν β − P ¯ CPT = ν β → ¯ ν α αβ = P ν α → ν β − P ν β → ν α + P ν β → ν α − P ¯ ν β → ¯ ν α = A T αβ + A CP βα = A T αβ − A CP A T αβ = A CP ⇒ = αβ αβ � � � ∆ m 2 kj L A T αβ ( L, E ) = 4 J αβ ; kj sin 2 E k>j violation of T symmetry depends only on Dirac phases � � A T ⇒ = 0 = observation of T violation needs measurement of oscillations αβ C. Giunti , Neutrino Mixing and Oscillations − 66

Two Generations ( k = 1 , 2 )    cos ϑ sin ϑ ∆ m 2 ≡ ∆ m 2 21 ≡ m 2 2 − m 2  U = 1 − sin ϑ cos ϑ � ∆ m 2 L � P ν α → ν β ( L, E ) = sin 2 2 ϑ sin 2 Transition Probability ( α � = β ) : 4 E Survival Probability ( α = β ) : P ν α → ν α ( L, E ) = 1 − P ν α → ν β ( L, E ) � P ν α → ν β � = 1 2 sin 2 2 ϑ Averaged Transition Probability: C. Giunti , Neutrino Mixing and Oscillations − 67

TYPES OF EXPERIMENTS observable if „ ∆ m 2 L « Two-Neutrino P ν α → ν β ( L, E ) = sin 2 2 ϑ sin 2 ⇒ ∆ m 2 L 4 E Mixing � 1 4 E SBL (high statistics) Reactor SBL: L ∼ 10 m , E ∼ 1 MeV ∆ m 2 � 0 . 1 eV 2 L/E � 1 eV − 2 ⇒ Accelerator SBL: L ∼ 1 km , E � 1 GeV = ATM & LBL Reactor LBL: L ∼ 1 km , E ∼ 1 MeV CHOOZ, PALO VERDE L/E � 10 4 eV − 2 Accelerator LBL: L ∼ 10 3 km , E � 1 GeV K2K, MINOS, CNGS L ∼ 10 2 − 10 4 km , E ∼ 0 . 1 − 10 2 GeV ⇓ Atmospheric: ∆ m 2 � 10 − 4 eV 2 Kamiokande, IMB, Super-Kamiokande, Soudan, MACRO L ∼ 10 8 km , E ∼ 0 . 1 − 10 MeV SUN Homestake, Kamiokande, GALLEX, SAGE, L E ∼ 10 11 eV − 2 ∆ m 2 � 10 − 11 eV 2 ⇒ = Super-Kamiokande, GNO, SNO 10 − 8 eV 2 � ∆ m 2 � 10 − 4 eV 2 10 − 4 � sin 2 2 ϑ � 1 Matter Effect (MSW) = ⇒ C. Giunti , Neutrino Mixing and Oscillations − 68

MSW effect (resonant transitions in matter) � U ∗ a flavor neutrino ν α with momentum p is described by | ν α ( p ) � = αk | ν k ( p ) � k � p 2 + m 2 H 0 | ν k ( p ) � = E k | ν k ( p ) � E k = k H = H 0 + H I H I | ν α ( p ) � = V α | ν α ( p ) � in matter V α = effective potential due to coherent interactions with medium forward elastic CC and NC scattering C. Giunti , Neutrino Mixing and Oscillations − 69

� � e � ; � ; � � ; � ; � e e � � e � � EFFECTIVE POTENTIAL IN MATTER W Z � � � e � e ; p; n e ; p; n e √ √ 2 V ( e − ) = − V ( p ) V NC = V ( n ) ⇒ NC = − V CC = 2 G F N e 2 G F N n NC NC ⇒ V e − V µ = V CC V e = V CC + V NC V µ = V τ = V NC (common phase) = V CC = − V CC V NC = − V NC antineutrinos: C. Giunti , Neutrino Mixing and Oscillations − 70

i d Schr¨ odinger picture: d t | ν α ( p, t ) � = H| ν α ( p, t ) � , | ν α ( p, 0) � = | ν α ( p ) � ϕ αβ ( p, t ) = � ν β ( p ) | ν α ( p, t ) � , flavor transition amplitudes: ϕ αβ ( p, 0) = δ αβ i d d t ϕ αβ ( p, t ) = � ν β ( p ) |H| ν α ( p, t ) � = � ν β ( p ) |H 0 | ν α ( p, t ) � + � ν β ( p ) |H I | ν α ( p, t ) � � � ν β ( p ) |H 0 | ν α ( p, t ) � = � ν β ( p ) |H 0 | ν ρ ( p ) � � ν ρ ( p ) | ν α ( p, t ) � � �� � ρ ϕ αρ ( p, t ) � � U ∗ U βk � ν k ( p ) |H 0 | ν j ( p ) � = ρj ϕ αρ ( p, t ) � �� � ρ k,j δ kj E k � � ν β ( p ) |H I | ν α ( p, t ) � = � ν β ( p ) |H I | ν ρ ( p ) � ϕ αρ ( p, t ) = V β ϕ αβ ( p, t ) � �� � ρ δ βρ V β �� � � i d U βk E k U ∗ d t ϕ αβ = ρk + δ βρ V β ϕ αρ ρ k C. Giunti , Neutrino Mixing and Oscillations − 71

E k = p + m 2 k ultrarelativistic neutrinos: E = p t = x 2 E V e = V CC + V NC V µ = V τ = V NC �� � � m 2 i d 2 E U ∗ k d x ϕ αβ ( p, x ) = ( p + V NC ) ϕ αβ ( p, x ) + U βk ρk + δ βe δ ρe V CC ϕ αρ ( p, x ) ρ k R x 0 V NC ( x ′ ) d x ′ ψ αβ ( p, x ) = ϕ αβ ( p, x ) e ipx + i ⇓ 0 V NC ( x ′ ) d x ′ � � R x i d − p − V NC + i d d x ψ αβ = e ipx + i ϕ αβ d x �� � � m 2 i d 2 E U ∗ k d x ψ αβ = U βk ρk + δ βe δ ρe V CC ψ αρ ρ k P ν α → ν β = | ϕ αβ | 2 = | ψ αβ | 2 C. Giunti , Neutrino Mixing and Oscillations − 72

evolution of flavor transition amplitudes in matrix form � � i d d x Ψ α = 1 U M 2 U † + A Ψ α 2 E � � � ψ αe � � A CC 0 0 � m 2 0 0 1 M 2 = A CC = 2 EV CC Ψ α = A = m 2 √ ψ αµ 0 0 0 0 0 2 = 2 2 EG F N e 0 0 0 ψ ατ m 2 0 0 3 effective effective VAC = U M 2 U † U M 2 U † + 2 E V matter M 2 = M 2 mass-squared mass-squared − − − − → MAT matrix matrix ↑ in vacuum in matter potential due to coherent forward elastic scattering � cos ϑ � sin ϑ simplest case: ν e → ν µ transitions with U = (two-neutrino mixing) − sin ϑ cos ϑ � � � � = 1 + 1 cos 2 ϑm 2 1 +sin 2 ϑm 2 2 cos ϑ sin ϑ ( m 2 2 − m 2 1 ) U M 2 U † = − ∆ m 2 cos2 ϑ ∆ m 2 sin2 ϑ 2 Σ m 2 ∆ m 2 sin2 ϑ ∆ m 2 cos2 ϑ cos ϑ sin ϑ ( m 2 2 − m 2 1 ) sin 2 ϑm 2 1 +cos 2 ϑm 2 2 2 ↑ irrelevant common phase Σ m 2 ≡ m 2 ∆ m 2 ≡ m 2 1 + m 2 2 − m 2 2 1 C. Giunti , Neutrino Mixing and Oscillations − 73

       − ∆ m 2 cos2 ϑ + 2 A CC ∆ m 2 sin2 ϑ  ψ ee  ψ ee i d  = 1   ∆ m 2 sin2 ϑ ∆ m 2 cos2 ϑ d x 4 E ψ eµ ψ eµ      ψ ee (0)  1  =  ⇒ initial ν e = ψ eµ (0) 0 P ν e → ν µ ( x ) = | ψ eµ ( x ) | 2 P ν e → ν e ( x ) = | ψ ee ( x ) | 2 = 1 − P ν e → ν µ ( x ) tan 2 ϑ ⇒ Effective Mixing Angle in Matter: Diagonalization = tan 2 ϑ M = A CC 1 − ∆ m 2 cos 2 ϑ e = ∆ m 2 cos 2 ϑ CC = ∆ m 2 cos 2 A R N R ⇒ √ Resonance ( ϑ M = π/ 4 ): ϑ = 2 2 EG F � ϑ − A CC ) 2 + (∆ m 2 sin 2 (∆ m 2 cos 2 ϑ ) 2 ∆ m 2 Effective Squared-Mass Difference: M = C. Giunti , Neutrino Mixing and Oscillations − 74

R N = N A e 90 � ' � � ' � e 2 � 1 80 70 60 50 40 30 � 4 # = 10 20 10 � ' � � ' � e 1 � 2 ν e = cos ϑ M ν 1 + sin ϑ M ν 2 0 0 20 40 60 80 100 M � 3 # N = N ( m ) e A ν µ = − sin ϑ M ν 1 + cos ϑ M ν 2 R N = N A e 14 � 2 � e 12 tan 2 ϑ 10 tan 2 ϑ M = A CC 1 − � � 8 � 1 ∆ m 2 cos 2 ϑ � � 2 � 6 4 � 1 2 » ` 2 � 6 � 3 ) � � m = 7 � 10 eV , # = 10 2 e 2 ∆ m 2 cos 2 ´ 2 ∆ m 2 eV ϑ − A CC M = 0 � 6 0 20 40 60 80 100 – 1 / 2 (10 � 3 ` ∆ m 2 sin 2 ´ 2 N = N ( m ) e A + ϑ M 2 � m C. Giunti , Neutrino Mixing and Oscillations − 75

       ψ ee  cos ϑ M sin ϑ M  ψ 1  =   − sin ϑ M ψ eµ cos ϑ M ψ 2       �   � − i d ϑ M 0  − ∆ m 2  ψ 1 0  ψ 1 i d A CC + 1   M d x  =  +    i d ϑ M d x 4 E 4 E ∆ m 2 ψ 2 0 ψ 2 0 M d x ↑ irrelevant common phase ↑ maximum near resonance          cos ϑ 0 − sin ϑ 0  cos ϑ 0  ψ 1 (0)  1 M M M  =   =  sin ϑ 0 cos ϑ 0 sin ϑ 0 ψ 2 (0) 0 M M M Z x R Z x R » „ « „ « – ∆ m 2 M ( x ′ ) ∆ m 2 M ( x ′ ) cos ϑ 0 d x ′ A R 11 + sin ϑ 0 d x ′ A R ψ 1 ( x ) ≃ M exp i M exp − i 21 4 E 4 E 0 0 Z x „ « ∆ m 2 M ( x ′ ) d x ′ × exp i 4 E x R Z x R Z x R » „ « „ « – ∆ m 2 ∆ m 2 M ( x ′ ) M ( x ′ ) cos ϑ 0 d x ′ A R 12 + sin ϑ 0 d x ′ A R ψ 2 ( x ) ≃ − i M exp i M exp 22 4 E 4 E 0 0 Z x „ « ∆ m 2 M ( x ′ ) d x ′ × exp − i 4 E x R C. Giunti , Neutrino Mixing and Oscillations − 76

ψ ee ( x ) = cos ϑ x M ψ 1 ( x ) + sin ϑ x M ψ 2 ( x ) neglect phases (averaged over energy spectrum) 11 | 2 + cos 2 ϑ x P ν e → ν e ( x ) = |� ψ ee ( x ) �| = cos 2 ϑ x M cos 2 ϑ 0 M |A R M sin 2 ϑ 0 M |A R 21 | 2 12 | 2 + sin 2 ϑ x + sin 2 ϑ x M cos 2 ϑ 0 M |A R M sin 2 ϑ 0 M |A R 22 | 2 11 | 2 = |A R 22 | 2 = 1 − P c 12 | 2 = |A R 21 | 2 = P c |A R |A R crossing probability � 1 � P ν e → ν e ( x ) = 1 cos2 ϑ 0 M cos2 ϑ x 2 + 2 − P c [Parke, PRL 57 (1986) 1275] M C. Giunti , Neutrino Mixing and Oscillations − 77

CROSSING PROBABILITY � � � � − π − π F P c = exp 2 γF − exp 2 γ sin 2 ϑ � � [Kuo, Pantaleone, PRD 39 (1989) 1930] − π F 1 − exp 2 γ sin 2 ϑ � ∆ m 2 sin 2 2 ϑ γ = ∆ m 2 � M / 2 E � adiabaticity parameter: = � � � � d ln A CC � 2 | d ϑ M / d x | 2 E cos2 ϑ R d x R A ∝ x F = 1 (Landau-Zener approximation) [Parke, PRL 57 (1986) 1275] � � 2 / � � 1 − tan 2 ϑ 1 + tan 2 ϑ A ∝ 1 /x F = [Kuo, Pantaleone, PRD 39 (1989) 1930] F = 1 − tan 2 ϑ A ∝ exp ( − x ) [Pizzochero, PRD 36 (1987) 2293, Toshev, PLB 196 (1987) 170, Petcov, PLB 200 (1988) 373] [Kuo, Pantaleone, RMP 61 (1989) 937] C. Giunti , Neutrino Mixing and Oscillations − 78

� � − x R ⊙ N e ( x ) ≃ N c N c e = 245 N A / cm 3 SUN: e exp x 0 = x 0 10 . 54 � 1 � ν e → ν e = 1 sun cos2 ϑ 0 2 − P c P 2 + M cos2 ϑ � � � � − π − π F − exp P c = exp 2 γF 2 γ sin 2 ϑ � � − π F 1 − exp 2 γ sin 2 ϑ ∆ m 2 sin 2 2 ϑ � � γ = � d ln A CC � 2 E cos2 ϑ d x R F = 1 − tan 2 ϑ √ A CC = 2 2 EG F N e   numerical | d ln A CC / d x | R for x ≤ 0 . 904 R ⊙ practical prescription: | d ln A CC / d x | R → 18 . 9  for x > 0 . 904 R ⊙ [Lisi et al., PRD 63 (2001) 093002] R ⊙ C. Giunti , Neutrino Mixing and Oscillations − 79

� � � � sun ν 2 → ν e − sin 2 ϑ P earth 1 − 2 P ν e → ν e sun P sun+earth Earth Matter Effect: = P ν e → ν e + ν e → ν e cos2 ϑ [Mikheev, Smirnov, Sov. Phys. Usp. 30 (1987) 759], [Baltz, Weneser, PRD 35 (1987) 528] 14 12 (A) 10 ρ (g/cm 3 ) 8 P earth ν 2 → ν e is usually calculated numerically ap- 6 proximating the Earth density profile with a 4 Data step function. 2 Our approximation 0 (B) 6 Effective massive neutrinos propagate as 5 N e / N A (cm − 3 ) plane waves in regions of constant density. 4 3 Wave functions of flavor neutrinos are joined 2 Data 1 at the boundaries of steps. Our approximation 0 0 1000 2000 3000 4000 5000 6000 r (Km) [Giunti, Kim, Monteno, NP B 521 (1998) 3] C. Giunti , Neutrino Mixing and Oscillations − 80

∆ m 2 ∼ 5 × 10 − 5 eV 2 , tan 2 ϑ ∼ 0 . 8 LMA (Large Mixing Angle): ∆ m 2 ∼ 7 × 10 − 8 eV 2 , tan 2 ϑ ∼ 0 . 6 LOW (LOW ∆ m 2 ): ∆ m 2 ∼ 5 × 10 − 6 eV 2 , tan 2 ϑ ∼ 10 − 3 SMA (Small Mixing Angle): ∆ m 2 ∼ 10 − 9 eV 2 , tan 2 ϑ ∼ 1 QVO (Quasi-Vacuum Oscillations): ∆ m 2 � 5 × 10 − 10 eV 2 , tan 2 ϑ ∼ 1 VAC (VACuum oscillations): 10 4 - LMA SMA 5 10 - 6 10 - ∆ m (eV ) 2 2 LOW 10 - 7 10 - 8 10 - 9 VAC 10 - 10 0.001 0.01 0.1 1 10 tan 2 θ [de Gouvea, Friedland, Murayama, PLB 490 (2000) 125] [Bahcall, Krastev, Smirnov, JHEP 05 (2001) 015] C. Giunti , Neutrino Mixing and Oscillations − 81

90 80 70 60 ∆ m 2 = 5 × 10 − 6 eV 2 50 solid line: 40 tan 2 ϑ = 5 × 10 − 4 (typical SMA) 30 20 ∆ m 2 = 7 × 10 − 5 eV 2 dashed line: M 10 # tan 2 ϑ = 0 . 4 0 (typical LMA) � 4 � 3 � 2 � 1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 � 3 N = N [ m ℄ A ∆ m 2 = 8 × 10 − 8 eV 2 dash-dotted line: � 2 � 3 10 10 tan 2 ϑ = 0 . 7 � 4 10 (typical LOW) � 3 10 � 5 10 � 5 10 � 6 10 � 4 10 � 7 10 � 5 10 � 8 10 � 9 ℄ ℄ ℄ 10 2 2 2 [eV [eV � 6 [eV 10 � 6 � 10 10 10 2 2 2 m m m � 7 � 11 10 10 0 1 0 1 2 3 4 � 4 � 3 � 2 � 1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 � 3 � 3 � 3 N = N [ m ℄ N = N [ m ℄ N = N [ m ℄ A A A typical SMA typical LMA typical LOW C. Giunti , Neutrino Mixing and Oscillations − 82

[Bahcall, Krastev, Smirnov, PRD 58 (1998) 096016] [Bahcall, Krastev, Smirnov, JHEP 05 (2001) 015] ∆ m 2 = 5 . 0 × 10 − 6 eV2 sin22 ϑ = 3 . 5 × 10 − 3 ∆ m 2 = 4 . 2 × 10 − 5 eV2 tan2 ϑ = 0 . 26 SMA: LMA: ∆ m 2 = 1 . 6 × 10 − 5 eV2 ∆ m 2 = 5 . 2 × 10 − 6 eV2 tan2 ϑ = 5 . 5 × 10 − 4 sin22 ϑ = 0 . 57 LMA: SMA: ∆ m 2 = 7 . 9 × 10 − 8 eV2 sin22 ϑ = 0 . 95 ∆ m 2 = 7 . 6 × 10 − 8 eV2 tan2 ϑ = 0 . 72 LOW: LOW: Just So 2 : ∆ m 2 = 5 . 5 × 10 − 12 eV2 tan2 ϑ = 1 . 0 ∆ m 2 = 1 . 4 × 10 − 10 eV2 tan2 ϑ = 0 . 38 VAC: C. Giunti , Neutrino Mixing and Oscillations − 83

IN NEUTRINO OSCILLATIONS DIRAC ∼ MAJORANA � � d ν α d t = 1 UM 2 U † + 2 EV Evolution of Amplitudes: αβ ν β 2 E   U (D) Dirac: difference: U (M) = U (D) D ( λ )  Majorana:   1 0 ··· 0 0 e iλ 21 ··· 0 D † = D − 1   D ( λ ) = . . . ⇒ ... . . . . . . ··· e iλN 1 0 0   m 2 0 ··· 0 1 m 2 0 2 ··· 0   M 2 = DM 2 = M 2 D DM 2 D † = M 2 ⇒ ⇒ = =   . . . . ... . . . . . ··· m 2 0 0 N U (M) M 2 ( U (M) ) † = U (D) DM 2 D † ( U (D) ) † = U (D) M 2 ( U (D) ) † C. Giunti , Neutrino Mixing and Oscillations − 84

1 AVERAGE OVER ENERGY SPECTRUM � ∆ m 2 L � � � ∆ m 2 L �� 0.8 = 1 P ν α → ν β ( L, E ) = sin 2 2 ϑ sin 2 2 sin 2 2 ϑ 1 − cos ( α � = β ) 0.6 4 E 2 E 0.4 0.2 0 � ! � 100 1000 10000 100000 � L (km) � P ∆ m 2 = 10 − 3 eV sin 2 2 ϑ = 1 � E � = 1 GeV ∆ E = 0 . 2 GeV � � ∆ m 2 L � � � � P ν α → ν β ( L, E ) � = 1 2 sin 2 2 ϑ 1 − cos φ ( E ) d E ( α � = β ) 2 E C. Giunti , Neutrino Mixing and Oscillations − 85 1

� � ∆ m 2 L � � � � P ν α → ν β ( L, E ) � = 1 2 sin 2 2 ϑ 1 − ( α � = β ) cos φ ( E ) d E 2 E 2 P max 1 sin 2 2 ϑ ≤ ν α → ν β � P ν α → ν β ( L, E ) � ≤ P max ⇒ experiment: = � � ∆ m 2 L � ν α → ν β 0.8 1 − cos φ ( E ) d E EX CLUDED REGION 2 E 0.6 � 1 0.4 10 0.2 0 2 # � 4 � 3 � 2 � 1 10 10 10 10 2 2 − − − − − → sin � m (eV) � 2 rotate REGION 10 and mirror (eV) 2 CLUDED � m 1 � 3 10 EX � 4 10 0 0.2 0.4 0.6 0.8 1 sin 2 # 2 C. Giunti , Neutrino Mixing and Oscillations − 86 1

Summary of Part 2: Neutrino Oscillations in Vacuum and in Matter detectable neutrinos are extremely relativistic ⇓ ( ∆ m 2 standard expression for the neutrino oscillation probabilities kj , U αk ) Neutrino Oscillations can test CPT, CP, T symmetries Matter Effects are important for Solar neutrinos and VLBL experiments in Neutrino Oscillations Dirac ∼ Majorana average over energy spectrum ⇓ constant flavor changing probability C. Giunti , Neutrino Mixing and Oscillations − 87

Part 3: Experimental Results and Theoretical Implications C. Giunti , Neutrino Mixing and Oscillations − 88

Neutrino Fluxes LAMPF = Los Alamos WANF = CERN CNGS = CERN → Gran Sasso [A. Geiser, Rept. Prog. Phys. 63 (2000) 1779] C. Giunti , Neutrino Mixing and Oscillations − 89

SOLAR NEUTRINOS Extreme ultraviolet Imaging Telescope (EIT) 304 ˚ A images of the Sun emission in this spectral line (He II) shows the upper chromosphere at a temperature of about 60,000 K [The Solar and Heliospheric Observatory (SOHO), http://sohowww.nascom.nasa.gov/] C. Giunti , Neutrino Mixing and Oscillations − 90

2 + � 2 ( pp ) p + p ! H + e + � p + e + p ! H + � ( pep ) e e X � X � X � X � 99.6% X � 0.4% X � X � X Standard Solar Model (SSM) � X � X � X � X � ? 2 3 H + p ! He + � pp and CNO cycles 12 13 13 13 + 13 - � X C + p ! N + � N ! C + e + � ( N) � X e � X � X � X � X � X � X � X 6 � X � X � X � X � 5 4 p + 2 e − → 4 He + 2 ν e + 26 . 731 MeV � X � X 85% 2 � 10 % ? � X �� - � � X X 15 12 4 13 14 ? N + p ! C + He C + p ! N + � CN ? ? 6 �� � 3 3 4 3 4 + 6 He + He ! He + 2 p He + p ! He + e + � e 99 : 9% ? 15% ? ( hep ) 15 pp I 15 15 + � 14 15 ( O) O ! N + e + � N + p ! O + � e 3 4 7 He + He ! Be + � 6 0 : 1% ? � P � P � P 15 16 17 14 4 � P N + p ! O + � O + p ! N + He � P � P � P 99.87% 0.13% � P � P 6 ? ? ? 7 7 8 7 � 7 ( Be) 16 17 17 Be + e ! Li + � Be + p ! B + � - 17 17 + e O + p ! F + � F ! O + e + � ( F) e ? ? 7 4 � 8 8 8 + ( B) Li + p ! 2 He B ! Be + e + � e pp I I ? � 8 4 Be ! 2 He pp I I I Current SSM: BP2000 [Bahcall, Pinsonneault, Basu, AJ 555 (2001) 990] [J.N. Bahcall, http://www.sns.ias.edu/˜jnb] C. Giunti , Neutrino Mixing and Oscillations − 91

[J.N. Bahcall, http://www.sns.ias.edu/~jnb ] C. Giunti , Neutrino Mixing and Oscillations − 92

[Castellani, Degl’Innocenti, Fiorentini, Lissia, Ricci, Phys. Rept. 281 (1997) 309, astro-ph/9606180] C. Giunti , Neutrino Mixing and Oscillations − 93

[Castellani, Degl’Innocenti, Fiorentini, Lissia, Ricci, Phys. Rept. 281 (1997) 309, astro-ph/9606180] C. Giunti , Neutrino Mixing and Oscillations − 94

[J.N. Bahcall, http://www.sns.ias.edu/~jnb ] predicted versus measured sound speed the rms fractional difference between the calculated and the measured sound speeds is 0.10% for all solar radii between between 0 . 05 R ⊙ and 0 . 95 R ⊙ and is 0.08% for the deep interior region, r < 0 . 25 R ⊙ , in which neutrinos are produced C. Giunti , Neutrino Mixing and Oscillations − 95

HOMESTAKE ν e + 37 Cl → 37 Ar + e − radiochemical experiment [Pontecorvo (1946), Alvarez (1949)] ⇒ Φ µ ≃ 4 m − 2 day − 1 Homestake Gold Mine (South Dakota), 1478 m deep, 4200 m.w.e. = steel tank, 6.1 m diameter, 14.6 m long ( 6 × 10 5 liters) 615 tons of tetrachloroethylene ( C 2 Cl 4 ), 2 . 16 × 10 30 atoms of 37 Cl (133 tons) energy threshold: E Cl ⇒ 8 B , 7 Be , pep , hep , 13 N , 15 O , 17 F th = 0 . 814 MeV = R exp Cl /R SSM ⇒ = 0 . 34 ± 0 . 03 1970–1994, 108 extractions = [APJ 496 (1998) 505] Cl R exp R SSM = 7 . 6 +1 . 3 Cl = 2 . 56 ± 0 . 23 SNU − 1 . 1 SNU Cl 1 SNU = 10 − 36 events atom − 1 s − 1 C. Giunti , Neutrino Mixing and Oscillations − 96

GALLIUM EXPERIMENTS SAGE, GALLEX, GNO ν e + 71 Ga → 71 Ge + e − radiochemical experiments [Kuzmin (1965)] threshold: E Ga ⇒ pp , 7 Be , 8 B , pep , hep , 13 N , 15 O , 17 F th = 0 . 233 MeV = R exp Ga /R SSM ⇒ = 0 . 56 ± 0 . 03 SAGE+GALLEX+GNO = Ga R exp R SSM = 128 +9 Ga = 72 . 4 ± 4 . 7 SNU − 7 SNU Ga C. Giunti , Neutrino Mixing and Oscillations − 97

SAGE: Soviet-American Gallium Experiment Baksan Neutrino Observatory, northern Caucasus, 3.5 km from entrance of horizontal adit ⇒ Φ µ ≃ 2 . 6 m − 2 day − 1 50 tons of metallic 71 Ga , 2000 m deep, 4700 m.w.e. = detector test: 51 Cr Source: R = 0 . 95 +0 . 11 +0 . 06 [PRC 59 (1999) 2246] − 0 . 10 − 0 . 05 R SAGE /R SSM ⇒ = 0 . 54 ± 0 . 05 1990 – 2001 = [astro-ph/0204245] Ga Ga R SAGE = 70 . 8 +6 . 5 R SSM = 128 +9 − 6 . 1 SNU − 7 SNU Ga Ga 400 L+K peaks K peak only 300 Capture rate (SNU) All runs combined 200 100 L K 0 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean extraction time C. Giunti , Neutrino Mixing and Oscillations − 98

GALLium EXperiment (GALLEX) Gran Sasso Underground Laboratory, Italy, overhead shielding: 3300 m.w.e. 30.3 tons of gallium in 101 tons of gallium chloride (GaCl 3 -HCl) solution R GALLEX /R SSM May 1991 – Jan 1997 = ⇒ = 0 . 61 ± 0 . 06 [PLB 477 (1999) 127] Ga Ga Gallium Neutrino Observatory (GNO) continuation of GALLEX, GNO30: 30.3 tons of gallium R GNO /R SSM May 1998 – Jan 2000 = ⇒ = 0 . 51 ± 0 . 08 [PLB 490 (2000) 16] Ga Ga R G+G Ga = 0 . 58 ± 0 . 05 R SSM Ga C. Giunti , Neutrino Mixing and Oscillations − 99

Kamiokande ν + e − → ν + e − water Cherenkov detector Sensitive to ν e , ν µ , ν τ , but σ ( ν e ) ≃ 6 σ ( ν µ,τ ) Kamioka mine (200 km west of Tokyo), 1000 m underground, 2700 m.w.e. 3000 tons of water, 680 tons fiducial volume, 948 PMTs threshold: E Kam ⇒ 8 B , hep ≃ 6 . 75 MeV = th R Kam Jan 1987 – Feb 1995 (2079 days) = ⇒ = 0 . 55 ± 0 . 08 νe [PRL 77 (1996) 1683] R SSM νe Super-Kamiokande continuation of Kamiokande, 50 ktons of water, 22.5 ktons fiducial volume, 11146 PMTs threshold: E Kam ⇒ 8 B , hep ≃ 4 . 75 MeV = th R SK 1996 – 2001 (1496 days) = ⇒ = 0 . 465 ± 0 . 015 νe [SK, PLB 539 (2002) 179] R SSM νe C. Giunti , Neutrino Mixing and Oscillations − 100