Diagonalization of fermion mass matrices I. Dirac fermions (e.g. charged leptons): N f R m ′† Ψ ′ � ab ¯ bR + h.c. = ¯ R + ¯ m ′ Ψ ′ aL Ψ ′ Ψ ′ L m ′ Ψ ′ Ψ ′ −L m = L a,b =1 Rotate Ψ ′ L and Ψ ′ R by unitary transformations: m = V † Ψ ′ Ψ ′ L m ′ V R = diag. L = V L Ψ L , R = V R Ψ R ; Diagonalized mass term: N f −L m = ¯ Ψ L ( V † � m i ¯ L m ′ V R )Ψ R + h.c. = Ψ iL Ψ Ri + h.c. i =1 Mass eigenstate fields: N f � m i ¯ Ψ i = Ψ iL + Ψ iR ; −L m = Ψ i Ψ i i =1 Invariant w.r.t. U (1) transfs. Ψ i → e iα i Ψ i – conservs individual ferm. numbers Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 11
Diagonalization of fermion mass matrices II. Majorana fermions: N f L m = − 1 bL + h.c. = 1 T C − 1 m ′ Ψ ′ aL ) c Ψ ′ � m ′ 2Ψ ′ ab (Ψ ′ L + h.c. L 2 a,b =1 Matrix m ′ is symmetric: m ′ T = m ′ . ⋄ Problem: prove this. Unitary transformation of Ψ ′ L : L m ′ U L = diag. Ψ ′ m = U T L = U L Ψ L , Diagonalized mass term: N f L m = 1 L m ′ U L )Ψ L + h.c. = 1 Li C − 1 Ψ Li + h.c. � 2[Ψ T L C − 1 ( U T m i Ψ T 2 i =1 Mass eigenstate fields: N f L m = − 1 � χ i = Ψ iL + (Ψ iL ) c ; m i ¯ χ i χ i 2 i =1 Not invariant w.r.t. U (1) transfs. Ψ Li → e iα i Ψ Li Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 12
Neutrino masses and lepton flavour violation For Dirac neutrinos the relevant terms in the Lagrangian are g La γ µ ν ′ e ′ La ) W − µ + ( m ′ e ′ Ra e ′ Lb + ( m ′ ν ′ Ra ν ′ −L w + m = √ 2(¯ l ) ab ¯ ν ) ab ¯ Lb + h.c. Diagonalization of mass matrices: e ′ e ′ ν ′ ν ′ L = V L e L , R = V R e R , L = U L ν L , R = U R ν R V † U † L m ′ L m ′ l V R = m l , ν U R = m ν ( m l,ν − diagonal mass matrices) g e L γ µ V † L U L ν L ) W − √ −L w + m = (¯ + diag . mass terms + h.c. µ 2 For m ′ ν = 0 : without loss of generality one can consider both CC term and m l term diagonal ⇒ the Lagrangian is invariant w.r.t. three separate U (1) transformations: e La,Ra → e iφ a e La,Ra , ν La,Ra → e iφ a ν La,Ra ♦ ( a = e, µ, τ ) Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 13
Neutrino masses and lepton flavour violation ⇒ For massles neutrinos three individual lepton numbers (lepton flavours) L e , L µ , L τ conserved. For massive Dirac neutrinos L e , L µ , L τ are violated ⇒ ν oscillations and µ → eγ , µ → 3 e , etc. allowed. But: the total lepton number L = L e + L µ + L τ is conserved. For massive Majorana neutrinos: individual lepton flavours L e , L µ , L τ and the total lepton number L are violated. In addition to neutrino oscillations and LFV decays 2 β 0 ν decay ( ∆ L = 2 process) is allowed. Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 14
Why are neutrinos so light ? In the minimal SM: m ν = 0 . Add 3 RH ν ’s N Ri : ν Li −L Y ⊃ Y ν ¯ l L N R H + h.c., l Li = e Li � H 0 � = v = 174 GeV ⇒ m ν = m D = Y ν v Y ν < 10 − 11 – Not natural ! m ν < 1 eV ⇒ Is it a problem? Y e ≃ 3 × 10 − 6 . But: with m ν � = 0 , huge disparity between the masses within each fermion generation ! A simple and elegant mechanism – seesaw (Minkowski, 1977; Gell-Mann, Ramond & Slansky, 1979; Yanagida, 1979; Glashow, 1979; Mohapatra & Senjanovi´ c, 1980) Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 15
Heavy N Ri ’s make ν Li ’s light : H + 1 −L Y + m = Y ν ¯ l L N R ˜ 2 M R N R N R + h.c., In the n L = ( ν L , ( N R ) c ) T basis: −L m = 1 2 n T L C M ν n L + h.c. , m T 0 D M ν = m D M R N Ri are EW singlets ⇒ M R can be ∼ M GUT ( M I ) ≫ m D ∼ v . Block diagonalization: M N ≃ M R , m ν ∼ (174 GeV) 2 D M − 1 m ν L ≃ − m T ♦ R m D ⇒ M R For m ν � 0 . 05 eV ⇒ M R � 10 15 GeV ∼ M GUT ∼ 10 16 GeV ! Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 16
The (type I) seesaw mechanism Consider the case of n LH and k RH neutrino fields: L m = 1 L + 1 L C − 1 m L ν ′ R C − 1 M ∗ 2 ν ′ T 2 N ′ T R m D ν ′ R N ′ L − N ′ R + h.c. m L and M R – n × n and k × k symmetric matrices, m D – an k × n matrix.
The (type I) seesaw mechanism Consider the case of n LH and k RH neutrino fields: L m = 1 L + 1 L C − 1 m L ν ′ R C − 1 M ∗ 2 ν ′ T 2 N ′ T R m D ν ′ R N ′ L − N ′ R + h.c. m L and M R – n × n and k × k symmetric matrices, m D – an k × n matrix. Introduce an n + k - component LH field ν ′ ν ′ L L = n L = ⇒ R ) c N ′ c ( N ′ L
The (type I) seesaw mechanism Consider the case of n LH and k RH neutrino fields: L m = 1 L + 1 L C − 1 m L ν ′ R C − 1 M ∗ 2 ν ′ T 2 N ′ T R m D ν ′ R N ′ L − N ′ R + h.c. m L and M R – n × n and k × k symmetric matrices, m D – an k × n matrix. Introduce an n + k - component LH field ν ′ ν ′ L L = n L = ⇒ R ) c N ′ c ( N ′ L L m = 1 2 n T L C − 1 M n L + h.c. , where m T m L D M = ( M : matrix ( n + k ) × ( n + k )) m D M R Problem: prove these formulas. Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 17
Block-diagonalization of M m T m L ˜ m L 0 V T M V = V T D V = n L = V χ ′ L , ˜ m D M R 0 M R
Block-diagonalization of M m T m L ˜ m L 0 V T M V = V T D V = n L = V χ ′ L , ˜ m D M R 0 M R Look for the unitary matrix V in the form � 1 − ρρ † ρ V = ( ρ : matrix n × k ) � − ρ † 1 − ρ † ρ
Block-diagonalization of M m T m L ˜ m L 0 V T M V = V T D V = n L = V χ ′ L , ˜ m D M R 0 M R Look for the unitary matrix V in the form � 1 − ρρ † ρ V = ( ρ : matrix n × k ) � − ρ † 1 − ρ † ρ Assume that characteristic scales of neutrino masses satisfy m L , m D ≪ M R ⇒ ρ ≪ 1 .
Block-diagonalization of M m T m L ˜ m L 0 V T M V = V T D V = n L = V χ ′ L , ˜ m D M R 0 M R Look for the unitary matrix V in the form � 1 − ρρ † ρ V = ( ρ : matrix n × k ) � − ρ † 1 − ρ † ρ Assume that characteristic scales of neutrino masses satisfy m L , m D ≪ M R ⇒ ρ ≪ 1 . Treat ρ as perturbation ⇒ ρ ∗ ≃ m T ˜ D M − 1 R , M R ≃ M R , D M − 1 m L ≃ m L − m T ˜ R m D Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 18
Type I seesaw mechanism – 1-gener. case A simple 1-flavour case ( n = k = 1 ). Notation change: M R → m R , N R → ν R . m L m D M = ( m L , m D , m R − real positive numbers ) m D m R
Type I seesaw mechanism – 1-gener. case A simple 1-flavour case ( n = k = 1 ). Notation change: M R → m R , N R → ν R . m L m D M = ( m L , m D , m R − real positive numbers ) m D m R Can be diagonalized as O T M O = M d where O is real orthogonal 2 × 2 matrix and M d = diag ( m 1 , m 2 ) . Introduce the fields χ L through n L = Oχ L :
Type I seesaw mechanism – 1-gener. case A simple 1-flavour case ( n = k = 1 ). Notation change: M R → m R , N R → ν R . m L m D M = ( m L , m D , m R − real positive numbers ) m D m R Can be diagonalized as O T M O = M d where O is real orthogonal 2 × 2 matrix and M d = diag ( m 1 , m 2 ) . Introduce the fields χ L through n L = Oχ L : ν L cos θ sin θ χ 1 L = n L = ( χ 1 L , χ 2 L − LH comp. of χ 1 , 2 ) ν c − sin θ cos θ χ 2 L L
Type I seesaw mechanism – 1-gener. case A simple 1-flavour case ( n = k = 1 ). Notation change: M R → m R , N R → ν R . m L m D M = ( m L , m D , m R − real positive numbers ) m D m R Can be diagonalized as O T M O = M d where O is real orthogonal 2 × 2 matrix and M d = diag ( m 1 , m 2 ) . Introduce the fields χ L through n L = Oχ L : ν L cos θ sin θ χ 1 L = n L = ( χ 1 L , χ 2 L − LH comp. of χ 1 , 2 ) ν c − sin θ cos θ χ 2 L L Rotation angle and mass eigenvalues: 2 m D tan 2 θ = , m R − m L �� m R − m L � 2 m 1 , 2 = m R + m L + m 2 ∓ D . 2 2 m 1 , m 2 real but can be of either sign Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 19
1-generation case – contd. 1 L C − 1 M n L + h.c. = 1 2 n T 2 χ T L C − 1 M d χ L + h.c. L m = 1 2 L C − 1 χ 2 L ) + h.c. = 1 2( m 1 χ T 1 L C − 1 χ 1 L + m 2 χ T = 2( | m 1 | χ 1 χ 1 + | m 2 | χ 2 χ 2 )
1-generation case – contd. 1 L C − 1 M n L + h.c. = 1 2 n T 2 χ T L C − 1 M d χ L + h.c. L m = 1 2 L C − 1 χ 2 L ) + h.c. = 1 2( m 1 χ T 1 L C − 1 χ 1 L + m 2 χ T = 2( | m 1 | χ 1 χ 1 + | m 2 | χ 2 χ 2 ) Here χ 1 = χ 1 L + η 1 ( χ 1 L ) c , χ 2 = χ 2 L + η 2 ( χ 2 L ) c . with η i = 1 or − 1 for m i > 0 or < 0 respectively.
1-generation case – contd. 1 L C − 1 M n L + h.c. = 1 2 n T 2 χ T L C − 1 M d χ L + h.c. L m = 1 2 L C − 1 χ 2 L ) + h.c. = 1 2( m 1 χ T 1 L C − 1 χ 1 L + m 2 χ T = 2( | m 1 | χ 1 χ 1 + | m 2 | χ 2 χ 2 ) Here χ 1 = χ 1 L + η 1 ( χ 1 L ) c , χ 2 = χ 2 L + η 2 ( χ 2 L ) c . with η i = 1 or − 1 for m i > 0 or < 0 respectively. ♦ Mass eigenstates χ 1 , χ 2 are Majorana states!
1-generation case – contd. 1 L C − 1 M n L + h.c. = 1 2 n T 2 χ T L C − 1 M d χ L + h.c. L m = 1 2 L C − 1 χ 2 L ) + h.c. = 1 2( m 1 χ T 1 L C − 1 χ 1 L + m 2 χ T = 2( | m 1 | χ 1 χ 1 + | m 2 | χ 2 χ 2 ) Here χ 1 = χ 1 L + η 1 ( χ 1 L ) c , χ 2 = χ 2 L + η 2 ( χ 2 L ) c . with η i = 1 or − 1 for m i > 0 or < 0 respectively. ♦ Mass eigenstates χ 1 , χ 2 are Majorana states! Interesting limiting cases: (a) m R ≫ m L , m D (seesaw limit) m L − m 2 → − m 2 D D m 1 ≈ for m L = 0 m R m R m 2 ≈ m R Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 20
1-generation case – contd. (b) m L = m R = 0 (Dirac case) 0 m − m 0 . M = → M d = m 0 0 m
1-generation case – contd. (b) m L = m R = 0 (Dirac case) 0 m − m 0 . M = → M d = m 0 0 m Diagonalized by rotation with angle θ = 45 ◦ . We have η 2 = − η 1 = 1 ; √ √ 2( ν c L + ν c R ) = − ( χ 1 + χ 2 ) c . χ 1 + χ 2 = 2( ν L + ν R ) , χ 1 − χ 2 = − ⇓
1-generation case – contd. (b) m L = m R = 0 (Dirac case) 0 m − m 0 . M = → M d = m 0 0 m Diagonalized by rotation with angle θ = 45 ◦ . We have η 2 = − η 1 = 1 ; √ √ 2( ν c L + ν c R ) = − ( χ 1 + χ 2 ) c . χ 1 + χ 2 = 2( ν L + ν R ) , χ 1 − χ 2 = − ⇓ 1 2 m ( χ 1 χ 1 + χ 2 χ 2 ) = 1 4 m [( χ 1 + χ 2 )( χ 1 + χ 2 )+[( χ 1 − χ 2 )( χ 1 − χ 2 )] = m ¯ ν D ν D , where ν D ≡ ν L + ν R .
1-generation case – contd. (b) m L = m R = 0 (Dirac case) 0 m − m 0 . M = → M d = m 0 0 m Diagonalized by rotation with angle θ = 45 ◦ . We have η 2 = − η 1 = 1 ; √ √ 2( ν c L + ν c R ) = − ( χ 1 + χ 2 ) c . χ 1 + χ 2 = 2( ν L + ν R ) , χ 1 − χ 2 = − ⇓ 1 2 m ( χ 1 χ 1 + χ 2 χ 2 ) = 1 4 m [( χ 1 + χ 2 )( χ 1 + χ 2 )+[( χ 1 − χ 2 )( χ 1 − χ 2 )] = m ¯ ν D ν D , where ν D ≡ ν L + ν R . | m 1 , 2 | ≈ m D ± m L + m R (c) m L , m R ≪ m D (pseudo-Dirac neutrino): . 2 Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 21
The 3 basic seesaw models λ i.e. tree level ways to generate the dim 5 operator M LLHH Fermion triplet: Right-handed singlet: Scalar triplet: (type-III seesaw) (type-I seesaw) (type-II seesaw) + + 1 µ ∆ 1 m ν = Y T Y Σ v 2 v 2 m ν = Y T Y N v 2 m ν = Y ∆ Σ N M 2 M Σ M N ∆ small if large small if large small if large M N M ∆ M Σ m ν m ν m ν (or if small) (or if small) (or if small) Y ν Y ∆ , µ Y Σ Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 22
� Access to the seesaw parameters from mass matrix data ν • Type II seesaw: mass matrix data ν µ ∆ gives full access to v 2 m ν ij = Y ∆ ij M 2 ∆ type II flavour structure • Type I or III seesaw model: mass matrix data: gives ν 1 + m ν ij = Y T access to 9 parameter Y Nkj v 2 Nik M Nk combinations of and Y N M N 3 masses of the N 9 real parameters 18 parameters 15 parameters in Yukawa matrix 6 phases Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 23
Neutrino oscillations Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 24
Neutrinos can oscillate ! A periodic change of neutrino flavour (identity): ν e → ν µ → ν e → ν µ → ν e ... Happens without any external influence! Dr. Jekyll / Mr. Hyde kind of story Neutrinos have two-sided (or even 3-sided) personality ! P ( ν e → ν µ ; L ) = sin 2 2 θ · sin 2 � � ∆ m 2 4 p L Hints of oscillations of solar neutrinos seen since the 1960s First unambiguous evidence – oscillations of atmospheric neutrinos (The Super-Kamiokande Collaboration, 1998) Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 25
A bit of history... Idea of neutrino oscillations: First put forward by Pontecorvo in 1957. Suggested possibility of ν ↔ ¯ ν oscillations by K 0 oscillations. analogy with K 0 ¯
A bit of history... Idea of neutrino oscillations: First put forward by Pontecorvo in 1957. Suggested possibility of ν ↔ ¯ ν oscillations by K 0 oscillations. analogy with K 0 ¯ Flavour transitions (“virtual transmutations”) first considered by Maki, Nakagawa and Sakata in 1962.
A bit of history... Idea of neutrino oscillations: First put forward by Pontecorvo in 1957. Suggested possibility of ν ↔ ¯ ν oscillations by K 0 oscillations. analogy with K 0 ¯ Flavour transitions (“virtual transmutations”) first considered by Maki, Nakagawa and Sakata in 1962. B. Pontecorvo S. Sakata Z. Maki M. Nakagawa 1913 - 1993 1911 – 1970 1929 – 2005 1932 – 2001 Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 26
Neutrino revolution Neutrino mass had been unsuccessfully looked for for almost 40 years (several wrong discovery claims) Since 1998 – an avalanche of discoveries : Oscillations of atmospheric, solar, reactor and accelerator neutrinos Neutrino oscillations imply that neutrinos are massive In the standard model neutrinos are massless ⇒ we have now the first compelling evidence of physics beyond the standard model ! Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 27
Oscillations discovered experimentally ! Zenith angle distributions Best fit 10 -3 10 -3 sin 2 2 � =1.0, � m 2 =2.0x10 -3 eV 2 ∆ m 2 in eV 2 ∆ m 2 in eV 2 � � � � � KamLAND 2-flavor oscillations Null oscillation Cl Cl 10 -4 10 -4 95% exclusion by rate Sub-GeV Multi-R Up stop � 10 -5 10 -5 � -like KamLAND KamLAND 10 -6 10 -6 95% allowed 95% allowed SNO SNO by rate+shape by rate+shape Sub-GeV e-like Sub-GeV � -like 10 -7 10 -7 SuperK SuperK Ga Ga Ga Ga 10 -8 10 -8 Multi-GeV Multi-R Multi-GeV � -like Up thru � � -like + PC 10 -9 10 -9 10 -10 10 -10 Multi-GeV e-like 10 -11 10 -11 10 -12 10 -12 ~13000km ~500km ~15km ~13000km ~500km 10 -4 10 -3 10 -2 10 -1 10 2 10 -4 10 -3 10 -2 10 -1 10 2 1 10 1 10 tan 2 ( Θ ) tan 2 ( Θ ) ν µ �����''&�!�1�&��&��-!&/&10 Neutrino Oscillation � � � � ( ) � → = � − ��� � 2 � �� � previous result (above 2.6 MeV) *��%����� ν µ ���������& � � � � � � 1.4 KamLAND data CHOOZ data ν µ ��������� short baseline best-fit osci. �������������� 1.2 best-fit osci. + Expected Geo ! experiment e 1 1st 2nd 3rd 2������������ <������������������ Ratio 0.8 �������������= 0.6 )��������� <����������� 0.4 ������������= 0.2 preliminary hypothetical single reactor 0 0 10 20 30 40 50 60 70 ���������� ���������� at 180 km L /E (km/MeV) 0 ! e KamLAND covers the 2nd and 3rd maximum characteristic of neutrino oscillation � *����+��,���$�� ������-��������� (�� �.%/0�1"����������������4 Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 28
Oscillations: a well known QM phenomenon Ψ 2 E 2 Ψ 1 ( t ) = e − i E 1 t Ψ 1 (0) Ψ 2 ( t ) = e − i E 2 t Ψ 2 (0) Ψ E 1 1 ( | a | 2 + | b | 2 = 1) ; Ψ(0) = a Ψ 1 (0) + b Ψ 2 (0) ⇒ Ψ( t ) = a e − i E 1 t Ψ 1 (0) + b e − i E 2 t Ψ 2 (0) Probability to remain in the same state | Ψ(0) � after time t : P surv = |� Ψ(0) | Ψ( t ) �| 2 = � | a | 2 e − i E 1 t + | b | 2 e − i E 2 t � � 2 � ♦ = 1 − 4 | a | 2 | b | 2 sin 2 [( E 2 − E 1 ) t/ 2] Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 29
Neutrino oscillations: theory Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 30
Leptonic mixing For m ν � = 0 weak eigenstate neutrinos ν e , ν µ , ν τ do not coincide with mass eigenstate neutrinos ν 1 , ν 2 , ν 3 Diagonalization of leptonic mass matrices: e ′ ν ′ L → V L e L , L → U L ν L . . . ⇒ g e L γ µ V † L U L ν L ) W − √ −L w + m = 2(¯ + diag . mass terms + h.c. µ Leptonic mixing matrix: U = V † L U L � � U ∗ ♦ ν αL = U αi ν iL ⇒ | ν αL � = αi | ν iL � i i ( α = e , µ , τ, i = 1 , 2 , 3) Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 31
Master formula for ν oscillations The standard formula for the oscillation probability of relativistic or quasi-degenerate in mass neutrinos in vacuum: 2 ∆ m 2 � � L U ∗ ij i U βi e − i � � � ♦ P ( ν α → ν β ; L ) = 2 p � � αi � � ( � = c = 1 ) Problem: prove that the RHS does not depend on the index j . Oscillation disappear when either U = 1 , i.e. U αi = δ αi (no mixing) or ∆ m 2 ij = 0 (massless or mass-degenerate neutrinos). Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 32
How is it usually derived? Assume at time t = 0 and coordinate x = 0 a flavour eigenstate | ν α � is produced: � | ν (0 , 0) � = | ν fl U ∗ αi | ν mass α � = � i i After time t at the position x , for plane-wave particles: � U ∗ αi e − ip i x | ν mass | ν ( t, � x ) � = � i i Mass eigenstates pick up the phase factors e − iφ i with φ i ≡ p i x = Et − � p� x � 2 � � ν fl � � P ( ν α → ν β ) = β | ν ( t, x ) � Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 33
How is it usually derived? Consider � x || � p ⇒ � p� x = px ( p = | � p | , x = | � x | ) Phase differences between different mass eigenstates: ∆ φ = ∆ E · t − ∆p · x Shortcuts to the standard formula 1. Assume the emitted neutrino state has a well defined momentum (same momentum prescription) ⇒ ∆p = 0 . i ≃ p + m 2 p 2 + m 2 � For ultra-relativistic neutrinos E i = ⇒ i 2p ∆ E ≃ m 2 2 − m 2 ≡ ∆ m 2 1 2 E ; t ≈ x ( � = c = 1) 2 E ⇒ The standard formula is obtained Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 34
How is it usually derived? 2. Assume the emitted neutrino state has a well defined energy (same energy prescription) ⇒ ∆ E = 0 . ∆ φ = ∆ E · t − ∆p · x ⇒ − ∆p · x i ≃ E − m 2 E 2 − m 2 � For ultra-relativistic neutrinos p i = ⇒ i 2p − ∆p ≡ p 1 − p 2 ≈ ∆ m 2 2 E ; ⇒ The standard formula is obtained ik ≃ 2 . 5 m E (MeV) 4 πE Stand. phase ⇒ ( l osc ) ik = ∆ m 2 ∆ m 2 ik eV 2 Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 35
Same E and same p approaches
Same E and same p approaches Very simple and transparent
Same E and same p approaches Very simple and transparent Allow one to quickly arrive at the desired result
Same E and same p approaches Very simple and transparent Allow one to quickly arrive at the desired result Trouble: they are both wrong Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 36
Kinematic constraints Same momentum and same energy assumptions: contradict kinematics! Pion decay at rest ( π + → µ + + ν µ , π − → µ − + ¯ ν µ ): For decay with emission of a massive neutrino of mass m i : � 2 � � � 1 − m 2 1 − m 2 i = m 2 + m 2 + m 4 µ µ E 2 π i i m 2 m 2 4 m 2 4 2 π π π � 2 � � � 1 − m 2 1 + m 2 i = m 2 − m 2 + m 4 µ µ p 2 π i i m 2 m 2 4 m 2 4 2 π π π m 2 � � E i = p i = E ≡ m π µ For massless neutrinos: 1 − ≃ 30 MeV 2 m 2 π To first order in m 2 i : � � 1 − m 2 E i ≃ E + ξ m 2 p i ≃ E − (1 − ξ ) m 2 ξ = 1 µ i i 2 E , 2 E , ≈ 0 . 2 m 2 2 π Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 37
Kinematic constraints Same momentum or same energy would require ξ = 1 or ξ = 0 – not the case! Also: would violate Lorentz invariance of the oscillation probability How can wrong assumptions lead to the correct oscillation formula ? Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 38
Problems with the plane-wave approach ⇒ Same momentum oscillation probabilities depend only on time. Leads to a paradoxical result – no need for a far detector ! “Time-to-space conversion” (??) – assumes neutrinos to be point-like particles (notion opposite to plane waves). Same energy – oscillation probabilities depend only on coordinate. Does not explain how neutrinos are produced and detected at certain times. Correspponds to a stationary situation. Plane wave approach ⇔ exact energy-momentum conservation. Neutrino energy and momentum are fully determined by those of external particles ⇒ only one mass eigenstate can be emitted! Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 39
♦ Consistent approaches:
♦ Consistent approaches: QM wave packet approach – neutrinos described by wave packets rather than by plane waves
♦ Consistent approaches: QM wave packet approach – neutrinos described by wave packets rather than by plane waves QFT approach: neutrino production and detection explicitly taken into account. Neutrinos are intermediate particles described by propagators P f ( k ) D f ( k ′ ) ν P i ( q ) D i ( q ′ ) Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 40
QM wave packet approach In QM propagating particles are described by wave packets! – Finite extensions in space and time. x ) = e i� p 0 � x Plane waves: the wave function at time t = 0 Ψ � p 0 ( � 1.5 1 0.5 0 –4 –2 2 4 x –0.5 –1 –1.5 Wave packets: superpositions of plane waves with momenta in an interval of width σ p around mom. p 0 ⇒ constructive interference in a spatial interval of width σ x around some point x 0 and destructive interference outside it. σ x σ p ≥ 1 / 2 – QM uncertainty relation Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 41
Wave packets W. packet centered at � x 0 = 0 at time t = 0 : d 3 p � p 0 ) e i� p � x Ψ( � x ; � p 0 , σ � p ) = (2 π ) 3 f ( � p − � Rectangular mom. space w. packet: 1 f 0.5 0 –4 –2 2 4 x –0.5 p p 0 –1 2 σ p Gaussian mom. space w. packet: 0.6 0.7 0.4 0.6 0.2 0.5 0.4 0 –4 –3 –2 –1 1 2 3 4 p 0.3 –0.2 0.2 –0.4 0.1 –0.6 0 1 2 3 4 5 6 7 8 p σ x σ p = 1 / 2 – minimum uncertainty packet Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 42
Propagating wave packets Include time dependence: d 3 p � p 0 ) e i� p� x − iE ( p ) t Ψ( � x, t ) = (2 π ) 3 f ( � p − � Example: Gaussian wave packets Momentum-space distribution: p 0 ) 2 1 � − ( � p − � � f ( � p − � p 0 ) = p ) 3 / 4 exp 4 σ 2 (2 πσ 2 p p � 2 = σ 2 p 2 � − � � Momentum dispersion: � � p . Coordinate-space wave packet (neglecting spreading): v g t ) 2 � � 1 − ( � x − � x, t ) = e i� p 0 � x − iE ( p 0 ) t σ 2 x = 1 / (4 σ 2 Ψ( � x ) 3 / 4 exp , p ) 4 σ 2 (2 πσ 2 x x � 2 = σ 2 x 2 � − � � � � x � = � v g t ; � � x . Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 43
QM wave packet approach The evolved produced state: � � | ν fl αi | ν mass αi Ψ S x, t ) | ν mass U ∗ U ∗ α ( � x, t ) � = ( � x, t ) � = i ( � � i i i i The coordinate-space wave function of the i th mass eigenstate (w. packet): d 3 p � Ψ S (2 π ) 3 f S p ) e i� p� x − iE i ( p ) t i ( � x, t ) = i ( � p = � Momentum distribution function f S i ( � p ) : sharp maximum at � P (width of the peak σ pP ≪ P ). ∂ 2 E i ( p ) � � E i ( P ) + ∂E i ( p ) 1 P ) 2 + . . . p − � p − � � � E i ( p ) = ( � P ) + ( � � � � � p 2 ∂� p 2 ∂� � P � p 0 α ≡ ∂ 2 E i ( p ) = m 2 v i = ∂E i ( p ) � p i � = , E 2 p 2 ∂� p E i ∂� i Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 44
Evolved neutrino state x g S x, t ) ≃ e − iE i ( P ) t + i � Ψ S P� i ( � i ( � x − � v i t ) ( α → 0 ) d 3 q q + � g S � (2 π ) 3 f S P ) e i� q ( � x − � v g t ) i ( � x − � v i t ) ≡ i ( � Problem: derive this result Center of the wave packet: � x − � v i t = 0 . Spatial length: σ xP ∼ 1 /σ pP ( g S i decreases quickly for | � x − � v i t | � σ xP ). x = � Detected state (centered at � L ): � | ν fl U ∗ βk Ψ D x ) | ν mass β ( � x ) � = k ( � � i k The coordinate-space wave function of the i th mass eigenstate (w. packet): d 3 p � x − � Ψ D (2 π ) 3 f D p ) e i� p ( � L ) i ( � x ) = i ( � Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 45
Oscillation probability Transition amplitude: A αβ ( T, � α ( T, � � αi U βi A i ( T, � L ) = � ν fl β | ν fl U ∗ L ) � = L ) i d 3 p � p� A i ( T, � (2 π ) 3 f S p ) f D ∗ p ) e − iE i ( p ) T + i� L L ) = i ( � ( � i Strongly suppressed unless | � L − � v i T | � σ x . E.g., for Gaussian wave packets: � � − ( � v i T ) 2 L − � A i ( T, � σ 2 x ≡ σ 2 xP + σ 2 L ) ∝ exp , xD 4 σ 2 x Oscillation probability: L ) = |A αβ | 2 = P ( ν α → ν β ; T, � � βk A i ( T, � k ( T, � U ∗ αi U βi U αk U ∗ L ) A ∗ ♦ L ) i,k Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 46
Phase difference Oscillations are due to phase differences of different mass eigenstates: � p 2 i + m 2 ∆ φ = ∆ E · T − ∆ p · L ( E i = i ) Consider the case ∆ E ≪ E (relativistic or quasi-degenerate neutrinos) ⇒ ∆ E = ∂E ∂p ∆ p + ∂E 1 ∂m 2 ∆ m 2 = v g ∆ p + 2 E ∆ m 2 ∆ φ = ( v g ∆ p + 1 2 E ∆ m 2 ) T − ∆ p · L = − ( L − v g T )∆ p + ∆ m 2 2 E T In the center of wave packet ( L − v g T ) = 0 ! In general, | L − v g T | � σ x ; if σ x ≪ l osc , | L − v g T | ∆ p ≪ 1 ⇒ Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 47
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered!
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered! Now instead of expressing ∆ E through ∆ p and ∆ m 2 express ∆ p through ∆ E and ∆ m 2 :
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered! Now instead of expressing ∆ E through ∆ p and ∆ m 2 express ∆ p through ∆ E and ∆ m 2 : ( L − v g T )∆ E + ∆ m 2 ∆ m 2 ∆ φ = − 1 ♦ L ⇒ L v g 2 p 2 p
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered! Now instead of expressing ∆ E through ∆ p and ∆ m 2 express ∆ p through ∆ E and ∆ m 2 : ( L − v g T )∆ E + ∆ m 2 ∆ m 2 ∆ φ = − 1 ♦ L ⇒ L v g 2 p 2 p – the result of the “same energy” approach recovered!
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered! Now instead of expressing ∆ E through ∆ p and ∆ m 2 express ∆ p through ∆ E and ∆ m 2 : ( L − v g T )∆ E + ∆ m 2 ∆ m 2 ∆ φ = − 1 ♦ L ⇒ L v g 2 p 2 p – the result of the “same energy” approach recovered! The reasons why wrong assumptions give the correct result:
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered! Now instead of expressing ∆ E through ∆ p and ∆ m 2 express ∆ p through ∆ E and ∆ m 2 : ( L − v g T )∆ E + ∆ m 2 ∆ m 2 ∆ φ = − 1 ♦ L ⇒ L v g 2 p 2 p – the result of the “same energy” approach recovered! The reasons why wrong assumptions give the correct result: Neutrinos are relativistic or quasi-degenerate with ∆ E ≪ E
∆ φ = ∆ m 2 2 E T , L ≃ v g T ≃ T – the result of the “same momentum” approach recovered! Now instead of expressing ∆ E through ∆ p and ∆ m 2 express ∆ p through ∆ E and ∆ m 2 : ( L − v g T )∆ E + ∆ m 2 ∆ m 2 ∆ φ = − 1 ♦ L ⇒ L v g 2 p 2 p – the result of the “same energy” approach recovered! The reasons why wrong assumptions give the correct result: Neutrinos are relativistic or quasi-degenerate with ∆ E ≪ E The size of the neutrino wave packet is small compared to the oscillation length: σ x ≪ l osc (more precisely: energy uncertainty σ E ≫ ∆ E ) Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 48
Oscillation probability in WP approach L ) = |A αβ | 2 = P ( ν α → ν β ; T, � � βk A i ( T, � k ( T, � U ∗ αi U βi U αk U ∗ L ) A ∗ L ) i,k d 3 p � p� A i ( T, � (2 π ) 3 f S p ) f D ∗ p ) e − iE i ( p ) T + i� L L ) = i ( � ( � i Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 49
Oscillation probability in WP approach Neutrino emission and detection times are not measured (or not accurately measured) in most experiments ⇒ integration over T : ∆ m 2 � L ˜ � ik U ∗ αi U βi U αk U ∗ βk e − i P ( ν α → ν β ; L ) = dT P ( ν α → ν β ; T, L ) = I ik 2 ¯ P i,k � dq ˜ 2 π f S i ( r k q − ∆ E ik / 2 v + P i ) f D ∗ I ik = N ( r k q − ∆ E ik / 2 v + P i ) i k ( r i q + ∆ E ik / 2 v + P k ) e i ∆ v × f S ∗ k ( r i q + ∆ E ik / 2 v + P k ) f D v qL r i,k ≡ v i,k v ≡ v i + v k , ∆ v ≡ v k − v i , v , N ≡ 1 / [2 E i ( P )2 E k ( P ) v ] , Here: 2 Problem: derive this result. Hint: use ∆ E ik ≃ v ∆ p ik + ∆ m 2 ik / 2 E and go to the shifted integration variable q ≡ p − P where P ≡ ( P i + P k ) / 2 . Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 50
When are neutrino oscillations observable? Keyword: Coherence Neutrino flavour eigenstates ν e , ν µ and ν τ are coherent superpositions of mass eigenstates ν 1 , ν 2 and ν 3 ⇒ oscillations are only observable if neutrino production and detection are coherent coherence is not (irreversibly) lost during neutrino propagation. Possible decoherence at production (detection): If by accurate E and p p 2 + m 2 ) which mass eigenstate � measurements one can tell (through E = is emitted, the coherence is lost and oscillations disappear! Full analogy with electron interference in double slit experiments: if one can establish which slit the detected electron has passed through, the interference fringes are washed out. Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 51
When are neutrino oscillations observable? Another source of decoherence: wave packet separation due to the difference of group velocities ∆ v of different mass eigenstates. If coherence is lost: Flavour transition can still occur, but in a non-oscillatory way. E.g. for π → µν i decay with a subsequent detection of ν i with the emission of e : � � | U µi | 2 | U ei | 2 P ∝ P prod ( µ ν i ) P det ( e ν i ) ∝ i i – the same result as for averaged oscillations. How are the oscillations destroyed? Suppose by measuring momenta and energies of particles at neutrino production (or detection) we can determine its energy E and momentum p with uncertainties σ E and σ p . From � p 2 i + m 2 E i = i : (2 Eσ E ) 2 + (2 pσ p ) 2 � 1 / 2 � σ m 2 = Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 52
When are neutrino oscillations observable? If σ m 2 < ∆ m 2 = | m 2 i − m 2 k | – one can tell which mass eigenstate is emitted. σ m 2 < ∆ m 2 implies 2 pσ p < ∆ m 2 , or σ p < ∆ m 2 / 2 p ≃ l − 1 osc . But: To measure p with the accuracy σ p one needs to measure the momenta of particles at production with (at least) the same accuracy ⇒ uncertainty of their coordinates (and the coordinate of ν production point) will be σ x , prod � σ − 1 > l osc p ⇒ Oscillations washed out. Similarly for neutrino detection. Natural necessary condition for coherence (observability of oscillations): L source ≪ l osc , L det ≪ l osc No averaging of oscillations in the source and detector Satisfied with very large margins in most cases of practical interest Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 53
Wave packet separation Wave packets representing different mass eigenstate components have different group velocities v gi ⇒ after time t coh (coherence time) they separate ⇒ Neutrinos stop oscillating! (Only averaged effect observable). Coherence time and length: ∆ v · t coh ≃ σ x ; l coh ≃ vt coh ≃ ∆ m 2 ∆ v = p i − p k 2 E 2 E i E k ∆ v σ x = 2 E 2 v l coh ≃ ∆ m 2 vσ x The standard formula for P osc is obtained when the decoherence effects are negligible. Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 54
A manifestation of neutrino coherence Even non-observation of neutrino oscillations at distances L ≪ l osc is a consequence of and an evidence for coherence of neutrino emission and detection! Two-flavour example (e.g. for ν e emission and detection): A prod / det ( ν 1 ) ∼ cos θ , A prod / det ( ν 2 ) ∼ sin θ ⇒ A prod ( ν i ) A det ( ν i ) ∼ cos 2 θ + e − i ∆ φ sin 2 θ � A ( ν e → ν e ) = i =1 , 2 Phase difference ∆ φ vanishes at short L ⇒ P ( ν e → ν e ) = (cos 2 θ + sin 2 θ ) 2 = 1 If ν 1 and ν 2 were emitted and absorbed incoherently) ⇒ one would have to sum probabilities rather than amplitudes: | A prod ( ν i ) A det ( ν i ) | 2 ∼ cos 4 θ + sin 4 θ < 1 � P ( ν e → ν e ) ∼ i =1 , 2 Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 55
Are coherence constraints compatible? Observability conditions for ν oscillations: Coherence of ν production and detection Coherence of ν propagation Both conditions put upper limits on neutrino mass squared differences ∆ m 2 : ∆ m 2 ∆ m 2 jk jk (1) ∆ E jk ∼ ≪ σ E ; (2) 2 E 2 L ≪ σ x ≃ v g /σ E 2 E
Are coherence constraints compatible? Observability conditions for ν oscillations: Coherence of ν production and detection Coherence of ν propagation Both conditions put upper limits on neutrino mass squared differences ∆ m 2 : ∆ m 2 ∆ m 2 jk jk (1) ∆ E jk ∼ ≪ σ E ; (2) 2 E 2 L ≪ σ x ≃ v g /σ E 2 E But: The constraints on σ E work in opposite directions: ∆ m 2 2 E 2 v g jk (1) ∆ E jk ∼ ≪ σ E ≪ (2) ∆ m 2 2 E L jk
Are coherence constraints compatible? Observability conditions for ν oscillations: Coherence of ν production and detection Coherence of ν propagation Both conditions put upper limits on neutrino mass squared differences ∆ m 2 : ∆ m 2 ∆ m 2 jk jk (1) ∆ E jk ∼ ≪ σ E ; (2) 2 E 2 L ≪ σ x ≃ v g /σ E 2 E But: The constraints on σ E work in opposite directions: ∆ m 2 2 E 2 v g jk (1) ∆ E jk ∼ ≪ σ E ≪ (2) ∆ m 2 2 E L jk Are they compatible? – Yes, if LHS ≪ RHS ⇒ 2 π L v g ≪ ( ≫ 1) – fulfilled in all cases of practical interest l osc ∆ v g Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 56
Are coherence conditions satisfied? The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic ν ’s ...)
Are coherence conditions satisfied? The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic ν ’s ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass
Are coherence conditions satisfied? The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic ν ’s ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass But: Is not automatically guaranteed in the case of “light” sterile neutrinos! m sterile ∼ eV − keV − MeV scale ⇒ heavy compared to the “usual” (active) neutrinos
Are coherence conditions satisfied? The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic ν ’s ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass But: Is not automatically guaranteed in the case of “light” sterile neutrinos! m sterile ∼ eV − keV − MeV scale ⇒ heavy compared to the “usual” (active) neutrinos Sterile neutrinos: hints from SBL accelerator experiments (LSND, MiniBooNE), reactor neutrino anomaly, keV sterile neutrinos, pulsar kicks, leptogenesis via ν oscillations, SN r -process nucleosynthesis, unconventional contributions to 2 β 0 ν decay ...
Are coherence conditions satisfied? The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic ν ’s ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass But: Is not automatically guaranteed in the case of “light” sterile neutrinos! m sterile ∼ eV − keV − MeV scale ⇒ heavy compared to the “usual” (active) neutrinos Sterile neutrinos: hints from SBL accelerator experiments (LSND, MiniBooNE), reactor neutrino anomaly, keV sterile neutrinos, pulsar kicks, leptogenesis via ν oscillations, SN r -process nucleosynthesis, unconventional contributions to 2 β 0 ν decay ... Production/detection coherence has to be re-checked – important implications for some neutrino experiments! Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 57
Neutrino oscillations: Coherence at macroscopic distances – L > 10,000 km in atmospheric neutrino experiments ! Evgeny Akhmedov ISAPP 2019 Summer School MPIK Heidelberg, May 28 – June 4, 2019 – p. 58
Oscillation probability in WP approach Neutrino emission and detection times are not measured (or not accurately measured) in most experiments ⇒ integration over T : ∆ m 2 � L ˜ � ik U ∗ αi U βi U αk U ∗ βk e − i P ( ν α → ν β ; L ) = dT P ( ν α → ν β ; T, L ) = I ik 2 ¯ P i,k
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