NEUTRINOS. Alexey Boyarsky PPEU 1

Transcription

1 NEUTRINOS Alexey Boyarsky PPEU 1

2 Discovery of neutrino Observed 14 6 C 14 7 N + e Two body decay: the electron has the same energy (almost) not observed! Energy is not conserved? Pauli s letter, Dec. 4, because of the wrong statistics of the N and Li 6 nuclei and the continuous beta spectrum, I have hit upon a desperate remedy to save the exchange theorem of statistics and the law of conservation of energy. Namely, the possibility that there could exist in the nuclei electrically neutral particles... which have spin 1/2 and obey the exclusion principle.... The continuous beta spectrum would then become understandable by the assumption that in beta decay a neutron is emitted in addition to the electron such that the sum of the energies of the neutron and the electron is constant... Alexey Boyarsky PPEU 2

3 Neutrino "I have done a terrible thing. I invented a particle that cannot be detected." W. Pauli Pauli (1930) called this new particle neutron Chadwick discovered a massive nuclear particle in 1932 neutron Fermi renamed it into neutrino (italian little neutral one ) Alexey Boyarsky PPEU 3

4 Neutrino β-decay is the decay of neutron n p + e + ν e inside the nucleus Electron capture: 22 11Na + e 22 10Ne + ν e Here ν e has opposite spin than that of ν e! The process inside a nucleus p + e n + ν e Alexey Boyarsky PPEU 4

5 β + -decay Also observed was β + -decay 22 11Na 22 10Ne + e + + ν e Again the particle ν e has the opposite spin to ν e! Formally β + decay would come from the p? n + e + + ν e but mass of proton m p < m n (mass of neutron)?! possible if neutrons are not free (nuclear binding energy) This reaction is the main source of solar neutrinos: 4p 4 He + 2e + + 2ν e MeV Alexey Boyarsky PPEU 5

6 Can neutrino be detected? Along with p? n + e + + ν e there can be a reaction ν e + p n + e + energetic neutrino can cause β + decay of a stable nucleus In 1934 estimated that the cross-section for such a reaction is Bethe & Peierls tremendously small: σ cm 2 for comparison: cross-section of photon scattering on non-relativistic electron (Thomson cross-section) is σ T homson cm 2 In 1942 Fermi had build the first nuclear reactor source of large number of neutrinos ( neutrinos/sec/cm 2 ) Flux of neutrinos from e.g. a nuclear reactor can initiate β + decays in protonts of water Positrons annihilate and two γ-rays and neutron were detected! Discovered by Cowan & Reines in 1956 Alexey Boyarsky PPEU 6

7 How to describe ν e and ν e Dirac equation describes evolution of 4-component spinor ψ ν (iγ µ µ m ν )ψ ν = 0 (where neutrinos mass m ν m e but not necessarily zero)! This would predict that there should be particle and anti-particle (neutrino and anti-neutrino): ψ ν = (χ, χ). Each of χ and χ has spin and spin components Are χ and χ distinct? So far people detected only neutral particles with spin up or down. Only two degrees of freedom? One possibility: one can put a solution of the Dirac equation, putting χ( ) = χ( ) 0. This seems to be impossible, because even if one starts with such a solution, it will change over the course of evolution. Alexey Boyarsky PPEU 7

8 How to describe ν e and ν e This can be done only if m ν = 0. Indeed, if m ν = 0, then Dirac equation splits into two equations: ( ) iσ µ χ( ) µ χ( ) = 0 (and the one with interchanged spins). If this is true, then the prediction is that there are particles ν e that always have definite spin projection and anti-neutrino with the opposite spin projection. These particles are distinct (there is an interaction that differentiates between two) Alternatively, identify χ( ) = χ( ) and vice versa. Overall, there Alexey Boyarsky PPEU 8

9 How to describe ν e are only two degrees of freedom: Majorana fermion ψ ν = and ν e ( χ iσ 2 χ ) where the matrix iσ 2 = components of the spinor χ ( 0 ) interchanges spin-up and spin-down The corresponding modification of the Dirac equation was suggested by Majorana (1937) iσ µ µ χ m ν iσ 2 χ = 0 (1) Such equation is possible only for trully neutral particles Indeed, if the particle is not neutral, i.e. if there is a charge under which χ ˆQ χe iαq then χ has the opposite charge and Alexey Boyarsky PPEU 9

10 How to describe ν e and ν e equation (1) does not make sense (relates two objects with different transformation properties) Notice that when m ν = 0 there is no difference between Majorana equatino for the massless particle and massless 2-component Dirac equation In the same year when Cowan & Reines did their experiment, people were not abe to detect the same reactor neutrinos via ν e Cl 37 18Ar + e (2) The conclusion was that ν e (that would be captured in reaction (2)) and ν e that was captured by proton in Cowan & Reines were different particles! Alexey Boyarsky PPEU 10

11 Muon Muon, discovered in cosmis rays heavier brother of electron Mass: m µ 105 MeV Alexey Boyarsky PPEU 11

12 Another type of neutrino Later it was observed that muon decays always via 3-body decay: µ e + ν e + ν µ... but never via two-body decay, e.g. µ e + γ not observed! or µ e + e + + e not observed! Pions decay to muons or electrons via two-body decay, emitting some neutral massless particle with the spin 1/2: π + µ + + ν µ π µ + ν µ π + e + + ν e π e + ν e Alexey Boyarsky PPEU 12

13 Universality of weak interactions Original Fermi theory (1934) L Fermi = G F 2 [ p(x)γ µ n(x)][ē(x)γ µ ν(x)] (3) Universality of weak interactions before 1957! L Fermi = G F 2 [ p(x)γ µ n(x)][ē(x)γ µ ν e (x)] }{{} β decay (4) + G F 2 [ µ(x)γ µ ν µ (x)[ē(x)γ µ ν e (x)] }{{} muon decay + G F 2 [ē(x)γ µ ν e (x)[ē(x)γ µ ν e (x)] }{{} electron-neutrino scattering... (pion decay, etc.) All processes are governed by the same Fermi coupling constant: G F GeV 2 Alexey Boyarsky PPEU 13

14 Parity transformation Parity transformation is a discrete space-time symmetry, such that all spatial coordinates flip x x and time does not change t t Show that: P-even P-odd time position x angular momentum momentum p mass density Force electric charge electric current magnetic field electric field For fermions parity is related to the notion of chirality Dirac equation without mass can be split into two non-interacting parts (iγ µ µ m)ψ = ( 0 m i( t + σ ) ) (ψl ) i( t σ ) 0 m ψ R = 0 Alexey Boyarsky PPEU 14

15 Parity transformation Show that if particle moves only in one direction p = (p x, 0, 0), then these two components ψ L,R are left-moving and right-moving along x-direction. One can define the γ 5 = iγ 0 γ 1 γ 2 γ 3. In the above basis (Peskin & Schroeder conventions) : and γ 5 = ( ) γ 5 ψ R,L = ±ψ R,L Show that for massless fermions one can define a conserved quantity helicity: projection of spin onto momentum: h (p s) p Show that left/right chiral particles have definite helicity ±1. Alexey Boyarsky PPEU 15

16 Parity transformation For fermions: Left-Handed Right-Handed. ψ L ( p) P ψ R ( p). The neutrino being massless particle would have two states: Left neutrino: spin anti-parallel to the momentum p Right neutrino: spin parallel to the momentum p This has been tested in the experiment by Wu et al. in 1957 Alexey Boyarsky PPEU 16

17 Parity violation in weak interactions Put nuclei into the strong magnetic field to align their spins Cool the system down (to reduce fluctuations, flipping spin of the Cobalt nucleus) The transition from 60 Co to 60 Ni has momentum difference J = 1 (spins of nuclei were known) Spins of electron and neutrino are parallel to each other Electron and neutrino fly in the opposite directions Parity flips momentum but does not flip angular momentum/spin/magnetic field Alexey Boyarsky PPEU 17

18 Parity violation in weak interactions B s B s p e È Ö ØÝ ØÖ Ò ÓÖÑ p ν p ν p e s s Ó ÖÚ ÆÇÌ Ó ÖÚ Alexey Boyarsky PPEU 18

19 Parity violation in weak interactions This result means that neutrino always has spin anti-parallel to its momentum (left-chiral particle) Parity exchanges left and right chiralities. left-polarized this means that As neutrino is always Particle looked in the mirror and did not see itself??? How can this be? Alexey Boyarsky PPEU 19

20 CP-symmetry Another symmetry, charge conjugation comes to rescue. Charge conjugation exchanges particle and anti-particle. Combine it with parity (CP-symmetry): P: ν L ν R impossible CP: ν L ν R possible. Anti-neutrino exists and is always rightpolarized Life turned out to be more complicated. CP-symmetry is also broken Alexey Boyarsky PPEU 20

21 Parity and weak interactions L Fermi = G F 2 [ p(x)γ µ (1 γ 5 )n(x)][ē(x)γ µ (1 γ 5 )ν e (x)] }{{} β decay (5) + G F 2 [ µ(x)γ µ (1 γ 5 )ν µ (x)[ē(x)γ µ (1 γ 5 )ν e (x)] }{{} muon decay + G F 2 [ē(x)γ µ (1 γ 5 )ν e (x)[ē(x)γ µ (1 γ 5 )ν e (x)] }{{} electron-neutrino scattering... (pion decay, etc.) Only left (spin opposed to momentum) neutrinos and right (spin co-aligned with momentum) anti-neutrinos are produced or detected in weak interactions The weak interactions conserve flavour lepton numbers Alexey Boyarsky PPEU 21

22 µ e + ν e + ν µ ν µ + n p + µ Detection of other neutrinos Muon neutrino ν µ has been eventually detected via the process: If the particle produced in muon decay were ν e it would not produce muon in the second reaction (but electron instead) Such a reaction was observed in 1962 by Lederman, Schwartz and Steinberger In (1975), the third lepton, τ, has been discovered. The third type of neutrino, ν τ as found in (2000) To this date there has not been a single detection of ν τ, although we do believe in its existence Alexey Boyarsky PPEU 22

23 Fermion number conservation Recall L = ψ/ ψ + m ψψ does not change if ψ ψe iα Nöther theorem guarantees fermion number conservation: J µ F = ψγ µ ψ µ J µ F = 0 If there are several flavours (ψ i ) then L = N i=1 ψ i / ψ i + m i ψi ψ i (6) we have N conserved fermion (flavour) numbers J µ i = ψ i γ µ ψ i µ J µ i = 0 As a consequence J µ F = i J µ i is also conserved Alexey Boyarsky PPEU 23

24 Flavour lepton numbers Define flavor lepton numbers L e, L ν, L τ : L e L µ L τ L e L µ L τ (ν e, e ) ( ν e, e + ) (ν µ, µ ) ( ν µ, µ + ) (ν τ, τ ) ( ν τ, τ + ) Total lepton number is L tot = L e + L µ + L τ. Symmetry of the Standard Model: number and total lepton number conserved flavour lepton Fermi interactions respect this symmetry L = ) [ ( ) ( ( νe 1 0 V e )] i/ + Fermi 0 ν µ V µ }{{ Fermi } weak interactions ( νe ν µ ) Alexey Boyarsky PPEU 24

25 Neutrino experiments The atmospheric evidence: disappearance of ν µ and ν µ SuperKamioka atmospheric neutrinos (ν µ ν τ ) The solar evidence: deficit 50% of solar ν e (ν e ν µ,τ ) SNO The reactor evidence: disappearance of ν e produced by nuclear reactors. Back to neutrinos KamLAND Alexey Boyarsky PPEU 25

26 How is this possible if weak interactions conserve flavour? Alexey Boyarsky PPEU 26

27 Mass and charge eigenstates Define charge eigenstates as those where interaction term is diagonal L = ) [ ( ) ( ( νe 1 0 V e )] i/ + Fermi 0 ν µ V µ }{{ Fermi } weak interactions ( νe ν µ ) For exampe, V Fermi G F n e if neutrinos propagate in the medium with high density of electrons (interior of the Sun) If neutrinos have mass, the mass term is not necessarily diagonal in this basis: L = ) [ ( ) ( ( νe 1 0 V e )] i/ + Fermi 0 ν µ V µ }{{ Fermi } weak interactions ( νe ν µ ) ) ( ) ( ) ( νe m11 m + 12 νe ν µ m 21 m 22 ν µ Alexey Boyarsky PPEU 27

28 Mass and charge eigenstates One can define mass eigenstates such that the kinetic plus mass term is diagonal in this basis L = ( ψ1 ψ 2 ) [i/ ( ) ( )] ( ) ( ) ( ) ( ) m1 0 ψ1 ψ1 V11 V + 12 ψ1 0 m 2 ψ 2 ψ 2 V 21 V 22 ψ 2 A unitary transformation rotates between these two choices of basis ( νe ν µ ) = ( cos θ ) sin θ sin θ }{{ cos θ } matrix U ( ψ1 ψ 2 ) Alexey Boyarsky PPEU 28

29 Neutrino oscillations Consider the simplest case: two flavours, two mass eigen-states. Matrix U is parametrized by one mixing angle θ ν e = cos θ 1 + sin θ 2 ν µ = cos θ 2 sin θ 1 Let take the initial state to be ν e (created via some weak process) at time t = 0: ψ 0 = ν e = cos θ 1 + sin θ 2 Then at time t > 0 ψ t = e ie 1t cos θ 1 + sin θ 2 e ie 2t We detect the particle later via another weak process (e.g. ν? + n p + µ /e ) Alexey Boyarsky PPEU 29

30 Neutrino oscillations The probability of conversion ν e ν µ is given by ( ) P (ν e ν µ ) = ν µ ψ t 2 = sin 2 (2θ) sin 2 (E2 E 1 )t 2 The probability to detect ν e is give by ( ) P (ν e ν e ) = ν e ψ t 2 = cos 2 (2θ) sin 2 (E2 E 1 )t 2 Alexey Boyarsky PPEU 30

31 Fermion number conservation? Apparent violation of flavour lepton number for neutrinos can be explained by the presene of the non-zero neutrino mass ν ] e L = ν µ [i/ V Fermi ν e ν µ + ν e ν µ m 11 m m 21 m ν e ν µ ν τ }{{} ν τ ν τ ν τ conserves flavour number In this case only one fermion current (total lepton fermion number) is conserved: J µ = ν i γ µ ν i (7) i=e,µ,τ while any independent J µ i = ν i γ µ ν i is not conserved. Alexey Boyarsky PPEU 31

32 Neutrino oscillations The prediction is: neutrinos oscillate, i.e. probability to observe a given flavour changes with the distances travelled: ( ) P α β = sin 2 (2θ) sin m2 L GeV E ev 2 km (8) Mass difference: m 2, Neutrino energy: E (kev-mev in stars, GeV in air showers, etc. Distance traveled: L Alexey Boyarsky PPEU 32

33 Neutrino oscillations Eq. (8) predicts that probability oscillates as a function of the ratio E/L. This is indeed observed: in this plot the distance between reactor and detector and energy is different, therefore E/L is different Alexey Boyarsky PPEU 33

34 3 neutrino generations Neutrino experiments determine two mass splittings between three mass eigenstates (m 1, m 2, m 3 ): m 2 solar = ev 2 and m 2 atm = ev 2 A 3 3 unitary transformation U relates mass eigenstates (ν 1, ν 2, ν 3 ) to flavour eigenstates ν e = U e1 U e2 U e3 U µ1 U µ2 U µ3 ν 1 ν 2 U τ1 U τ2 U τ3 ν 3 ν µ ν τ Any unitary 3 3 martix has 9 real parameters: λ 1 u 12 e iδ 12 u 13 e iδ 13 U = Exponent i u 12 e iδ 12 λ 2 u 23 e iδ 23 u 13 e iδ 13 u 23 e iδ 23 λ 3 How many of them can be measured in experiments? Alexey Boyarsky PPEU 34

35 Neutrino mixing matrix Recall that neutrinos ν e,µ,τ couple to charged leptons Invariant under ν e ν e e iα simultaneously with e e e iα, etc. All other terms in the Lagrangian have the form ψ /Dψ or m ψψ i.e. are invariant if ψ ψe iα (here ψ is any of ν e, ν µ, ν τ, e, µ, τ) Additionally, we can rotate each of the ν 1,2,3 by an independent phase 5 of 9 parameters of the mixing matrix U can be absorbed in the redefinitions of ν 1,2,3 and ν e,µ,τ (6th phase does is overall redefinition of all fields does not change U). Alexey Boyarsky PPEU 35

36 Neutrino mixing matrix The rest 9 5 = 4 parameters are usually chosen as follows: 3 mixing angles θ 12, θ 23, θ 13 and 1 phase φ (since 3 3 real orthogonal matrix has 3 parameters only) U = c 12 c 13 c 13 s 12 s 13 U = c 23 s 12 e iφ c 12 s 13 s 23 c 12 c 23 e iφ s 12 s 13 s 23 c 13 s 23 s 23 s 12 e iφ c 12 c 23 s 13 c 12 s 23 e iφ c 23 s 12 s 13 c 13 c 23 (9) where one denotes cos θ 12 = c 12, sin θ 23 = s 23, etc. Three rotations plus one phase φ: cos θ 0 cos θ 23 sin θ e iφ sin θ 13 cos θ 12 sin θ sin θ 12 cos θ sin θ 23 cos θ 23 e iφ sin θ 13 0 cos θ Alexey Boyarsky PPEU 36

37 Problems for mixing matrix U 1. Show that for two flavour and two mass eigenstates the matrix U has 1 real free parameter (a mixing angle) 2. Show that if any of the angle θ 12, θ 23 or θ 13 is equal to zero, the matrix U can be chosen real Alexey Boyarsky PPEU 37

38 Discrete symmetries of the Standard Model Charge conjugation C : Particle Antiparticle In general, all elementary particles can be divided into three groups: Truly neutral, like photon, intermediate Z-boson, Majorana neutrino. They do not carry any charges Particles: like proton, neutron, and electron Antiparticles: antiproton, antineutron, and positron Naturally, a matter is a substance which consists of particles, and antimatter is a substance consisting of antiparticles. Parity P : Left-Handed Right-Handed. ν( p) P ν( p). Parity is broken in weak interactions (only left neutrinos and right antineutrinos participate in weak interactions) Alexey Boyarsky PPEU 38

39 T, CP, CPT,... Time reversal T : P α β T P β α CP : ν( p) CP ν( p) P α β CP Pᾱ β CP was believed to be the exact symmetry of nature after parity violations were discovered However: CP-violation in kaon decays (1964 Cronin, Fitch,... ) In a small fraction of cases ( 10 3 ), long-lived K L (a mixture of K 0 and K 0 decays into pair of two pions, what is forbidden by CP-conservation. If CP were exact symmetry, an equal number of K 0 and K 0 would produce an equal number of electrons and positrons in the reaction K 0 π e + ν e, K0 π + e ν e, Alexey Boyarsky PPEU 39

40 T, CP, CPT,... However, the number of positrons is somewhat larger ( 10 3 ) than the number of electrons. CPT: P α β CP T P β ᾱ CPT theorem: particles and antiparticles have the same mass, the same lifetime, but all their charges (electric, baryonic, leptonic, etc) are opposite. All known processes conserve CPT Alexey Boyarsky PPEU 40

41 Problems for discrete symmetries 1. s-quark carries a quantum number called strangness (S s = 1 s ). K + = u s. What is charge conjugated of K +? 2. Neutral kaon K 0 = d s. 3. Is K 0 a truly neutral particle (i.e. does it coincide with its own antiparticle)? 4. Is π 0 meson a truly neutral particle? Why? 5. Violation of P and CP are experimentally observed, while all processes are CPT symmetric. What other discrete symmetries are broken as a consequence of these experimental facts? Alexey Boyarsky PPEU 41

42 CP-violation in neutrino oscillations The matrix that rotates mass to flavour eigenstates: U = c 12 c 13 c 13 s 12 s 13 c 23 s 12 e iφ c 12 s 13 s 23 c 12 c 23 e iφ s 12 s 13 s 23 c 13 s 23 s 23 s 12 e iφ c 12 c 23 s 13 c 12 s 23 e iφ c 23 s 12 s 13 c 13 c 23 ν α = U αi ν i (α is a flavour index e, µ, τ and i is the mass index 1, 2, 3). Then probability P να ν β i U aiuiβ 2 CP-violation in neutrino oscillations? ( ) P να ν β P να ν β Im U αk UkβU αju jβ k<j } {{ } sin φ 0 Alexey Boyarsky PPEU 42

43 Neutrino mass from the point of view of the Standard Model? Alexey Boyarsky PPEU 43

44 Majorana mass terms Mass eigenstates ν 1,2,3 are freely propagating massive fermions Majorana mass term: L Majorana = νc 1 ν2 c ν3 c m m m 3 ν 1 (10) ν 2 ν 3 m 1, m 2, m 3 can be complex... or in terms of charge eigenstates: L Majorana = M αβ ν α ν β, α, β = e, µ, τ matrix M αβ is non-diagonal Can such a term exist in the Standard Model? Alexey Boyarsky PPEU 44

45 W -boson interactions Recall that Fermi-interactions are just a low energy limit of the interaction, mediated by a vector boson : L Fermi L W = g(w + µ J µ + h.c.) 1957 The W ± are charged vector bosons and the intraction currents have Alexey Boyarsky PPEU 45

46 W -boson interactions lepton and hadron contributions: J λ lepton = ν e γ λ (1 γ 5 )e + ν µ γ λ (1 γ 5 )µ +... J λ hadron = ūγ λ (1 γ 5 )d + other quarks Charged nature of the W -bosons leads to two new interaction vertices involving photon. notice that the diagram (a) depends both minimal term e that has the structure of ew W A and non-minimal term of the form eκw W A The interaction vertex W W γ has the following form (for the choice of Alexey Boyarsky PPEU 46

47 W -boson interactions momentum as shown in Fig. (a) above) V λµν (k, p, q) =e (k p) ν η µλ + (p q) λ η µν + (q k) µ η }{{ νλ } V λµν YM (k,p,q) + e(1 κ)(q λ η µν q µ η νλ ) (11) The interaction vertex W W γγ has the structure independent on κ, but proportional on e 2 rather than e. V µνρσ = e 2 (2η µν η ρσ η µρ η νσ η µσ η νρ ) (12) Alexey Boyarsky PPEU 47

48 Z-boson In addition to W there is another massive neutral vector boson New vector boson couples to electrons and to neutrinos in the parity violating way and that also couples to W + W. New boson (Z-bosons) interacts with ν: L ννz = 1 2 g ννz νγ µ (1 γ 5 )νz µ (13) New boson also interacts with W + W and the vertex W W Z is similar to the vertex W W γ As result there are two processes contributing to νν W + W scattering Alexey Boyarsky PPEU 48

49 Z-boson Alexey Boyarsky PPEU 49

50 Interactions of intermediate vector bosons The kinetic term for W -boson is L W = 1 2 (D µwν D µ Wν )(D µ W ν + D µ W ν + )+ M W 2 2 W µ + Wµ +ef µν W µ + Wν (14) where D µ W ν ± = ( µ iea µ )W ν ± Similarly for Z-boson L Z = 1 2 ( µz ν ν Z µ ) 2 + ZW W + ZZW W (15) Alexey Boyarsky PPEU 50

51 Symmetry between e and ν e? Consider 2 different fermion fields ψ (1) and ψ (2) which are physically equivalent for some interaction (good historical example is n and p for strong interactions). The Dirac equation is (iγ µ µ m) ψ (1) + (iγ µ µ m) ψ (2) + V int (ψ (1) ) + V int (ψ (2) ) (16) We can compose two-component field Ψ = Dirac equation using Ψ as ( ) ψ (1) ψ (2) and rewrite the (iγ µ µ m) Ψ + L int ( Ψ) = 0 (17) Probability P = [ ] d 3 x ψ (1) (x)γ 0 ψ (1) (x) + ψ (2) γ 0 ψ (2) = d 3 x Ψ + Ψ Alexey Boyarsky PPEU 51

52 Symmetry between e and ν e? and Dirac equation (17) are invariant under global transformations: Ψ(x) Ψ (x) = U Ψ(x) (18) that leaves the length of the two-dimensional isovector invariant. Ψ Such transformation is called unitary transformation and the matrix U in Eq. is 2 2 complex matrix which obeys conditions U + U = 1 and det(u) = 1. Alexey Boyarsky PPEU 52

53 Majorana mass term Majorana mass term couples ν and its charge conjugated However, neutrino is a part of the SU(2) doublet L = therefore a Majorana mass term ν c ν ( L H)(L H) Λ ( ν e) and (19) This is operator of dimension 5 (similar to G F ) ν e ν µ ν e ν µ H H Λ F ei N I F µi and it means that a new particle is present Alexey Boyarsky PPEU 53

54 Problems on neutrino mass term 1. Write the Dirac equation for fermions ψ i = (ν i, ν ir ), i = 1, 2, 3 2. The Majorana equation can be written for 2-component fermion as iσ µ µ ν + mν c = 0 (20) (where σ are Pauli matrices, σ 0 = 1 and ν c = iσ 2 ν ) Show that its Lagrangian is Lorentz invariant (hint: use expression (10) as a mass term). 3. Show that the mass m in (20) can be complex number, yet the fermion ν has the dispersion relation E 2 = p 2 + m 2 4. Show that rotation of each of the ν 1,2,3 by a phase leaves the term (15) invariant (if one performs additional rotation of ν ir ) Alexey Boyarsky PPEU 54

55 Problems on neutrino mass term 5. Show that rotation of each of the ν 1,2,3 changes phase of the masses in (10) (which does not affect propagation in view of the exercise 3). 6. Show that of three Majorana masses in (10) one can be made real without changing the form of the matrix U in (16). Alexey Boyarsky PPEU 55

56 See-saw Lagrangian Add right-handed neutrinos N I to the Standard Model ν e N 1 N c 1 L right = i N I / N I + ν µ F H N 2 + N c 2 ν τ }{{} Dirac mass M D M }{{} Majorana mass N 1 N 2... ν α = HL α, where L α are left-handed lepton doublets Active masses are given via usual see-saw formula: (m ν ) = m D 1 M I m T D ; m D M I Neutrino mass matrix 7 parameters. Dirac+Majorana mass matrix 11 (18) parameters for 2 (3) sterile neutrinos. Two sterile neutrinos are enough to fit the neutrino oscillations data. Alexey Boyarsky PPEU 56

57 Problems about see-saw Lagrangian 1. Demonstrate that knowing the masses of all neutrinos does not allow to fix the scale of masses m D and M M. 2. Consider the Lagrangian with only one flavour and introduce one singlet right handed neutrino ν R, and add both Majorana mass term to it and Dirac mass term via Higgs mechanism L seesaw = L SM + λ N Le H c (ν R ) M M ( ν c R) (ν R ) + h.c.. (21) Suppose, that the Dirac mass m D = λ N v is much smaller than Majorana mass M M, m D M M. Find the spectrum (mass eigenstates) in (21). Identify linear combinations of ν and N that are mass mass eigenstates and rewrite the Lagrangian in this basis. Do not forget that the left double L e participates in electric and weak interactions. 3. Generalize the above see-saw Lagrangian (21) for the number of SM lepton flavors other than one. Alexey Boyarsky PPEU 57

58 Problems about see-saw Lagrangian 4. Can one obtain the observed mass splittings (see e.g. PDG) by adding only one right-handed neutrino in the three-flavour generalization of the Lagrangian (21)? 5. Generalize the above see-saw Lagrangian (21) for all three SM lepton flavors and N generations of right-handed neutrinos. How many new parameters appears in the see-saw Lagrangian for the case of N = 1, 2, 3? Alexey Boyarsky PPEU 58

59 The simple model CP-violating processes L = L SM + N 1 / N 1 + N 2 / N 2 + λ 1 N1 HL + λ 2 N2 HL + M N M N 2 2 (22) N 1 λ L L H L N1 λ N1 λ N2 N2 c H H H L H L Tree level decay of N 1 L + H (the first graph): Γ = λ 1 2 M 1 8π complex phase does not contributes! Alexey Boyarsky PPEU 59

60 CP-violating processes N 1 λ L L H L N1 λ N1 λ N2 N2 c H H H L H L Sum of the matrix elements of all three graphs gives: Γ(N 1 LH) λ 1 + Aλ 1λ 2 2 2, Γ(N 1 LH ) λ 1 + Aλ 1 λ where A is some CP-conserving number Γ(N 1 LH) Γ(N 1 LH ) Γ(N 1 LH) + Γ(N 1 LH ) Im(λ2 2) Alexey Boyarsky PPEU 60