Neutrino Phenomenology. Boris Kayser ISAPP July, 2011 Part 1

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1 Neutrino Phenomenology Boris Kayser ISAPP July, 2011 Part 1 1

reaction Spin: p + p "d + e + +# 1 2 1 2 1 1 2 1

2 What Are Neutrinos Good For? Energy generation in the sun starts with the reaction Spin: p + p "d + e + +# Without the neutrino, angular momentum would not be conserved. Uh, oh 2

The Neutrinos Neutrinos and photons are by far the most abundant elementary particles in the universe. There are 340 neutrinos/cc. The neutrinos are spin 1/2, electrically neutral, leptons.

3 The Neutrinos Neutrinos and photons are by far the most abundant elementary particles in the universe. There are 340 neutrinos/cc. The neutrinos are spin 1/2, electrically neutral, leptons. The only known forces they experience are the weak force and gravity. This means that their interactions with other matter have very low strength. Thus, neutrinos are difficult to detect and study. Their weak interactions are successfully described by the Standard Model. 3

4 The Neutrino Revolution (1998 ) Neutrinos have nonzero masses! Leptons mix! 4

5 These discoveries come from the observation of neutrino flavor change (neutrino oscillation). 5

6 The Physics of Neutrino Oscillation Preliminaries 6

7 The Neutrino Flavors We define the three known flavors of neutrinos, ν e, ν µ, ν τ, by W boson decays: e µ τ W ν e W ν µ W ν τ As far as we know, neither W µ Short Journey ν µ e Detector nor any other change of flavor in the ν interaction ever occurs. With α = e, µ, τ, ν α makes only α ( e e, µ µ, τ τ). 7

8 Neutrino Flavor Change If neutrinos have masses, and leptons mix, we can have µ τ π W ν µ Long Journey ν τ Detector Give ν time to change character ν µ ν τ The last decade has brought us compelling evidence that such flavor changes actually occur. 8

9 Flavor Change Requires Neutrino Masses There must be some spectrum of neutrino mass eigenstates ν i : ν 3 (Mass) 2 ν 2 ν 1 Mass (ν i ) m i 9

10 Flavor Change Requires Leptonic Mixing The neutrinos ν e,µ,τ of definite flavor (W eν e or µν µ or τν τ ) must be superpositions of the mass eigenstates: ν α > = Σ U* αi ν i >. i Neutrino of flavor Neutrino of definite mass m i α = e, µ, or τ PMNS Leptonic Mixing Matrix There must be at least 3 mass eigenstates ν i, because there are 3 orthogonal neutrinos of definite flavor ν α. 10

11 This mixing is easily incorporated into the Standard Model (SM) description of the νw interaction. For this interaction, we then have Semi-weak coupling L SM = " g 2 Left-handed! L# $ % " & L# W % + & L# $ % + ( (! L# W % ) #=e,µ,' = " g 2 ( #=e,µ,' i = 1,2,3 (! L# $ % " U #i & Li W % + & Li $ % * U #i! + L# W % ) Taking mixing into account If neutrino masses are described by an extension of the SM, and there are no new leptons, U is unitary. Then ( ) ' U #i Amp W "! # + + $# ;$ # "! % & + W 3 ( * U%i = ) %#, as observed. i=1 11

12 The Meaning of U W + W + # +! " g 2 U * "i " i g 2 U "i " i! " # U = e µ " " 1 " 2 " 3 # U e1 U e2 U e3 & % ( % U µ1 U µ2 U µ3 ( $ % U "1 U " 2 U " 3 '( The e row of U: The linear combination of neutrino mass eigenstates that couples to e. The ν 1 column of U: The linear combination of charged-lepton mass eigenstates that couples to ν 1. 12

13 The Physics of Neutrino Oscillation Boris Kayser ISAPP July,

14 Neutrino Flavor Change (Oscillation) in + l α (e.g. µ) Vacuum ( Approach of ) B.K. & Stodolsky - l β (e.g. τ) Amp ν W (ν α ) (ν β ) W Source Target l α + l β - = Σ Amp i W U αi * ν i Prop(ν i ) U W βi Source Target 2

15 Amp [ν α ν β ] = ΣU αi * Prop(ν i )U βi What is Propagator (ν i ) Prop(ν i )? In the ν i rest frame, where the proper time is τ i, Thus, i τ i ν i (τ i ) >= m i ν i (τ i ) >. ν i (τ i ) >= e im iτ i ν i (0) >. Then, the amplitude for propagation for time τ i is Prop(ν i ) < ν i (0) ν i (τ i ) >= e im iτ i. 3

16 In the laboratory frame Time t ν Distance L The experimenter chooses L and t. They are common to all components of the beam. For each ν i, by Lorentz invariance, m i τ i = E i t - p i L. (E i, p i ) x (t, L) = m i τ i = E i t p i L. 4

17 Neutrino sources are ~ constant in time. Averaged over time, the e -ie 1t e -ie 2t interference is < e -i(e 1-E 2 )t > t = 0 unless E 2 = E 1. Only neutrino mass eigenstates with a common energy E are coherent. (Stodolsky) 5

18 For each mass eigenstate, p i = E 2 m 2 i = E m2 i 2E. Then the phase in the ν i propagator exp[-im i τ i ] is m i τ i = E i t p i L = ~ Et (E m 2 i /2E)L Irrelevant overall phase = E(t L) + m i 2 L/2E. } 6

19 Amp [ν α ν β ] l α + l β - = Σ Amp i W U αi * ν i e im2 i Prop(ν i ) L 2E U W βi Source Target = i U αie im2 i L 2E Uβi 7

20 Probability for Neutrino Oscillation in Vacuum P (ν α ν β )= Amp(ν α ν β ) 2 = = δ αβ 4 i>j R(U αiu βi U αj U βj) sin 2 ( m 2 ij +2 i>j I(U αiu βi U αj U βj) sin( m 2 ij L 2E ) L 4E ) where m 2 ij m 2 i m 2 j 8

21 For Antineutrinos We assume the world is CPT invariant. Our formalism assumes this. 9

22 P (ν α ν β ) CPT = P (ν β ν α )=P (ν α ν β ; U U ) Thus, ( ) ( ) P (ν α ν β )= = δ αβ 4 i>j R(U αiu βi U αj U βj) sin 2 ( m 2 ij +2 i>j I(U αiu βi U αj U βj) sin( m 2 ij ( ) L 2E ) L 4E ) A complex U would lead to the CP violation P (ν α ν β ) P (ν α ν β ). 10

23 Must we assume all mass eigenstates have the same E? No, we can take entanglement into account, and use energy conservation. The oscillation probabilities are still the same. 11

24 Comments 1. If all m i = 0, so that all Δm ij 2 = 0, 2. If there is no mixing, ( ) ( ) P (ν α ν β ) = δ αβ Flavor change ν Mass W l α but W l β α ν i ν j i always same ν i U α i U β α,i = 0, so that P (ν ( ) α ( ) ν β ) = δ αβ. Flavor change Mixing 12

25 3. One can detect (ν α ν β ) in two ways: See ν β α in a ν α beam (Appearance) See some of known ν α flux disappear (Disappearance) 4. Including ħ and c m 2 L 4E =1.27 m2 (ev 2 ) L(km) E(GeV) sin 2 [1.27 m 2 (ev) 2 L(km) becomes appreciable when E(GeV) ] its argument reaches O(1). An experiment with given L/E is sensitive to m 2 (ev 2 ) > E(GeV). L(km) 13

26 5. Flavor change in vacuum oscillates with L/E. Hence the name neutrino oscillation. {The L/E is from the proper time τ.} ( ) ( ) 6. P (ν α ν β ) depends only on squared-mass splittings. Oscillation experiments cannot tell us (mass) 2 ν 3 } ν 2 2 ν 1 }Δm 21 2 Δm 32?? 0 14

27 7. Neutrino flavor change does not change the total flux in a beam. It just redistributes it among the flavors. All β ( ) ( ) P (ν α ν β )=1 But some of the flavors β α could be sterile. Then some of the active flux disappears: φ νe + φ νµ + φ ντ <φ Original 15

28 Important Special Cases For, Three Flavors e im2 1 L 2E Amp (ν α ν β )= i U αi U βie im2 i L 2E e im2 1 L 2E = U +U α1 β1 α3 Uβ3e 2i 31 + U α2 Uβ2e 2i 21 (U α3 Uβ3 + U α2 Uβ2) }{{} Unitarity =2i[U α3 U β3e i 31 sin 31 + U α2 U β2e i 21 sin 21 ] where ij m 2 ij L 4E (m2 i m 2 j) L 4E. 16

29 P (ν α ν β )= 2E Amp (ν α ν β ) ( ) ( ) e im2 1 L ( ) ( ) 2 =4[ U α3 U β3 2 sin U α2 U β2 2 sin U α3 U β3 U α2 U β2 sin 31 sin 21 cos( 32 + δ 32 )]. ( ) Here δ 32 arg(u α3 U β3u α2u β2 ), a CP violating phase. Two waves of different frequencies, and their CP interference. 17

30 When the Spectrum Is ν 3 ν 2 ν 1 Invisible if Δm 2 or Δm 2 { m 2 = O(1). LE ν 2 ν 1 ν 3 For, ( ) ( ) P (ν α ν β ) = 4 U α3 U β3 2 sin 2 ( m 2 L 4E ). For no flavor change, ( ) ( ) P (ν α ν α ) = 1 4 U α3 2 (1 U α3 2 ) sin 2 ( m 2 L 4E ). Experiments with flavor content of ν 3. m 2 L E = O(1) can determine the 18

31 When There are Only Two Flavors and Two Mass Eigenstates ν 2 U = Uα1 U β1 U α2 U β2 ν 1 Δm 2 cos θ = sin θ sin θ cos θ Majorana CP phase e iξ Mixing angle ( ) ( ) For, P (ν α ν β ) = sin 2 2θ sin 2 ( m 2 L 4E ). ( ) ( ) For no flavor change, P (ν α ν α )=1 sin 2 2θ sin 2 ( m 2 L 4E ). 19

32 Neutrino Flavor Change in Matter Coherent forward scattering from ambient matter can have a big effect. V W =+ 2G F N e ( for ν e ) V Z = 2 2 G F N n #e/vol (+ for ν α ) #n/vol 20

33 Neutrino propagation in matter is conveniently treated via a Schrödinger Equation: i ν(t) =Hν(t) t Matrix in flavor space Multi-component { in flavor space To illustrate, we describe the case When Only Two Neutrinos Count Amp. to be a ν e ν(t) = [ fe (t) f µ (t) Amp. to be a ν µ ] ν ; H = ν [ e ν µ ] e ν µ A 2x2 matrix in ν e -ν µ space 21

34 i t [ fe (t) f µ (t) ] = [ H ] [ fe (t) f µ (t) ] In general <ν α H ν β >=< i U αiν i H j U βjν j >= j In vacuum U αj U βj p 2 + m 2 j Momentum of the beam In flavor change, only relative phases, hence relative energies, matter. In H, any multiple of the Identity Matrix I may be omitted. 22

35 In Vacuum ν 2 U = ν [ e ν µ ν 1 ν 2 cos θ sin θ sin θ cos θ ν 1 Δm 2 It follows that, omitting a piece I, ] ; ν e = ν 1 cos θ + ν 2 sin θ ν µ ν 1 ( sin θ)+ν 2 cos θ H Vac = m2 4E [ cos 2θ sin 2θ sin 2θ cos 2θ ]. With Schrödinger s Equation, this gives the usual P(ν e ν µ ). 23

36 The eigenvalues of H Vac are With c cos θ, s sin θ, ± m2 4E ±λ. ν e = ν 1 c + ν 2 s t ν(t) =ν 1 ce iλt + ν 2 se iλt P (ν e ν µ )= <ν µ ν(t) > 2 = sc( e iλt + e iλt ) 2 = sin 2 2θ sin 2 ( m 2 L 4E ) 24

37 H M = H Vac + V W 2 In Matter [ ν e ν µ 1 0 H M = H Vac + V W 0 0 [ ] ν e ν µ ] + + V W 2 [ ν e ν µ ] 1 0 ν e V Z 0 1 ν µ I, so drop [ ] H M = m2 4E [ (cos 2θ x) sin 2θ sin 2θ cos 2θ x ], with x V W /2 m 2 /4E = 2 2GF N e E m 2. 25

38 The Effective Splitting and Mixing in Matter If we define m 2 M m 2 sin 2 2θ + (cos 2θ x) 2 and then sin 2 2θ M H M = m2 M 4E sin 2 2θ sin 2 2θ + (cos 2θ x) 2 [ ] cos 2θM sin 2θ M sin 2θ M cos 2θ M,. This is H Vac with (Δm 2, θ) (Δm M 2, θ M ). Thus, Δm M 2 and θ M are the effective splitting and mixing angle in matter. 26

39 Travel Through the Earth The matter density encountered en route is ~ constant. Thus, H M is position-independent, just like H Vac. Therefore, in the earth (but not too deep), P M (ν e ν µ ) = sin 2 2θ M sin 2 ( m 2 M L 4E ) In matter 27

40 The Size and Consequence of the Matter Effect The matter effect depends on x = 2 2G F N e E m 2 E. The denominator contains a Sign In the earth s mantle, for Δm 2 = Δm 2 (atmospheric) 2.4 x 10-3 ev 2, x E 11GeV Since V W (ν) = -V W (ν), x(ν) = -x(ν). 2 2 Thus Δm M ΔmM and sin 2 2θ M sin 2 2θ M. The matter effect causes an asymmetry between ν and ν oscillation. This must be separated from the genuine CP asymmetry. 28

41 Evidence For Flavor Change Neutrinos Solar Reactor (L ~ 180 km) Atmospheric Accelerator (L = 250 and 735 km) Evidence of Flavor Change Compelling Compelling Compelling Compelling Stopped µ + Decay ( LSND ) L 30 m Does MiniBooNE see this too?? Very recent evidence to be discussed soon. 14

42 Recent Evidence From Gran Sasso Oscillation Prediction 15

43 KamLAND Evidence for O s c i l l a t o r y Behavior 16

44 The KamLAND detector studies ν e produced by Japanese nuclear power reactors ~ 180 km away. For KamLAND, x Matter < Matter effects are negligible. ( ) " e P " e # " e The survival probability,, should oscillate as a function of L / E following the vacuum oscillation formula. In the two-neutrino approximation, we expect ( ) =1$ sin 2 2% sin &m 2 ev 2 P " e # " e ' ) ( *, E(GeV ) + ( ) L(km). 17

45 Survival probability P(ν e ν e ) of reactor ν e L 0 = 180 km is a flux-weighted average travel distance. P(ν e ν e ) actually oscillates! 18

46 What We Have Learned 19

47 The (Mass) 2 Spectrum ν 3 ν 2 ν 1 } Δm 2 sol (Mass) 2 Δm 2 atm or Δm 2 atm ν ν 2 } Δm 2 sol 1 ν 3 Normal Inverted Δm 2 sol = ~ 7.5 x 10 5 ev 2, Δm 2 atm = ~ 2.3 x 10 3 ev 2 20

48 The Absolute Scale of Neutrino Mass ν 3 Δm2 atm } ν Oscillation (Mass) 2 0 ν 2 1?? Δm 2 sol } Cosmology, β Decay, How far above zero is the whole pattern? Oscillation Data Δm 2 atm < Mass[Heaviest ν i ] 21

49 The Upper Bound From Cosmology (See Sergio Pastor) Cosmological Data + Cosmological Assumptions An Upper Bound (and maybe someday a value) for m i. Σ 22

50 The Upper Bound From Tritium Cosmology is wonderful, but there are known loopholes in its argument concerning neutrino mass. The absolute neutrino mass can in principle also be measured by the kinematics of β decay. Tritium decay: 3 H" 3 He + e # + $i ; i =1, 2, or 3 BR( 3 H" 3 He + e # + $ i ) % U ei 2 3 H" 3 He + e # + $i In, the bigger m i is, the smaller the maximum electron energy is. There are 3 separate thresholds in the β energy spectrum. 23

51 The β energy spectrum is modified according to ( E 0 " E) 2 2 #[ E 0 " E] $ % U ei ( E 0 " E) ( E 0 " E) 2 2 " m i # ( E0 " m i ) " E i [ ] Maximum β energy when there is no neutrino mass β energy Present experimental energy resolution is insufficient to separate the thresholds. Measurements of the spectrum bound the average neutrino mass 2 2 m " = # U ei mi Presently: m " < 2 ev i Mainz & Troitzk 24

52 Leptonic Mixing This has the consequence that Mass eigenstate ν i > = Σ U αi ν α >. e, µ, or τ α Flavor eigenstate Leptonic Mixing Matrix Flavor-α fraction of ν i = U αi 2. When a ν i interacts and produces a charged lepton, the probability that this charged lepton will be of flavor α is U αi 2. 25

53 The spectrum, showing its approximate flavor content, is sin 2 θ 13 ν 3 ν 2 ν 1 } Δm 2 sol (Mass) 2 Δm 2 atm or Δm 2 atm ν 2 ν 1 } Δm 2 sol ν 3 sin 2 θ 13 Normal Inverted ν e [ U ei 2 ] ν µ [ U µi 2 ] ν τ [ U τi 2 ] 26

54 ν 3 Bounded by reactor exps. with L ~ 1 km From max. atm. mixing, " 3 # " µ +" $ 2 (Mass) 2 Δm 2 atm { { From ν µ (Up) oscillate but ν µ (Down) don t In LMA MSW, P sol (ν e ν e ) = ν e fraction of ν 2 ν 2 ν 1 }Δm 2 sol From distortion of ν e (reactor) and ν e (solar) spectra { From max. atm. mixing, ν 1 + ν 2 includes (ν µ ν τ )/ 2 ν e [ U ei 2 ] ν µ [ U µi 2 ] ν τ [ U τi 2 ] 27

55 The 3 X 3 Unitary Mixing Matrix Caution: We are assuming the mixing matrix U to be 3 x 3 and unitary. L SM = " g 2! L# $ % " & L# W % + & L# $ % + ( (! L# W % ) #=e,µ,' = " g 2 ( #=e,µ,' i = 1,2,3 (! L# $ % " U #i & Li W % + & Li $ % * U #i! + L# W % ) (CP)(! L" # $ & U "i % Li W $ ) (CP)&1 = % Li # $ + U "i! L" W $ Phases in U will lead to CP violation, unless they are removable by redefining the leptons. 28

56 U αi describes W + ν i α U αi α W + H ν i When ν i e iϕ ν i, U αi e iϕ U αi When α e iϕ α, U αi e iϕ U αi Thus, one may multiply any column, or any row, of U by a complex phase factor without changing the physics. Some phases may be removed from U in this way. 29

57 Exception: If the neutrino mass eigenstates are their own antiparticles, then Charge conjugate ν i = ν i c = Cν i T One is no longer free to phase-redefine ν i without consequences. U can contain additional CP-violating phases. 30

58 How Many Mixing Angles and CP Phases Does U Contain? Real parameters before constraints: 18 Unitarity constraints * $ U "i U#i = % "# i Each row is a vector of length unity: 3 Each two rows are orthogonal vectors: 6 Rephase the three α : 3 Rephase two ν i, if ν i ν i : 2 Total physically-significant parameters: 4 Additional (Majorana) CP phases if ν i = ν i : 2 31

59 How Many Of The Parameters Are Mixing Angles? The mixing angles are the parameters in U when it is real. U is then a three-dimensional rotation matrix. Everyone knows such a matrix is described in terms of 3 angles. Thus, U contains 3 mixing angles. Mixing angles Summary CP phases if ν i ν i CP phases if ν i = ν i

60 The Mixing Matrix Atmospheric Cross-Mixing Solar # & # c 13 0 s 13 e "i* & # c % ( % ( 12 s 12 0& % ( U = % 0 c 23 s 23 ( )% () % "s 12 c 12 0 ( $ % 0 "s 23 c 23 '( % $ "s 13 e i* 0 c ( 13 ' $ % 0 0 1' ( c ij cos θ ij s ij sin θ ij Hints?? θ 12 θ sol 34, θ 23 θ atm 39-51, θ 13 < ~ 12 δ would lead to P(ν α ν β ) P(ν α ν β ). CP But note the crucial role of s 13 sin θ 13. # e i+ 1 /2 0 0& % ) 0 e i+ 2 /2 ( % 0( % $ 0 0 1( ' Majorana CP phases 33

61 Recent Indication Of Non-Zero θ 13 In an experiment where L/E is too small for the small 2 splitting # 2 m2 $ 2 m1 to be seen, "m 21 ( ) 2 $ 4U µ3 U e3 sin 2 ( %m31 P " µ #" e & ' ) + 4E * 2 L = sin 2 2, 13 sin 2, 23 sin 2 & 2 L ) (%m ' E * T2K has looked for experiment: " µ # " e in a long-baseline 34

62 Source π µ + ν µ Near Detector The T2K experiment ν 295 km E = 0.6 GeV The far detector is near the first "m 2 31 oscillation maximum. Far Detector Super-K Expectation Observation T2K sees 6 ν e candidate events in the far detector, whereas 1.5 are expected if θ 13 = 0. 35

63 θ 13 could be large. These take the Δm 2 21 contributions and matter effects into account. 36

64 MINOS, not designed to look for " µ # " e, sees 62 candidate events where 50 are expected if θ 13 = 0. While not highly significant by itself, this result is consistent with that from T2K. 37

65 There Is Nothing Special About θ 13 All mixing angles must be nonzero for CP in oscillation. For example P (" µ # " e ) $ P (" µ # " e ) = 2cos% 13 sin2% 13 sin2% 12 sin2% 23 sin& ) ' sin (m 2 L, ) sin (m 2 L, ) sin (m 2 L, * 4E - * 4E - * 4E - In the factored form of U, one can put δ next to θ 12 instead of θ

66 The Majorana CP Phases The phase α i is associated with neutrino mass eigenstate ν i : U αi = U 0 αi exp(iα i /2) for all flavors α. Amp(ν α ν β ) = Σ U αi* exp( im i2 L/2E) U βi i is insensitive to the Majorana phases α i. Only the phase δ can cause CP violation in neutrino oscillation. 39

Neutrino Phenomenology. Boris Kayser INSS August, 2013 Part 1

Neutrino Phenomenology. Boris Kayser INSS August, 2013 Part 1 Neutrino Phenomenology Boris Kayser INSS August, 2013 Part 1 1 What Are Neutrinos Good For? Energy generation in the sun starts with the reaction Spin: p + p "d + e + +# 1 2 1 2 1 1 2 1 2 Without the neutrino,