A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy Invisibles network INT Training lectures June 25 – 29, 2012

Modern version

also to set notations Oscillation and flavor conversion are consequence of - the lepton mixing and - production of mixed (flavor) states

  e mass  m 2 31  m 2 21 Normal mass hierarchy |U e3 | 2 |U  3 | 2 |U  3 | 2 |U e1 | 2 |U e2 | 2 tan   23 = |U  3 | 2 / |U  3 | 2 sin   13 = |U e3 | 2 tan   12 = |U e2 | 2 / |U e1 | 2  m 2 31 = m m 2 1  m 2 21 = m m 2 1 Mixing parameters f = U PMNS mass U PMNS = U 23 I  U 13 I  U 12 flavor Mixing matrix: e   = U PMNS Mass states can be enumerated by amount of electron flavor

e e Charged current weak interactions Charged current weak interactions Kinematics of specific reactions Kinematics of specific reactions  decays, energy conservation Beam dump, D - decay  decays, chirality suppression     What about neutral currents? Difference of the charged lepton masses Non-trivial interplay of Mixing in CC  mixing in produced states Breaking of coherence

What is the neutrino state produced in the Z-decay in the presence of mixing? f = [ ] 1313 f H Z 2 = H Z 2 Z0Z0 i i Do neutrinos from Z 0 - decay oscillate? If the flavor of one of the neutrino is fixed, another neutrino oscillates P = sin 2 2  sin 2 [  (L 1 + L 2 )/ l ] Two detectors experiment: detection of both neutrinos L 1 L2L2 Z is flavor blind

Challenging theory of neutrino oscillations New experimental setups - LBL long tunnels - Long leaved parents Without paradoxes New aspects Simple and straightforward and still correct

[1] Neutrino production coherence and oscillation experiments. E. Akhmedov, D. Hernandez, A. Smirnov, JHEP 1204 (2012) 052, arXiv: [hep-ph] [2] Neutrino oscillations: Entanglement, energy-momentum conservation and QFT. E.Kh. Akhmedov, A.Yu. Smirnov, Found. Phys. 41 (2011) arXiv: [hep-ph] [3] Paradoxes of neutrino oscillations. E. Kh. Akhmedov, A. Yu. Smirnov Phys. Atom. Nucl. 72 (2009) arXiv: [hep-ph] [4] Active to sterile neutrino oscillations: Coherence and MINOS results. D. Hernandez, A.Yu. Smirnov, Phys.Lett. B706 (2012) arXiv: [hep-ph] [5] Neutrino oscillations: Quantum mechanics vs. quantum field theory. E. Kh. Akhmedov, J. Kopp, JHEP 1004 (2010) 008 arXiv: [hep-ph]

``Mesonium and antimesonium’’ Zh. Eksp.Teor. Fiz. 33, 549 (1957) [Sov. Phys. JETP 6, 429 (1957)] translation mentioned a possibility of neutrino mixing and oscillations B. Pontecorvo Oscillations imply non-zero masses (mass squared differences) and mixing Pisa, 1913 Proposal of neutrino oscillations by B Pontecorvo was motivated by rumor that Davis sees effect in Cl-Ar detector from atomic reactor

Lagrangian l     (1 -  5 ) l  W +  g 2 Amplitudes, probabilities of processes QFT QM Observables, Number of events, etc.. Starting from the first principles - ½ m L L T C L - l L m l l R + h.c.

Initial conditions Recall, the usual set-up asymptotic states described by plane waves single interaction region Formalism should be adjusted to specific physics situation Approximations, if one does not want to consider whole history of the Universe to compute signal in Daya Bay - enormous simplification Truncating the process

production region detection region baseline L Two interaction regions in contrast to usual scattering problem External particles Neutrinos: propagator Finite space and time phenomenon Finite space-time integration limits wave packets for external particles encode info about finite production and detection regions formalism should be adjusted to these condition E. Akhmedov, A.S. i S D QFT but

Wave packet formalism. Consistent description of oscillations requires consideration of wave packets of neutrino mass states. B. Kayser, Phys. Rev D 48 (1981) years later, GGI lectures: The highest level of sophistication: to use proper time for neutrino mass and get correct result! Key point: phases of mass eigenstates should be compared in the same space-time point If not - factor of 2 in the oscillations phase

production region detection region baseline L External particles How external particles Should be described? wave packets for external particles detection/production areas are determined by localization of particles involved in neutrino production and detection not source/detector volume (still to integrate over) i S D Describe by plane waves but introduce finite integration

production region detection region baseline L External particles wave packets for external particles Unique process, neutrinos with definite masses are described by propagators, Oscillation pattern – result of interference of amplitudes due to exchange of different mass eigenstates i S D Very quickly converge to mass shell Real particles – described by wave packets

production region detection region baseline L factorization If oscillation effect in Production/detection regions can be neglected i S D Production propagation and Detection can be considered as three independent processes r D, r S << l

i l i       i W  Scattering N Eigenstates of the Hamiltonian in vacuum U li l     (1 -  5 ) i  W +  + h.c. g 2 interaction constant Lagrangian of interactions Wave packet compute the wave function of neutrino mass eigenstates wave functions of accompanying particles Without flavor states

|  (x,t)  =  k U  k *  k (x, t)| k  After formation of the wave packet (outside the production region) Suppose  be produced in the source centered at x = 0, t = 0  k ~ dp f k (p – p k ) e ipx – iE k (p)t E k (p)= p 2 + m k 2 f k (p – p k ) - the momentum distribution function peaked at p k - the mean momentum - dispersion relation In general E k (p) = E k (p k ) + (dE k /dp) (p - p k ) + (dE k 2 /dp 2 ) (p - p k ) 2 + … pk pk pk pk Expanding around mean momentun - group velocity of k v k = (dE k /dp) = (p/E k ) pk pk pk pk describes spread of the wave packets

Inserting into  k ~ e g k (x – v k t) E k (p k ) = p k 2 + m k 2 Depends on mean characteristics p k and corresponding energy: E k (p) = E k (p k ) + v k (p - p k ) (neglecting spread of the wave packets)  k ~ dp f k (p – p k ) e ipx – iE k (p)t ip k x – iE k (p k )t g k (x – v k t) = dp f k (p) e ip(x – v k t) Shape factor Phase factor ikik e  k = p k x – E k t Depends on x and t only in combination (x – v k t) and therefore Describes propagation of the wave packet with group velocity v k without change of the shape

2 x 1 xx e = cos  1  sin    = - sin  1  cos   cos  sin  e opposite phase 1 = cos  e  sin   2 = cos    sin   | (x,t)  = cos  g 1 (x – v 1 t)e | 1  + sin  g 2 (x – v 2 t)e | 2  i2i2 i1i1  =  2  -  1 Interference of the same flavor parts Oscillation phase  =  0 Main, effective frequency

One needs to compute the state which is produced g k (x – v k t) the shape factors mean momenta p k i.e. compute - Fundamental interactions - Kinematics - characteristics of parent and accompanying particles If heavy neutrinos are present but can not be produced for kinematical reasons, flavor states in Lagrangian differe from the produced states, etc.. Process dependent

What happens? Due to different masses (dispersion relations)  phase velocities Due to different group velocities Due to presence of waves with different momenta and energy in the packet

 i = - E i t + p i x group velocity  =  E/v g (v g t - x) + x  m 2 2E  =  Et -  px  p = (dp/dE)  E + (dp/dm 2 )  m 2 = 1/v g  E + (1/2p)  m 2 p i = E i 2 – m i 2  x  E 0 ~  m 2 /2E  x  m 2 /2E usually- small  =  2  -  1 standard oscillation phase Dispersion relation enters here Effect of averaging over size of the wave packet

2 x 1 xx e = cos  1  sin    = - sin  1  cos   cos  sin  e opposite phase | (x,t)  = cos  g 1 (x – v 1 t)| 1  + sin  g 2 (x – v 2 t)e | 2  ii  =  2  -  1 Interference pattern depends on relative (oscillation) phase Oscillation phase  =  Main, effective frequency  +

2 x 1 xx cos  sin  e  =  0 - Destructive interference of the muon parts - Constructive interference of electron parts

2 x 1 xx cos  sin  e - Destructive interference of the electron parts - Constructive interference of muon parts  =   +

As important as production Should be considered symmetrically with production Detection effect can be included in the generalized shape factors g k (x – v k t)  G k (L – v k t) x  L - distance between central points of the production and detection regions HOMEWORK…

A( e ) =  e | (x,t)  = cos 2  g 1 (x – v 1 t) + sin 2  g 2 (x – v 2 t) e ii Amplitude of (survival) probability P( e ) = dx |  e | (x,t)>| 2 = = cos 4  + sin 4  + 2sin 2  cos 2  cos  dx g 1 (x – v 1 t) g 2 (x – v 2 t) Probability depth of oscillations  E  m 2  = = If g 1 = g 2 P( e ) = sin 2  cos 2  (1 - cos  ) = 1 - sin 2 2  sin 2  ½   m 2 x 2E  x l l = Oscillation length (integration over the detection area) inteference

P(  )= sin 2 2  sin 2  x l All complications – absorbed in normalization or reduced to partial averaging of oscillations or lead to negligible corrections of order m/E << 1 Oscillations - effect of the phase difference increase between mass eigenstates Admixtures of the mass eigenstates i in a given neutrino state do not change during propagation Flavors (flavor composition) of the eigenstates are fixed by the vacuum mixing angle

In the configuration space: separation of the wave packets due to difference of group velocities 2 x  v gr =  m 2 /2E 2 1 no overlap: Coherence is determined by conditions of detection xx separation:  v gr L =  m 2 L/2E 2  v gr L >  x coherence length: L coh =  x E 2 /  m 2 In the energy space: period of oscillations  E = 4  E 2 /(  m 2 L) Averaging (loss of coherence) if energy resolution  E is bad:  E <  E If  E >  E - restoration of coherence even if the wave packets separated

g i (x,t) e = dx S dt S M   (x S, t S )   (x S, t S ) exp[ ip  i ( x - x S ) – iE i  (t – t S )]   (x S, t S ) = exp[ –½  t S ] g  (x S, t S ) exp[ -i   (x S, t S )] E. Akhmedov, D. Hernandez, A.S. Pion decay: integration over production region iiii part of matrix element   (x S, t S ) = g  (x’ - x S, t’ - t S ) exp[ i   (x’ - x S, t’ - t S )] g  (x S, t S ) ~  (x S - v  t S ) Pion wave function: plane wave for neuutrino determined by detection of muon Muon wave function: usually:

L lplp x  D. Hernandez, AS p targetdecay tunnel detector absorber E. Kh Akhmedov, D. Hernandez, AS arXiv:  = l p The length of the wave packet emitted in the forward direction g = g 0 exp (x -  )  (x, [0,  ]) Shape factor  2(v - v  ) Doppler effect v – v  v  box function wave packet

P = P + [cos  L + K] sin 2 2  2(1 +  2 ) 1 1 – e -lp-lp K =  sin  L - e [cos(  L -  p ) -  sin (  L -  p )] -lp-lp L lplp  L =  m 2 L/2E  p =  m 2 l p /2E  =  m 2 /2E  decoherence parameter x Equivalence Coherent -emission - long WP  Incoherent -emission - short WP x D. Hernandez, AS MINOS:  ~ 1  beam ?

x t detector pion target XDXD   lplp detector Coherent emission of neutrino by pion along its trajectory XTXT

x t detector pion target XDXD   lplp detector If pion ``pointlike – result coincides with usual incoherent computations; otherwise – integration – corrections ~   XTXT integration

x t detector pion target XDXD lplp detector 1 2    XTXT Q

d  d t i = H  M M + 2E H = + V(t)  e     M is the mass matrix V = diag (V e, 0, 0) – effective potential M M + = U M diag 2 U + M diag 2 = diag (m 1 2, m 2 2, m 3 2 ) Mixing matrix in vacuum If loss of coherence and other complications related to WP picture are irrelevant – point-like picture

Re e + , P = Im e + , e + e - 1/2 B = (sin 2  m, 0, cos2  m ) 2  l m  = ( B x P ) d  P dt Coincides with equation for the electron spin precession in the magnetic field  = e , Polarization vector:  P =  +  Evolution equation: i = H  d  d t d  d t i = (B  )  Differentiating P and using equation of motion

= P = (Re e + , Im e + , e + e - 1/2) B = (sin 2  m, 0, cos2  m ) 2  l m  = ( B x ) d dt Evolution equation  = 2  t/ l m - phase of oscillations P = e + e = Z + 1/2 = cos 2  Z /2 probability to find e e ,

source Far detector Near detector L - baseline   disappearance  appearance Oscillation probability – periodic function of - Distance L and - Inverse energy 1/E

     e e  e

Distortion of the  - wave packet due to oscillations leads to shift of the center of mass of the packet   Key point  p = 2  l (v – v  ) = 2  l p / l equals the phase acquired over the region of formation of the wave packets The oscillation phase which describes distortion

P(    ) l decay ~ l  E ~  Coherence: D. Henandez, A.S. MINOS, ND with decoherence

i  ~ |U il | 2 l i       i W  Scattering Analogy with quark sector? mass state N No beams of neutrino mass eigenstates  ~ |U  i    In practice: No neutrino mass spectrometer The problem: neutrinos are in the coherent (flavor) states lead to new effects which allow to measure mixing parameters Standard methods do not work

Search for new realizations of the oscillation setup Phenomenological consequences Manipulating with oscillation setup Academic interest? Theory of oscillations: Quantum mechanics at macroscopic distances Energy-momentum conservation Coherence of production Kinematic entanglment Collective effects Interplay of oscillations with other non-standard effects Identify origins of neutrino mass Coherence in propagation

Space-time localization Energy-momentum is conserved in whole the system which includes also particles whose interactions localize neutrino parents (walls, etc). energy-momentum uncertainty, DE, D p Generic feature of oscillation set-up E-p are not conserved in the sub-system: particles which Directly involved in the process of neutrino production E-p can be conserved for individual components of the wave packets. Uncertainty is restored when the probability is convoluted with wave packets Nuclei decay n

          i = 1,2  f       Exact energy-momentum conservation: kinematic entanglement Final state:  i is the state which (kinematically) corresponds mass eigenstate i  f  i   i    e e i  i ( ) Does recoil (muon) phase contribute to the oscillation phase? For muon: p 2 = E 2 – m 2 2p  p = 2E  E  p = E/p  E = 1/v g  E  =  Et –  px =  E(t – x/v g ) ~  E  x x = v g t Related to uncertainty of the phase due to localization Evolved: v g = p/E no mass difference between two muon components ii()ii()

 Flavor of neutrino state is fixed at the production This does not depend on where and how muon is detected and on what is the phase difference between the muon components Neutrino phase difference is determined by neutrino parameters production region

s What is the neutrino state produced in the Z-decay in the presence of mixing? f = [ ] 1313 f H Z 2 = H Z 2 Z0Z0 i i Do neutrinos from Z 0 - decay oscillate? If the flavor of one of the neutrino is fixed, another neutrino oscillates P = sin 2 2  sin 2 [  (L 1 + L 2 )/ l ] Two detectors experiment: detection of both neutrinos L 1 L2L2 Z is flavor blind

n2n2 n1n1 Due to difference of masses n 1 and n 2 have different phase velocities effects of the phase difference increase which changes the interference pattern Df = D v phase t Flavors of mass eigenstates do not change Admixtures of mass eigenstates do not change: no n 1 n 2 transitions Determined by q Df = 0 nene

A Yu Smirnov P (n m ) = 1 - cos= sin 2 2 q sin 2 A p 2 2p x l n p x l n Features of neutrino oscillations in vacuum: Oscillations -- effect of the phase difference increase between mass eigenstates Admixtures of the mass eigenstates n i in a given neutrino state do not change during propagation Flavors (flavor composition) of the eigenstates are fixed by the vacuum mixing angle

mass is generated by L L H H 11 Sun Atmosphere The Earth Geo-nu SN1987A Accelerators Reactors Rad. Sources Introduction of neutrino mass and mixing may have negligible impact on the rest of SM other effects ~  -2 All well established/confirmed results are described by + Universe (indirectly) of three neutrinos with rather peculiar pattern

e   e   LL L I W = 1/2 I 3W = 1/2 are neutrino flavor states form doublets (charged currents) with definite charged leptons, l l W Neutral current interaction l l l = e,  Z Conservation of lepton numbers L e, L , L  l    (1 -  5 ) l  W +  + h.c.  L  = ½(1 -  5 )  R  = ½(1 +  5 ) g 2 Chiral components ?  l    (1 -  5 ) l  Z  g 4

MSW-effect Oscillations Can be resonantly enhanced in matter

M l = M M l = U lL m l diag U lR + M = U L m diag U L T U  PMNS  = U lL + U L (Majorana neutrinos) Flavor basis:M l = m l diag U  PMNS  = U L Diagonalization: Mixing matrix Mass spectrum m diag = (m 1, m 2, m 3 ) Origin of mixing: off diagonal mass matrices l     (1 -  5 ) U PMNS mass CC in terms of mass eigenstates:

c 12 c 13 s 12 c 13 s 13 e -i  s 12 c 23 + c 12 s 23 s 13 e i  c 12 c 23 - s 12 s 23 s 13 e i  - s 23 c 13 s 12 s 23 - c 12 c 23 s 13 e i  c 12 s 23 + s 12 c 23 s 13 e i  c 23 c 13 U PMNS =  is the Dirac CP violating phase c 12 = cos  , etc.  12 is the ``solar’’ mixing angle  23 is the ``atmospheric’’ mixing angle  13 is the mixing angle determined by T2K, Daya Bay, DC, RENO … U PMNS = U 23 I  U 13 I  U 12 I  = diag (1, 1, e i  )

LBL experiments: - Long life-time parents: pions, muons, nuclei; - large decay pipes, straight lines of storage rings New oscillation setups Higher precision … Why the standard oscillation formula is so robust? Corrections required ? Neutrino anomalies as consequences of modified description of oscillations Nature of neutrino mass differs from nature of electron or top quark…

e   e   LL L I W = 1/2 I 3W = 1/2 are neutrino flavor states form doublets (charged currents) with definite charged leptons, l l W Neutral current interaction l l l = e,  Z Conservation of lepton numbers L e, L , L  l     (1 -  5 ) l  W +  + h.c.  L  = ½(1 -  5 )  R  = ½(1 +  5 ) g 2 Chiral components ?  l     (1 -  5 ) l  Z  g 4

production region detection region baseline L External particles How external particles Should be described? wave packets for external particles detection/production areas are determined by localization of particles involved in neutrino production and detection not source/detector volume (still to integrate over) i S D Describe by plane waves But introduce finite integration

production region detection region Neutrinos real particles described by the wave packets Which encode information about production and detection as well on oscillations in the production region on mass shell i

Flavor neutrino states:   e flavor is characteristic of interactions m1m1 m2m2 m3m3 Flavor states Mass eigenstates =  Mass eigenstates  e - correspond to certain charged leptons - interact in pairs n  p + e - + e    +  