A Yu Smirnov International Centre for Theoretical Physics

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

n 2 m n 1 m resonance if density changes slowly x - the amplitudes of the wave packets do not change - flavors of the eigenstates follow the density change

Sun, Supernova Initial state: From high to low densities n(0) = ne = cosqm 0 n 1 m(0) + sinqm 0 n 2 m(0) Adiabatic evolution to the surface of the Sun (zero density): Final state: Mixing angle in matter in initial state n 1 m(0) n 1 n 2 m(0) n 2 -if n(f) = cosqm 0 n 1 + sinqm 0 n 2 e Probability to find ne P = |< n | n(f) >|2 = (cosq 0)2 + (sinq 0)2 e m m averaged over oscillations = 0. 5[ 1 + cos 2 qm 0 cos 2 q ] P = sin 2 q + cos 2 qm 0

ne = cos qm n 1 m + sinqmn 2 m nm = - sin qmn 1 m + cos qmn 2 m inversely n 2 m = sin qm ne + cos qm nm n 1 m = cosqmne - sinq nm qm coherent mixtures of mass eigenstates ne n 2 m n 1 m nm n 2 m n 1 m Wave packets flavor composition of the mass eigenstates n 2 m n 1 m flavors of eigenstates Vacuum qm q n 1 m n 1 n 2 m n 2

resonance Him sin 2 2 q 12 = 0. 825 n 2 m V. Rubakov, private comm. N. Cabibbo, Savonlinna 1985 H. Bethe, PRL 57 (1986) 1271 ne nm n 1 m Dependence of the neutrino eigenvalues on the matter potential (density) ln / l 0 Large mixing ln 2 E V = l 0 Dm 2 sin 2 2 q 13 = 0. 08 ln = cos 2 q l 0 Crossing point - resonance - the level split is minimal - the oscillation length is maximal ne n 3 m n 2 m nt ln / l 0 Small mixing

Mass hierarchy, 2 -3 mixing and CP-phase with Huge Atmospheric Neutrino Detectors. E. Kh. Akhmedov, Soebur Razzaque, A. Yu. Smirnov. e-Print: ar. Xiv: 1205. 7071 [hep-ph] 1 -3 leptonic mixing and the neutrino oscillograms of the Earth. Evgeny K. Akhmedov, Michele Maltoni, Alexei Yu. Smirnov , JHEP 0705 (2007) 077 e-Print: hep-ph/0612285 Neutrino oscillograms of the Earth: Effects of 1 -2 mixing and CP-violation. Evgeny Kh. Akhmedov, Michele Maltoni, Alexei Yu. Smirnov, JHEP 0806 (2008) 072 e-Print: ar. Xiv: 0804. 1466 [hep-ph] and references therein

Can be treated on the same footing matter with constant density, (unless near topological defects) Mass diagonal Mixes flavor states Flavor states oscillate Flavor diagonal Mixes mass eigenstates Mass states oscillate in matter Flavor states and mass states change roles when matter and vacuum exchange

Collective flavor trasformation MSW flavor conversion inside the star How flavor diagonal interactions can lead to flavor off-diagonal elements of the Hamiltonian? ve i t c e l col f o n Origi s t effec Propagation in vacuum Oscillations Inside the Earth

nn ne ne ne Z 0 nb nb ne t-channel ne (p) nb nb (q) ne u-channel ne (q) Refraction in neutrino gases nb nb A = 2 GF (1 – ve vb ) ne velocities elastic forward scattering J. Pantaleone ne (p) nb nb Z 0 can lead to the coherent effect Momentum exchange flavor mixing Collective flavor transformations

ne projection na ne coherent J. Pantaleone S. Samuel V. A. Kostelecky nb nb background ne nt nn – scattering in u-channel due to Z 0 - exchange 1. Momentum exchange flavor exchange 2. Coherence if the background is in mixed state: projection |nib = Fie |ne + Fit | nt Coherent flavor changing transition Probe neutrino = background neutrino Potential depends on transition probability

ne projection ne coherent J. Pantaleone S. Samuel V. A. Kostelecky nb nb background ne nt projection Flavor exchange between the beam (probe) and background neutrinos Hnn = 2 GF Si (1 – ve vib ) If the background is in the mixed state: |nib = Fie |ne + Fit | nt Bet ~ Si Fie*Fit sum over particles of bg. w. f. give projections Contribution to the Hamiltonian in the flavor basis |Fie|2 Fie. Fit * Fie*Fit |Fit|2

Ensemble of neutrino polarization vectors Pw dt Pw =(- w. B + l. L + m. P) x Pw Vacuum mixing term B = (sin 2 q , 0, cos 2 q) w=D m 2 /2 E Usual matter potential L = (0, 0, 1) l = V = 2 G F ne Negative frequencies for antineutrinos Collective vector + inf P = dw Pw - inf m = 2 GF nn (1 – cos qnn) The term describes collective effects

qn -zenith angle Q Q = p - qn - nadir angle Oscillations in multilayer medium core-crossing trajectory Q = 33 o Applications: flavor-to-flavor transitions - accelerator - atmospheric - cosmic neutrinos core mantle 9

Adiabaticity breaking at the borders of layers

Equation of motion (= spin in magnetic field) z d P = (B x P) dt B P ne where ``magnetic field’’ vector: 2 p B = l (sin 2 qm, 0, cos 2 qm) m nt, P = (Re ne+ nt, Im ne+ nt, ne+ ne - 1/2) x Phase of oscillations y f = 2 pt/ lm Probability to find ne Pee = ne+ne = PZ + 1/2 = cos 2 q. Z/2

1 mantle 1 2 mantle 2 21

1 mantle core 2 2 1 3 4 mantle core mantle 3 mantle 4 22

mantle 1 core 3 4 2 2 mantle 4 3 1 23

a). b). a). Resonance in the mantle b). Resonance in the core c). Parametric ridge A d). Parametric ridge B c). d). e). Parametric ridge C f). Saddle point e). f).

1 - Pee MSW-resonance peak, 1 -3 mixing Parametric ridges 1 -3 mixing MSW-resonance peaks 1 -2 mixing 5 p/2 3 p/2 Parametric peak 1 -2 mixing MSW-resonance peak, 1 -3 mixing

For E > 0. 1 Ge. V ~ H = U 13 T U 12 T Hdiag U 12 U 13 nf = U 23 Id ~ n Propagation basis Hdiag = diag (H 1 m, H 2 m, H 3 m) Id = diag (1, 1, eid ) ne CP-violation and 2 -3 mixing - excluded from dynamics of propagation ne nm ~ n 2 nt ~ n 3 projection CP appears in projection only For instance: Ae 2 Ae 3 ne ne ~ n 2 nm n~3 nt propagation A 22 A 33 projection A 23 A(ne nm) = cosq 23 Ae 2 eid + sinq 23 Ae 3

E Kh Akhmedov, S Razzaque, A. Y. S. for hierarchy determination, neglect 1 -2 mixing effects P(ne nm) = s 232|Ae 3|2 P(n m nm) = 1 – ½ sin 2 2 q 23 - s 234|Ae 3|2 +½ Reduces the average probability sin 2 2 q 23 (1 - ½ |Ae 3|2) Reduces the depth of oscillations interference cos f Modifies phase f = arg (A 22 A 33*) P(n m nt) = ½ sin 2 2 q 23 - s 232 c 232|Ae 3|2 -½ sin 2 2 q 23 (1 - |Ae 3|2) ½ cos f

MSW resonance in core 2 nd and 3 rd parametric peaks ne nt nm MSW resonance in mantle

and physics of oscillations P. Lipari , T. Ohlsson M. Blennow M. Chizhov, M. Maris, S. Petcov T. Kajita …

ex d e d clu

1 - Pee e d xclu MSW-resonance peaks 1 -3 frequency Parametric ridges 1 -3 frequency Parametric peak 1 -2 frequency MSW-resonance peaks 1 -2 frequency 5 p/2 3 p/2

e-e e- m For 2 n system normal inverted neutrino antineutrino m-m 25

Solar magic lines Three grids of lines: Atmospheric magic lines Interference phase lines

d = 60 o Standard parameterization

d = 130 o

d = 315 o

Due to specific form of matter potential matrix (only V ee = 0) P(n e nm) = |cos q 23 Ae 2 e id + sin q 23 Ae 3|2 ``solar’’ amplitude dependence on d and q 23 is explicit ``atmospheric’’ amplitude For maximal 2 -3 mixing P(ne nm)d = |Ae 2 Ae 3| cos (f - d ) P(nm nm)d = - |Ae 2 Ae 3| cosf cos d P(nm nt)d = - |Ae 2 Ae 3| sinf sind S=0 f = arg (Ae 2* Ae 3)

P. Huber, W. Winter V. Barger, D. Marfatia, K Whisnant, A. S. Explicitly P(ne nm) = c 232|AS|2 + s 232|AA|2 + 2 s 23 c 23|AS||AA|cos(f + d) f = arg (AS AA*) Pint = 2 s 23 c 23|AS||AA|cos(f + d) Dependence on d disappears, interference term is zero if Pint = 0 AS = 0 AA = 0 - solar magic lines - atmospheric magic lines (f + d ) = p/2 + 2 p k - interference phase condition f (E, L) = - d + p/2 + p k depends on d 31

For nm nm channel Pint ~ 2 s 23 c 23|AS||AA|cosf cosd - The survival probabilities is CP-even functions of d - no CP-violation - dependences on phases factorize Dependence on d disappears AS = 0 Pint = 0 AA = 0 f = p/2 + p k Form the phase line grid interference phase does not depends on d 32

d - true (experimental) value of phase df - fit value Interference term: D P = P(d) - P(df) = Pint(d) - Pint(df) For ne nm channel: DP = 2 s 23 c 23 |AS| |AA| [ cos(f + d) - cos (f + df)] AS = 0 DP = 0 (along the magic lines) AA = 0 (f + d ) = - (f + df) + 2 p k f (E, L) = - ( d + df)/2 + p k int. phase condition depends on d

Int. phase line moves with d-change Grid (domains) does not change with d DP

DP

DP

contours of constant oscillation probability in energy- nadir (or zenith) angle plane 100 ne nm , nt Ice. Cube E, Ge. V 0. 005 10 ICAL, INO 1 IC Deep Core Nu. Fac 2800 CNGS 0. 03 0. 10 LENF PINGU-1 T 2 KK MINOS NOv. A T 2 K Hyper. K OK for CP LAr 0. 1

5 Ge. V 0 1 – 1. 0 0 : Energy range Baselines: 0 – 13000 km Matter effects: 3 which is not completely explored and largely unused – 15 g/cm 3 nue, numu t n e t n o c r o v Fla Lepton number nu -a ntinu which change with energy and zenith angle Discovery of neutrino oscillations Measurements of 2 -3 mixing and mass splitting Bounds on new physics - sterile neutrinos - non-standards interaction -violation of fundamental symmetries, CPT

High statistics solve the problems ies of t n i a t r tion a ic if t Unce n e id r s o fluxe Flav l a n i g i or Reconstr uction of direct ion s c i t s i t a t S lution Energy reso TITAND? E. Kh Akhmedov M. Maltoni A. Y. S. JHEP 05, (2007) 077 [hep-ph/0612285] JHEP 06 (2008) 072 [ar. Xiv: 0804. 1466] PRL 95 (2005) 211801 ar. Xiv: 0506064 unpublished, see M Maltoni talks A. Y. S. , hep-ph/0610198. E. Kh Akhmedov, S Razzaque, A. S. in preparation E Kh Akhmedov, A Dighe, P. Lipari, A Y. Smirnov , Nucl. Phys. B 542 (1999) 3 -30 hep-ph/9808270 Y. Suzuki Developments of new detection methods?

Original fluxes Integration averaging Detection different flavors: ne and nm neutrinos and antineutrinos Screening 2) s 3 factors (1 - r 2 (1 - ke ) (1 – km) averaging and smoothing effects reconstruction of neutrino energy and direction identification of flavor Reduces CPasymmetry

Triple suppression Ne. IH - Ne. NH ~ (PA - PA) (1 – km ) [r s 232 - (1 – k e)/(1 - km)] CP asymm etry Neutrino antineutrino factor can be avoided Flavor suppression (screening factors) unavoidable PA = |Ae 3|2 ka = (s Fa)/(s Fa) Nm. IH - Nm. NH ~ (Pmm - Pmm) (1 – km) - r-1(1 – ke) (Pem - Pem)]

Precision Ice. Cube Next Generation Upgrade

Digital Optical Module Ice. Cube : 86 strings (x 60 DOM) 100 Ge. V threshold Gton volume Deep Core IC : - 8 more strings (480 DOMs) - 10 Ge. V threshold - 30 Mton volume PINGU: 18, 20, 25 ? new strings (~1000 DOMs) in Deep. Core volume Existing Ice. Cube strings Existing Deep. Core strings New PINGU strings D. Cowen

Denser array PINGU v 2 20 new strings (~60 DOMs each) in 30 MTon Deep. Core volume Few Ge. V threshold in inner 10 Mton volume Energy resolution ~ 3 Ge. V Existing Ice. Cube strings Existing Deep. Core strings New PINGU-I strings 125 m

Effective area, effective volume

ne nm nt 1 -3 mixing ? bi-maximal tri-maximal n 2 n 1 Dm 232 Dm 221 FLAVOR Normal mass hierarchy MASS n 3 n 2 n 1 Dm 223 n 3 FLAVOR Inverted mass hierarchy xing i m l a m i x a ~ Tri-bim Symmetr y? 2 -3 2 Dm 32 = 2. 3 x 10 e. V nm - nt symmetry Dm 221 = 8 x 10 -5 e. V 2 Two large mixings

E. Kh Akhmedov, S Razzaque, A. Y. S. Asymmetry, statistical significance Syst e Quick estimation of sign matics ifica r significance educes by f Number of bins in acto Effective average r 2 resolution domains significance in individual bin Stot ~ s n 1/2

2 Ge. V, 11. 250 E. Akhmedov, S. Razzaque, A. Y. Smirnov ar. Xiv: 1205. 7071 Smearing with Gaussian reconstruction functions characterized by (half) widths ( s. E , sq ) 3 Ge. V, 150 4 Ge. V, 22. 50

s. E = 0. 2 E sq ~ 1/E 0. 5 Degeneracy

Flavor mixing in neutrino-neutrino scattering from flavor diagonal interactions Propagation in the Earth - neutrino image of the Earth: resonance enhancement of oscillations, parametric effects Magic lines, CP-violation domains Determination of neutrino parameters with huge atmospheric neutrino detectors

Integration over resolution (reconstruction domain) tracks

For the best fit values of parameters

Normal mass hierarchy EH EL 0. 1 Ge. V 6 Ge. V Resonance region E High energy range

neutrinosphere usual matter potential: R = 20 – 50 km l = V = 2 G F ne neutrino potential: m = 2 GF (1 – cos x) nn n r x nn ~ 1/r 2 x ~ 1/r n m ~ 1/r 4 in neutrinosphere in all neutrino species: electron density: nn ~ 1033 cm-3 ne ~ 1035 cm-3 l >> m for large r

Introducing negative frequencies for antineutrinos Pw = P- w w>0 dt Pw =(w. B + m. D) x Pw + inf D = dw sw Pw where sw = sign(w) - inf Equation of motion for D: integrating equation of motion with sw inf dt D = B x M where M = dw sw w Pw - inf In another form: where dt Pw = Hw(m) x Pw Hw =(w. B + m. D)

Kostelecky & Samuel Pastor, Raffelt, Semikoz B n If m |D| >> w - the individual vectors form large the self-interaction term dominates dt P w ~ m D x P w D does not depend on w n - evolution is the same for all modes – Pw are pinned to each other M = wsyn D synchronization frequency wsyn = dw sw w. Pw dw sw. P w dt D = wsyn B x D D - precesses around B with synchronization frequency

ne = cos qm n 1 m + sinqmn 2 m nm = - sin qmn 1 m + cos qmn 2 m coherent mixtures of mass eigenstates ne n 2 m n 1 m nm n 2 m n 1 m n 2 m = sin qmne + cos qm nm n 1 m = cosqmne - sinq nm qm flavor composition of the mass eigenstates Vacuum qm q wave packets n 2 m n 1 m flavors of eigenstates ne The relative phases of the mass states in ne and nm are opposite inversely nm n 2 m n 1 m Interference of the parts of wave packets with the same flavor depends on the phase difference Df between n 1 and n 2