A. Yu. Smirnov International Centre for Theoretical Physics, - PowerPoint PPT Presentation

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

L. Wolfenstein, Phys. Rev. D17 (1978) 2369 L. Wolfenstein, in ``Neutrino-78'', Purdue Univ. C3, 1978. L. Wolfenstein, Phys. Rev. D20 (1979) 2634 V. D. Barger, K. Whisnant, S. Pakvasa, R.J.N. Phillips, Phys.Rev. D22 (1980) 2718 S.P. Mikheev, A.Yu. Smirnov, Sov. J. Nucl.Phys. 42 (1985) 913-917, Yad.Fiz. 42 (1985) 1441-1448 S.P. Mikheev, A.Yu. Smirnov, Nuovo Cim. C9 (1986) 17-26 S.P. Mikheev, A.Yu. Smirnov, Sov. Phys. JETP 64 (1986) 4-7, Zh.Eksp.Teor.Fiz. 91 (1986) 7-13, arXiv:0706.0454 [hep-ph] S.P. Mikheev, A.Yu. Smirnov, 6 th Moriond workshop, Tignes, Jan. 1986 p. 355

H.A. Bethe, Phys.Rev.Lett. 56 (1986) 1305 A. Messiah, 6 th Moriond workshop, Tignes Jan. 1986 p.373 S. J. Parke, Phys.Rev.Lett. 57 (1986) 1275-1278 W.C. Haxton, Phys.Rev.Lett. 57 (1986) 1271-1274 S. P. Rosen, J. M. Gelb, Phys.Rev. D34 (1986) 969 P. Langacker, S.T. Petcov, G. Steigman, S. Toshev, Nucl.Phys. B282 (1987) 589 The MSW effect and matter effects in neutrino oscillations. A.Yu. Smirnov, Phys. Scripta T121 (2005) 57-64, hep-ph/0412391 A. Y. Smirnov, hep-ph/0305106 P.C. de Holanda, A.Yu. Smirnov, Astropart.Phys. 21 (2004) 287, hep- ph/0309299 Quantum field theoretic approach to neutrino oscillations in matter. E. Kh. Akhmedov, A. Wilhelm, arXiv:1205.6231 [hep-ph]

Adiabatic conversion Loss of coherence

L. Wolfenstein, 1978 for n e n m at low energies Re A >> Im A n e e inelelastic interactions can be neglected Elastic forward W V e , V m scattering n e e potentials Refraction index: difference of potentials n - 1 = V / p V = V e - V m = 2 G F n e V ~ 10 -13 eV inside the Earth for E = 10 MeV ~ 10 -20 inside the Earth n – 1 = < 10 -18 inside the Sun l 0 = 2 p Refraction length: V

derivation At low energies: neglect the inelastic scattering and absorption effect is reduced to the elastic forward scattering (refraction) described by the potential V: y is the wave function H int ( n ) = < y | H int | y > = V n n of the medium n e e CC interactions with electrons W G F n e H int = n g m (1 - g 5 ) n e g m (1 - g 5 ) e e 2 < e g 0 (1 - g 5 ) e> = n e - the electron number density For unpolarized < e g e> = n e v medium at rest: < e g g 5 e > = n e l e - averaged polarization vector of e V = 2 G F n e

Effective H 0 H = H 0 + V Hamiltonian depend n 1m , n 2m Eigenstates n 1 , n 2 on n e , E H 1m , H 2m m 12 /2E, m 22 /2E instantaneous Eigenvalues n mass q n e n 1 n f n 2m q m n H n 1m q n 2 Mixing angle determines flavors n m (flavor composition) of eigenstates q m of propagation

n e d n f i = H tot n f n f = n m dt H tot = H vac + V is the total Hamiltonian M 2 is the vacuum (kinetic) part H vac = 2E V e 0 V e = 2 G F n e matter part V = H tot 0 0 D m 2 D m 2 n e n e - cos 2q + V e sin 2 q 2E 4E d i = dt D m 2 n m n m sin 2 q 0 4E

Diagonalization of the Hamiltonian: sin 2 2 q sin 2 2 q m = V = 2 G F n e ( cos2 q - 2 EV/ D m 2 ) 2 + sin 2 2 q Mixing is maximal if D m 2 Resonance V = cos 2 q H e = H m 2E condition sin 2 2 q m = 1 Difference of the eigenvalues H 2m - H 1m = D m 2 ( cos2 q - 2 EV/ D m 2 ) 2 + sin 2 2 q 2E

In resonance: sin 2 2 q m n n sin 2 2 q m = 1 sin 2 2 q 13 = 0.08 sin 2 2 q 12 = 0.825 Flavor mixing is maximal l n = l 0 cos 2 q Vacuum Refraction ~ ~ oscillation length length l n / l 0 ~ n E Resonance width: D n R = 2n R tan2 q Resonance layer: n = n R +/- D n R

resonance H im sin 2 2 q 12 = 0.825 n e n 2m V. Rubakov, private comm. N. Cabibbo, Savonlinna 1985 n m H. Bethe, PRL 57 (1986) 1271 n 1m Dependence of the neutrino eigenvalues l n / l 0 on the matter potential (density) Large mixing l n 2E V = l 0 D m 2 sin 2 2 q 13 = 0.08 n e l n l 0 = cos 2 q n 3m n t l n / l 0 n 2m Crossing point - resonance - the level split is minimal Small - the oscillation length is maximal mixing

Normal mass hierarchy E H E L E 0.1 GeV 6 GeV High energy range Resonance region

H 0  H = H 0 + V n k  n mk n 1m eigenstates eigenstates of H of H 0 q  q m (n) x n 2m Resonance - maximal mixing in matter – oscillations with maximal depth q m = p /4 Constant density medium: the same dynamics Resonance condition: D m 2 Mixing changed V = cos2 q 2E phase difference changed

Oscillation P( n e -> n a ) = sin 2 2 q m sin 2 p L probability l m half-phase f constant density Amplitude of oscillatory factor oscillations q m (E, n ) - mixing angle in matter q m  q l m (E, n ) – oscillation length in matter In vacuum: l m  l n l m = 2 p /(H 2m – H 1m ) sin 2 2 q m = 1 MSW resonance condition Maximal effect: f = p/2 + p k

l n = 4 p E Oscillation D m 2 length in vacuum 2 p Refraction - determines the phase produced l 0 = length by interaction with matter 2 G F n e l n = l 0 /cos2 q ) (maximum at l n /sin2 q l m shifts with respect resonance energy: 2 p l m = l n (E R ) = l 0 cos2 q H 2m - H 1m l 0 converges to the ~ l n refraction length E E R

Constant density

Constant density n n e n e Detector Source layer of length L F(E) F 0 (E) Depth of oscillations determined by sin 2 2 q m as well as the oscillation length, l m depend on neutrino energy For neutrinos propagating in the mantle of the Earth

Large mixing sin 2 2 q = 0.824 Layer of length L k = p L / l 0 n thin layer thick layer k = 1 k = 10 F (E) F 0 (E) sin 2 2 q m E/E R E/E R

Small mixing sin 2 2 q = 0.08 k = 10 thick layer thin layer k = 1 F (E) F 0 (E) sin 2 2 q m E/E R E/E R

In maximum 1 B 1 2 2

Varying density

In non-uniform medium the Hamiltonian H tot = H tot (n e (t)) depends on time: n f = n e d n f i = H tot n f n m dt n m = n 1m q m = q m (n e (t)) n 2m Inserting n f = U( q m ) n m off=diagonal d q m n 1m n 1m 0 terms imply i d dt i = d q m transitios dt -i H 2m - H 1m n 2m n 2m dt n 1m n 2m off-diagonal elements can be neglected d q m if no transitions between eigenstates << H 2m - H 1m dt propagate independently

External conditions d q m (density) change slowly Adiabaticity condition << H 2m - H 1m dt the system has time to adjust them transitions between The eigenstates the neutrino eigenstates n 1m  n 2m propagate independently can be neglected Shape factors of the eigenstates do not change if vacuum mixing is small Crucial in the resonance layer: D r R > l R - the mixing changes fast oscillation length in resonance l R = l n / sin2 q - level splitting is minimal width of the res. layer D r R = n R / (dn/dx) R tan2 q If vacuum mixing is large, the point n(a.v.) -> n R0 > n R of maximal adiabaticity violation n R0 = D m 2 / 2 2 G F E is shifted to larger densities

Adiabaticity H 2m - H 1m k = condition: d q m k > 1 dt most crucial in the resonance where k R = D r R the mixing angle in matter changes fast l R D r R = h n tan2 q is the width of the resonance layer n h n = dn/dx is the scale of density change is the oscillation length in resonance l R = l n /sin2 q D m 2 sin 2 2 q h n Explicitly: k R = 2E cos2 q

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

Sun, Supernova From high to low densities n (0) = n e = cos q m0 n 1m (0) + sin q m0 n 2m (0) Initial state: Mixing angle in matter in initial Adiabatic evolution n 1m (0)  n 1 state to the surface of n 2m (0)  n 2 the Sun (zero density): -i f Final state: n (f) = cos q m0 n 1 + sin q m0 n 2 e Probability to find n e P = |< n e | n (f) >| 2 = (cos q cos q m0 ) 2 + (sin q sin q m0 ) 2 averaged over oscillations = 0.5[ 1 + cos 2 q m0 cos 2 q ] P = sin 2 q + cos 2 q cos 2 q m0

Adiabatic conversion survival probability distance

The picture is universal in terms of variable y = (n R - n ) / D n R no explicit dependence on oscillation parameters, density distribution, etc. only initial value y 0 matters resonance layer production survival probability point y 0 = - 5 resonance oscillation band averaged probability (distance) (n R - n) / D n R A Yu Smirnov

If density n e (t) changes fast SN shock waves d q m ~ | H 2m - H 1m | dt the off-diagonal terms in the Hamiltonian can not be neglected transitions n 1m n 2m n 1m n 2m Admixtures of in a given neutrino state change penetration under barrier: ``Jump probability’’ D H E n P 12 = e H H 2m E n ~ 1/h n is the energy associated to change of parameter D H (density) H 1m P 12 = e - pk R /2 Landau-Zener n

Pure adiabatic conversion Partialy adiabatic conversion n e n m

Different degrees of freedom Non-uniform medium or/and medium Vacuum or uniform medium with varying in time parameters with constant parameters Change of mixing in medium = Phase difference increase change of flavor of the eigenstates between the eigenstates q m f In non-uniform medium: interplay of both processes

Passing through the matter filter n F (E) Constant density Monotonously changing F 0 (E) density k = p L/ l 0 E/E R E/E R

Can be resonantly enhanced in matter