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Neutrino microscopes

Neutrino microscopes. Alexei Yu. Smirnov International Centre for Theoretical Physics. Oscillations in low density medium - Attenuation effect Next order corrections Improved perturbation theory - Neutrino microscopes. Surprises?.

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Neutrino microscopes

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  1. Neutrino microscopes Alexei Yu. Smirnov International Centre for Theoretical Physics • Oscillations in low density • medium • - Attenuation effect • Next order corrections • Improved perturbation • theory • - Neutrino microscopes

  2. Surprises? l0 ~ 1/GFn ~ several 103 km refraction length 1). Probing density profile structures (of the Earth) r > ln = 4pE/Dm2 oscillation length r ~ the smaller the energy the smaller the scale n r ~ E r << l0 2). Matter effect is the lowest order effect 3). Adiabatic perturbation theory works 4). Attenuation effect: the better energy resolution the deeper structures can be probed 5). Improved perturbation theory: strange limitations

  3. In the low density medium P.C. de Holanda, Wei Liao, A.S, Nucl. Phys. B702:307 (2004) [hep-ph/0404042] A. Ioannisian, A.S., Phys. Rev. Lett. 93:241801 (2004) [hep-ph/0404060] A. Ioannisian et al., Phys. Rev.D71:033006 (2005) [hep-ph/0407138]

  4. In the low density medium V(x) << Dm2/ 2E Potential << kinetic energy 2 E V(x) D m2 e (x) ~ (1 -3) 10-2 Small parameter: e (x) = << 1 perturbation theory in e (x) Solar neutrinos Inside the Earth For LMA oscillation parameters applications to Supernova neutrinos Oscillations appear in the first order in e (x) Relevant channel : mass-to-flavor n2 -> ne P2e = sin2q + freg

  5. e - perturbation theory For regeneration effect in the Earth 2EVE Dm2 e (r) =

  6. Weak matter effect: mixing in matter ~ mixing in vacuum, qm ~ q theory in terms of mass eigenstates nmass = (n1, n2 )T Mass states mix in matter: cos q’ sin q’ - sin q’ cos q’ nmass = U’ nm U’ = nm = (n1m, n2m )T eigenstates in matter q’ = q’ (V) - mixing angle of mass states in matter e (x) sin 2q sin 2q’ = = e (x) sin 2qm ( cos 2q - e (x)) 2 + sin2 2q small id nmass /dx = H(x) nmass Evolution equation: 0 0 0 Dm(x) H(x)= U’(x) U’(x) + Dm2 2E Dm(x) = ( cos 2q - e (x)) 2 + sin2 2q

  7. S-matrix S-matrix in the basis of mass eigenstates nmass = (n1, n2 )T S(x0 -> xf) = (Un’ Dn Un’ +) ... (Uj’ Dj Uj’ +) ... (U1’ D1 U1’ +) In j-th layer: V Mixing matrix of mass states: cos qj’ sin qj’ - sin qj’ cos qj’ Vj Uj’ = qj’ = q’ (Vj) Dj Evolution matrix of the eigenstates in matter: Uj’ 1 0 0 e j n Dj = i Fmj x0 x xf Phase: Following procedure of the numerical computations . . . Fmj = D x Dm(Vj)

  8. Each block can be reduced to (Uj’ Dj Uj’ +) = Dj + Gj contains info about density change leads to transitions 0 1 1 0 i Fmj Gj = 0.5 (e - 1) sin2qj’ + O (e 2) ~ Dm(Vj) D x S(x0 -> xf) = Dn ... Dj ... D1 + Sj Dn ... Dj+1 Gj Dj-1 ... D1 + O(Gj Gk) + ... Dj = O(1) Gj = O (e ) expansion in power of Gj Limit n -> infty, D x -> 0 Sj Fmj = Sj Dx Dm(Vj) -> dx Dm(x) Sj D x -> dx xf x0 Adiabatic phase Fm(x0 -> xf) = dx Dm(x)

  9. S-matrix S-matrix in the basis of mass eigenstates nmass = (n1, n2 )T 1 0 0 e S(x0 -> xf) = - iFm(x0 -> xf) - iFm(x0 -> x) 0 e e 0 xf x0 + 0.5 i sin2q dx V(x) - iFm(x -> xf) + O(V2) The amplitude of the oscillation transition na -> nb Aa->b (x0 -> xf) = < nb|S(x0 -> xf)|na >

  10. Integral formula Mass-to-flavor transition: n2 --> ne P2e = sin2q + freg xf x0 Regeneration factor freg = 0.5 sin22q dx V(x) sin Fm(x -> xf) xf x0 xf x Dm2 2E 2EV(y) 2 Dm2 freg = 0.5 sin22q dx V(x) sin dy cos 2q - - sin22q V(x) Fm(x -> xf) Integration limits: x0 xf x The phase is integrated from a given point to the final point

  11. Analytic result 2E sin22q Dm2 freg = sinF0/2 Sj = 0 …n-1 DVj sinFj/2 j DVj j+1 F0 fj = 0.5(F0 - Fj) Defining Fj 2E sin22q Dm2 x freg = Sj = 0 …n-1DVj[sin2F0/2 cosfj - 0.5 sinF0 sinfj] fj If fj is large - averaging effect. This happens for remote structures, e.g. core

  12. Analytic vs. numerical results Regeneration factor as function of the zenith angle E = 10 MeV, Dm2 = 6 10-5 eV2, tan2q = 0.4

  13. Adiabatic perturbation theory Adiabatic condition: lm(x) 4ph(x) h(x) the height of distibution g (x) = << 1 At the borders of layers h(x) -> 0 Still the theory works!

  14. Attenuation effect Effects on Supernova neutrinos Features of regeneration of the solar neutrinos Oscillation tomography of the Earth

  15. Sensitivity to density profile For mass-to-flavor transition V(x) is integrated with sin Fm(d) d = xf - x the distance from structure to the detector stronger averaging effects weaker sensitivity to structure of density profile larger Fm(d) larger d Integration with the energy resolution function R(E, E’): freg = dE’ R(E, E’) freg(E’) xf x0 The effect of averaging: freg = 0.5 sin22q dx V(x) F(xf - x) sin Fm(x -> xf) averaging factor For box-like R(E, E’) with width DE: ln E p d DE p d DE ln E F(d) = sin

  16. Attenuation effect The width of the first peak d < ln E/DE Attenuation factor F ln is the oscillation length The sensitivity to remote structures is suppressed: Effect of the core of the Earth is suppressed Small structures at the surface can produce stronger effect d, km The better the energy resolution, the deeper penetration

  17. Attenuation length d = ln E/DE d ~4pE E/DEDm2 DE/E Core ln d, km Solar neutrinos E = 10 MeV can not be seen 300 km 10 - 20% 1500 – 3000 Supernova neutrinos E = 30 MeV can be seen 9000 900 km 10% ln is the oscillation length

  18. Averaging regeneration factor Regeneration factor averaged over the energy intervals E = (9.5 - 10.5) MeV (a), and E = (8 - 10) MeV (b). No enhancement for core crossing trajectories in spite of larger densities

  19. In e order ~ 1 e n2 ->ne e n2 ne xf x0 neglect oscillations n2 ne x flavor oscillations ~ 1 ne ->n2 e e ne n2 x0 xf x flavor oscillations ne n2 neglect oscillations

  20. Next order corrections - Next order correction - e increases with energy

  21. Second order correction Mass-to-flavor transition: n2 --> ne P2e = sin2q + freg xf x xc In symmetric density profile xc - is the center of trajectory freg = sin22q sinFm(xc -> xf) I + cos2q I2 + . . . Expansion in I, where xf xc Fm(xc -> x) adiabatic phase from the center to a given point x I = dx V(x) cos Fm(xc -> x) Estimate: O(I) - term is absent (suppressed) for trajectories where sinFm(xc -> xf) = k p 2EVmax Dm2 I < = emax

  22. Relative errors d = (fappr - fexact) / f0 f0 = 0.5 ef sin22q at the surface A. Ioannisian et al, Phys. Rev. D (2005) Second order First order For the neutrino trajectory which crosses the center of the Earth

  23. Improving theory further

  24. Improved perturbation theory 2EV Dm2 increases with energy Improved perturbation theory: expansion with respect to some average potential V0 2E DV 2E(V - V0) Dm2 Dm2 e’ = = The solution in the basis of eigenstates in matter with potential V0 ( n10, n20 ) The S-matrix for n10, n20, S0, can be obtained from the S-matrix in the basis of mass eigenstates by substitution V -> DV, q -> qm0 Fm(xc -> x) do not change S0 = S(DV, qm0) qm0 = qm0 (V0) - mixing in matter with the potential V0

  25. Regeneration factor P2e = sin2q + freg For symmetric profile freg = e0 sin22qm0 sinFm(xc -> xf) + B1(e0) sinFm(xc -> xf)ID + B2(e0)ID2 where B1 = sin22qm0 (cos2q0’ +e0 cos 2qm0) B2 = sin22qm0/2(cos 2qm0 + cos2q0’ - 2sin2q- 2e0 sin22qm0)I2 xf xc 2EV0 Dm2 e0 = ID = dx DV(x) cos Fm(xc -> x) xc - is the center of trajectory q0’ =qm0 - q Fm(xc -> x) adiabatic phase from the center to a given point x

  26. Relative errors d (10 MeV/E)2 5% 1% 50 MeV x 25 20 MeV x 4 10 MeV x 1 0 20% 1.67 2.5 3.0 N0e The reduced error (in second order) d/E2 as function of 1/E for different values of the density shift N0e

  27. Expansion parameter |B1|, |B2| < 1 Essentially the parameter of expansion is the integral ID Integrating by parts: ID ~ 2E(Vf – V0)/Dm2 sin Fm(xc – xf) - - 2E/Dm2 dx dV(x)/dx sin Fm(xc -> x) xf xc does not depend on shift of V The dependence on V0 due to the boundary condition Vf - potential at the surface The strongest suppression for V0 ~ Vf (also related to the attenuation effect) xc - is the center of trajectory Fm(xc -> x) adiabatic phase from the center to a given point x

  28. Where we are now?

  29. Day-Night asymmetry 2 (N – D) N + D ACC = Binned Day-Night CC event asymmetries as functions of the electron energy (statistical uncertainties only) ACC = - 0.021 +/- 0.063 +/- 0.035 Expected: + (0.025 - 0.030) Previous: - 0.07 Gradient with energy: OK Systematic shift of the night Spectrum?

  30. Day-Night asymmetry Day and Night CC energy spectra (statistical uncertainties only) ACC = freg /sin2q Gradient with energy: OK Systematic shift of the night Spectrum?

  31. Summary Integral formula for oscillations in low density medium which is valid for arbitrary density profile For LMA parameters it can be applied to the solar and supernova neutrino oscillations inside the Earth -> oscillation tomography of the Earth probing structures of 20 – 2000 km size Attenuation effect: the sensitivity to remote structures decreases. The better the energy resolution the weeker attenuation With next order corrections: results can be applied up to 40 – 50 MeV Expansion parameter - integral ID improvement when the shift potential: V0 ~ V(surface) Neutrino microscopes => large neutrino telescopes with low energy thresholds and good energy resolution

  32. Oscillations inside the Earth 1). Incoherent fluxes of n1 and n2 arrive at the surface of the Earth 2). In matter the mass states oscillate 3). the mass-to-flavor transitions, e.g. n2 --> ne are relevant Regeneration factor: P2e = sin2q + freg Pee = 0.5[ 1 + cos 2qm0cos 2q ] - cos2qm0freg 4). The oscillations proceed in the weak matter regime: 2EV(x) Dm2 e (x)= << 1

  33. Oscillations inside the Earth Oscillations in multilayer medium Solar and supernova neutrinos mass to flavor transitions n2 mantle Accelerator neutrinos LBL experiments atmospheric neutrinos: flavor to flavor transitions core Regeneration of the ne flux Variety of possibilities depending on - trajectory, - neutrino energy and - channel of oscillations

  34. Solar and SN neutrinos inside the Earth 1). Incoherent fluxes of n1 and n2 arrive at the surface of the Earth 2). In matter the mass states oscillate 3). the mass-to-flavor transitions, e.g. n2 --> ne are relevant Regeneration factor: P2e = sin2q + freg Pee = sin2q + cos 2qm0cos2q - cos2qm0freg 4). The oscillations proceed in the weak matter regime: 2EV(x) Dm2 e (x)= << 1

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