MAGNUS APPROXIMATION FOR NEUTRINO OSCILLATIONS WITH … · MAGNUS APPROXIMATION FOR NEUTRINO OSCILLATIONS WITH THREE FLAVORS IN MATTER Alexis A. Aguilar-Arévalo in collaboration

MAGNUS APPROXIMATION FOR NEUTRINO OSCILLATIONS

WITH THREE FLAVORS IN MATTER

Alexis A. Aguilar-Arévalo

in collaboration with J.C. D'Olivo

Instituto de Ciencias Nucleares,Universidad Nacional Autónoma de México

TAUP 2009, Rome, Italy, July 1-5, 2009

Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009 1

MAGNUS APPROXIMATION FOR NEUTRINO OSCILLATIONS

WITH THREE FLAVORS IN MATTER

Alexis A. Aguilar-Arévalo

in collaboration with J.C. D'Olivo

Instituto de Ciencias Nucleares,Universidad Nacional Autónoma de México

TAUP 2009, Rome, Italy, July 1-5, 2009

2

Outline:● Magnus expansion● Calculation of 3 evolution operator (1st order)● Examples (exp, powlaw profiles)

Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009

Magnus expansion

Schrödinger eq. for the evolution operator:

W. Magnus (Commun.Pure Appl.Math. 7, 649, 1954) solution of the form:

The 's are antiHermitian truncating the series at any order yields a unitary approximation for the evolution operator U.

. . .

3

More details: Blanes et al. Phys. Rep. 470: 151–238 (2009)


3 mixing

Neutrino state at time t :

flavor eigenstate basis mass eigenstate basis

Relation between amplitudes via mixing matrix U (PMNS):

Standard factorization of PMNS matrix with one CPV phase .

4Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009

3 evolution equation

Hamiltonian in the flavor basis:

Interaction with matter incorporated through V(t) ~ (e number density)

Diagonalization at any time t defines the instantaneous eigenstate basis:

in which the evolution operator satisfies the equation

It is convenient to work with a real symmetric matrix instead:


approximate diagonalizationThe real symmetric matrix has 3 real eigenvalues

Mass Hierarchy In each region the characteristic equation approximates the exact one:

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We obtain good approximations by diagonalizing approximate forms of the matrix in two regions: low density ( ) and high density ( )


approximate diagonalization

x x x x x

x x x x x

Via the orthogonal transformation:


back to the evolution equationBoth diagonalizations have the same form and lead to a U

m(t) of the form:

and a final change of representation:

redefinition:


two 2 problems

~0

~0

The Hamiltonian for the problem we solve:

A factorized solution is possible:

MAD = Maximum Angle Derivative

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~I


factorized solution

We solve each 2 factor with a 1st order Magnus approximation (D'Olivo, PRD 45,924,1992)


analytical approximationsOnly remains to calculate the K

1,3 factors. Evaluate up to t=T (border V=0):

Perform linear expansion of phase l,h around MAD points:

This integral can be approximately solved as (PRD 45,924,1992):


average survival probability

Applying the total evolution operator:

For the initial state of a e :

and the (averaged) survival probability has the form:


exponential & powerlaw profiles

Exp: Ne(t0)= 6.0x1025 cm3

Pow: Ne(t0)= 3.5x1034 cm3

13

Pow SN M=14 M⊙(mass of ejecta) [Shiguyama, and Nomoto APJ 360, 42, 1990] .


exponential profile

14

m221=7.59105 eV2, sin2 2

12= 0.87

m231=2.4103 eV2, sin2 2

13= 0.01

SNOBx 8B

Bx 7Be pp pred.

E=10 MeV,m2

31/m221=32,

sin2 212

/ sin2 213

= 0.87 / 0.01


powerlaw density profile

E=15 MeV, m231=2.4103 eV2, Tan2

13= 4104

+ Earth mantle (m~ 4.5g/cm3 L~ 8500 km) [as in Fogli et.al. PRD 65, 073008 (2002)]

Iso(3 survival probability) contours:


Summary

We use the Magnus expansion to find an approximate analytical expression for the evolution operator for 3 neutrinos propagating in a medium with arbitrary varying density.

Our solution is given as the product of two evolution operators for 2 neutrino systems, each calculated with a 1st order Magnus approximation.

Our result works well in both, the adiabatic and the non adiabatic regimes.

As an example, we calculated the averaged survival probability for the exponential((r)~exp(r) and the powerlaw ((r)~r3) density profile.

We are working on applying this calculation to other contexts (e.g. realistic Earth matter density profile in the case of atmospheric and accelerator 's).


BACKUPS

adding Earth matter effects (mantlecoremantle)

L~8500 km

m~ 4.5g/cm3 .

Final state substitution [T.K. Kuo and J. Pantaleone, Rev. Mod. Phys., 61, 937 (1989)]:

Independent of the mass hierarchy.

Use identity:

Want approximation of integral:

generalized Laguerre polynomials

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MAGNUS APPROXIMATION FOR NEUTRINO OSCILLATIONS WITH … · MAGNUS APPROXIMATION FOR NEUTRINO OSCILLATIONS WITH THREE FLAVORS IN MATTER Alexis A. Aguilar-Arévalo in collaboration