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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
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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.
. . .
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More details: Blanes et al. Phys. Rep. 470: 151–238 (2009)
Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 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:
5Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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 ( )
Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
approximate diagonalization
x x x x x
x x x x x
Via the orthogonal transformation:
7Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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:
8Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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
Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
factorized solution
We solve each 2 factor with a 1st order Magnus approximation (D'Olivo, PRD 45,924,1992)
10Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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):
11Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
average survival probability
Applying the total evolution operator:
For the initial state of a e :
and the (averaged) survival probability has the form:
12Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
exponential & powerlaw profiles
Exp: Ne(t0)= 6.0x1025 cm3
Pow: Ne(t0)= 3.5x1034 cm3
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Pow SN M=14 M⊙(mass of ejecta) [Shiguyama, and Nomoto APJ 360, 42, 1990] .
Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
exponential profile
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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
Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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:
15Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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).
16Alexis AguilarArévalo, ICNUNAM TAUP 2009 Rome, Italy, July 15, 2009
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.