One of the important goals for future neutrino telescopes is to identify the flavors of astrophysical neutrinos and therefore determine the flavor ratio. The flavor ratio of astrophysical neutrinos observed on the Earth depends on both the initial flavor ratio at the source and flavor transitions taking place during propagations of these neutrinos. We propose a model independent parametrization for describing the above flavor transitions. A few flavor transition models are employed to test our parametrization. The observational test for flavor transition mechanisms through our parametrization is discussed.
Recent developments of neutrino telescopes [1][2][3][4][5] have inspired numerous efforts of studying neutrino flavor transitions utilizing astrophysical neutrinos as the beam source [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].
Given the same neutrino flavor ratio at the source, some flavor transition models predict rather different neutrino flavor ratios on the Earth compared to those predicted by the standard neutrino oscillations [10]. In this article, we propose a scheme to parametrize flavor transition mechanisms of astrophysical neutrinos propagating from the source to the Earth. As will be shown later, such a parametrization is very convenient for classifying flavor transition models which can be tested by future neutrino telescopes.
To test flavor transition mechanisms, it is necessary to measure the flavor ratio of astrophysical neutrinos reaching to the Earth. The possibility for such a measurement in IceCube has been discussed in Ref. [8]. It is demonstrated that the ν e fraction can be extracted from the measurement of the muon track to shower ratio by assuming flavor independence of the neutrino spectrum and the equality of ν µ and ν τ fluxes on the Earth due to the approximate ν µ -ν τ symmetry [26,27]. Taking a neutrino source with fluxes of ν e and ν µ given by E 2 νe dN νe /dE νe = 0.5E 2 νµ dN νµ /dE νµ = 10 -7 GeV cm -2 s -1 , which is roughly the order of the Waxman-Bahcall bound [28], and thresholds for muon and shower energies taken to be 100 GeV and 1 TeV, respectively, the ν e fraction can be determined to an accuracy of 25% at IceCube for 1 yr of data taking, or equivalently to an accuracy of 8% for a decade of data taking. Such an accuracy is obtained for a ν e fraction in the vicinity of 1/3. The accuracies corresponding to other central values of the ν e fraction are also presented in Ref. [8].
The ν µ to ν τ event ratio can also be measured in IceCube. However, the accuracy of this measurement is limited by the low statistics of ν τ events.
Neutrino flavor ratio represented by the ternary plot.-To study neutrino flavor transitions, we describe the neutrino flavor composition at the source by a normalized flux Φ 0 = (φ 0,e , φ 0,µ , φ 0,τ ) T satisfying the condition [29] φ 0,e + φ 0,µ + φ 0,τ = 1, φ 0,α ≥ 0, for α = e, µ, τ,
where each φ 0,α is the sum of neutrino and antineutrino fluxes. Any point on or inside the triangle shown in Fig. 1 represents a specific flavor ratio characterizing the source. The triangular region bounded by vertices (1, 0, 0) T , (0, 1, 0) T , and (0, 0, 1) T contains all possible source types in terms of flavor ratios. The pion source and the muon-damped source with flavor compositions Φ 0,π = (1/3, 2/3, 0) T and Φ 0,µ = (0, 1, 0) T , respectively, are explicitly marked on the figure [30].
The net effect of flavor transition processes occurring between the source and the Earth is represented by the matrix P such that
where Φ = (φ e , φ µ , φ τ ) T is the flux of neutrinos reaching to the Earth. We note that our convention implies P αβ ≡ P (ν β → ν α ).
Q matrix parametrization for flavor transitions of astrophysical neutrinos.-Since the triangular region in Fig. 1 represents all possible neutrino flavor composition at the source, it is convenient to parametrize Φ 0 by [31] Φ
where
represents the center of the triangle, while aV The above parametrization for Φ 0 is also physically motivated. The vector V 1 gives the normalization for the neutrino flux since the sum of components in V 1 /3 is equal to unity, while the sum of components in V 2 and that in V 3 are both equal to zero. The vector aV 2 determines the difference between ν µ and ν τ flux, φ 0,µ -φ 0,τ , while preserving their sum, φ 0,µ + φ 0,τ . Finally the vector bV 3 determines the sum of ν µ and ν τ flux, φ 0,µ + φ 0,τ , while preserving their difference φ 0,µ -φ 0,τ .
Following the same parametrization, we write the neutrino flux reaching to the Earth as
It is easy to show that where
In other words, Q is related to P by a similarity transformation where columns of the transformation matrix A correspond to vectors V 1 , V 2 , and V 3 , respectively.
The parameters κ, ρ and λ are related to the flux of each neutrino flavor by
with the normalization φ e + φ µ + φ τ = 3κ. Since we have chosen the normalization φ 0,e + φ 0,µ + φ 0,τ = 1 for the neutrino flux at the source, the conservation of total neutrino flux during propagations corresponds to κ = 1/3. In general flavor transition models, κ could be less than 1/3 as a consequence of (ordinary) neutrino decaying into invisible states or oscillating into sterile neutrinos. To continue our discussions, it is helpful to rewrite Eq. ( 7)
It is then clear from Eqs. ( 5) and ( 8) that, for fixed a and b, the first row of matrix Q determines the normalization for the total neutrino flux reaching to the Earth, the second row of Q determines the breaking of ν µ -ν τ symmetry in the arrival neutrino flu
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