Analytic approximations for three neutrino oscillation parameters and probabilities in matter

Abstract

The corrections to neutrino mixing parameters in the presence of matter of constant density are calculated systematically as series expansions in terms of the mass hierarchy ∆m 2 21 /∆m 2 31 . The parameter mapping obtained is then used to find simple, but nevertheless accurate formulas for oscillation probabibilities in matter including CP-effects. Expressions with one to one correspondence to the vacuum case are derived, which are valid for neutrino energies above the solar resonance energy. Two applications are given to show that these results are a useful and powerful tool for analytical studies of neutrino beams passing through the Earth mantle or core: First, the "disentanglement problem" of matter and CP-effects in the CP-asymmetry is discussed and second, estimations of the statistical sensitivity to the CP-terms of the oscillation probabilities in neutrino factory experiments are presented.

Status of three-neutrino oscillations after the SNO-salt data

“…9 the ∆χ 2 from the global data as a function of this parameter. Also shown in this figure is the ∆χ 2 as a function of the parameter combination α sin 2θ 12 , since to leading order in the long baseline ν e → ν µ oscillation probability solar parameters appear in this particular combination [52]. We obtain the following best fit values and 3σ intervals:…”

LeptonicCPviolation

“…(3.1). Approximate expressions for the oscillation probabilities in matter of constant density have been obtained for ∆m 2 21 ≪ A, ∆m 2 31 (Asano and Minakata, 2011;Cervera et al, 2000;Freund, 2001). Treating θ 13 and r as small parameters, one has for the golden channel ν e → ν µ (Cervera et al, 2000) P eµ ≃ T 1 sin 2 2θ 13 + r (T 2 + T 3 ) sin 2θ 13 + r 2 T 4 , (3.25) at second order in sin 2θ 13 and r. The terms T i in the above equation are (Huber et al, 2006a)…”

Why is the neutrino oscillation formula expanded in Δm 21 2 /Δm 31 2 still accurate near the solar resonance in matter?

“…Then we show the α-expansion of the result from Freund's calculation [11] and compare the approximate result with the exact result (though complicated but numerically programmable) to see how much it deviates from the exact result. We will show that the α-expansion result of effective neutrino parameters are quite inaccurate and even invalid near the solar resonance but the final result (i.e.…”

“…OnceŨ is computed, we can extract effective mixing angles from the standard parametrization ofŨ . All effective parameters (masses and mixing angles) expanded in α are given below [11]:…”

“…The formula was originally derived in [10,11] as a series expansion in α. But the problem is that due to the non-perturbative behavior near the solar resonance, the expansion is expected to be valid only when the neutrino energy is well above the solar resonance, …”