Solar Neutrino Zenith Angle Distribution and Uncertainty in Earth Matter Density

We estimate in this paper the errors in the zenith angle distribution for the charged current events of the solar neutrinos caused by the uncertainty of the earth electron density. In the model of PREM with a 5% uncertainty in the earth electron density we numerically calculate the corrections to the correlation between [N ]5/[N ]2 and [N ]2/[N ]3, and find the errors notable. Forthcoming results from SNO[1] include a measurement of the day-night asymmetry (ADN )[2, 3, 4, 5]. This measurement is crucial to confirming the matter conversion solution to the solar neutrino problem. And the analysis on the zenith angle distribution of the events during the night may provide some insights to distinguish the various MSW solutions, i.e. LMA, LOW and SMA[6]. In the calculation of the regenerated νe flux, the electron density of the earth matter with which the neutrinos interact is a critical quantity. The uncertainty in Earth-matter density and chemical component can be a major cause of the error in ADN and the zenith angle distribution. So it will be interesting to estimate these errors. Furthermore since the experimental value of the ADN is around ∼ 0.047[2] and the theoretical expectations on the zenith angle distributions are small in magnitude[6], it is necessary to perform a quantitative estimation on these errors. In this paper, we follow the procedure outlined in [7] and study the uncertainty in the earth matter density, then investigate its implications on the predictions of ADN and the zenith angle distributions. We quantify the uncertainties of the earth matter in terms of two parameters: one is δNe/Ne, the variation in magnitude of the density which generally is expected to be around a few percent; the second one is δx which specifies the limitation on the spacial dimension by geophysics experiments and inverting calculations used in the fit of the earth density models. In general the scale δx is not much larger than the neutrino oscillation length, e.g., in the case with the parameters of the favored LMA solution, so its effect might arise beyond the linear order. We will show in this paper that this effect causes an sizable error in the zenith angle distributions. To begin with we consider a two-neutrino mixing model for simplicity. As discussed in[8, 4] the neutrino can be treated as a incoherent mixture of two mass eigenstates. In the day-time the survival probability for νe is given by, P = P1 cos 2 θ + (1− P1) sin θ , (1) where the mixing angle is defined through, ν1 = cos θ νe − sin θ νμ , ν2 = sin θ νe + cos θ νμ , (2) and P1 is the probability of the νe → ν1 conversion inside the sun[9, 6]. During the night time, the presence of the earth matter leads to a zenith angle dependent regeneration of the νe, P = P1 + (1 − 2P1)P2e = P − 2Xfreg , (3) where P2e is the probability of the ν2 → νe conversion inside the Earth, X = P1 − 1/2. And freg(θz) ≡ P2e(earth matter)− P2e(vacuum) , (4) is the regeneration factor which vanishes in the absence of the Earth matter effect. Defining f̄reg as the regeneration factor integrated over the zenith angle, one has the day-night asymmetry, ADN ≡ P − P 1 2 (P N + PD) = −2Xf̄reg 0.5 + (cos 2θ − f̄reg)X . (5)