Nu-Pert

master	version	coverage	doi

Overview

This code provides a very precise, yet still useful analytically, means of calculating the neurtino transition probabilities in uniform matter densities. We have included our results to various orders in a perturbative expansion. Our perturbative expansion is proprotional to the ratio of the two measured Delta m²'s. It is smaller than 1.5% and is exactly zero in vacuum.

Zeroth order is more than sufficient for any modern neutrino experiment with precision on the order of 10^-4. First order improves this by about three more orders of magnitude. Second order is also included through a different code that provides about three more orders of magnitude in precision, and, in some cases, hits the limit of double precision. These results contain all orders in the matter potential and in s₁₃.

About the code

Equation numbers are provided where possible which reference arXiv:1604.08167 unless otherwise stated. The various namespaces throughout the code refer to the various steps and means of calculating the transition probabilities.

First, the Hat namespace is in reference to the intermediate basis after the 23 and 13 rotations. The Hat namespace refers to the results from the arXiv:1505.01826 paper by S. Parke and H. Minakata and is described in this paper in subsection 2.3 of arXiv:1604.08167. This namespace contains the eigenvalues and the angle phi (see eqs. 2.3.3, 2.3.4, and 2.3.5). This namespace also includes the explicit formulas for P(nu_e -> nu_e) and P(nu_e -> nu_mu) in this basis (see eqs. (2.11) and (2.14) from arXiv:1505.01826).

Next is the Check namespace which is in reference to the final basis after an additional 12 rotation described in subsection 2.4 of arXiv:1604.08167. This namespace contains the final eigenvalues at zeroth and second order (zeroth order also contains all first order corrections), and the angle psi (see eqs. 2.4.5, 3.1.3, 2.4.7, and 2.4.9). It also contains the W and V matrices as well as a transition probability calculation (see subsection 3.2 and eq. 4.0.1). The probability calculated in this namespace is based on multiplying out the V matrix. This is identical to the GF namespace method at zeroth order, but at first order it contains some higher order terms. This method is coded through second order, but could be expanded to higher orders.

The GF namespace refers to the general form of the first order transition probabilities presented in eqs. 4.2.1 and 4.2.2, and explicitly shown in table 1. This namespace provides an order-by-order calculation for zeroth or first order in that the first order calculation provides no manifestly higher order terms.

The vacuum values of the oscillation parameters are described in Constant.cpp. Since there is no consistent way for all channels to switch the mass ordering, we take the simple choice of changing the sign on Delta m²₃₁.

See Examples.cpp for several examples.

Some further notes

The antineutrino transition probabilities are the same as the neutrino transition probabilities except with E -> -E. That is, L/E -> -L/E and a -> -a.

P(nu_a -> nu_b) is related to P(nu_b -> nu_a) by sending L -> -L.

Support

If you have questions or encounter any problems when running Nu-Pert, please use github's issue tracker.

This code is free to use, copy, distribute, and modify. If you use this code or any modification of this code, we request that you reference the relevant publication, arXiv:1604.08167.