Conventions

These are the conventions Magνs follows: the parametrization, signs, units and indexing that another code may choose differently. A convention applied wrongly but consistently passes every internal test, so each one is stated here.

At a glance

What	Convention
Probability matrix	`P[i][f]` is \(P(\nu_i \to \nu_f)\), initial flavor first; flavors in the order \((e, \mu, \tau, s_1, s_2)\); a batched call puts the point index first, `(n_points, d, d)`
Mixing matrix	The PDG parametrization, \[\mathbb{R} = \mathbb{R}_{23}(\theta_{23})\,\mathbb{R}_{13}(\theta_{13},\delta_{\rm CP})\, \mathbb{R}_{12}(\theta_{12}),\] with \(\mathbb{R}_{e3} = \sin\theta_{13}\,e^{-i\delta_{\rm CP}}\)
Mixing angles	Given as sines by default; `angles='sin2'`, `'rad'` or `'deg'` changes that
CP phases	Radians (degrees under `angles='deg'`)
Mass splittings	`D21` \(= m_2^2 - m_1^2\), `D31` \(= m_3^2 - m_1^2\), in eV²; the ordering is the sign of `D31`
Inverted ordering from NuFIT	NuFIT quotes \(\Delta m^2_{32}\); `load_nufit_params(..., 'IO')` returns `D31` \(= \Delta m^2_{32} + \Delta m^2_{21}\)
Antineutrinos	`nubar=True` conjugates the mixing matrix and flips the sign of the matter potential. A Lorentz-violating term is conjugated too, and changes sign when `n_liv` is even: an operator of dimension `n_liv + 3` is CPT-odd when `n_liv + 3` is odd, as in the Standard-Model Extension (Kostelecký & Mewes, Phys. Rev. D 85, 096005 (2012))
Matter potential	\(V_{\rm CC} = +\sqrt{2} G_F n_e\) on the \(\nu_e\) entry; \(Y_e = 0.5\) by default, and one value per layer in the Earth (0.4656 core, 0.4957 mantle)
Units	Natural units: energies in eV, distances in eV^-1, densities in eV⁴; see Units

The Hamiltonian and the probability

The state of \(n\) flavors evolves under an \(n \times n\) Hermitian Hamiltonian \(\mathbb{H}\). The solution of the Schrödinger equation defines the evolution operator \(\mathbb{U}(l_1, l_0)\), and the probability that a neutrino born as \(\nu_\alpha\) is detected as \(\nu_\beta\) is

(1)\[P_{\nu_\alpha \to \nu_\beta}(l_1, l_0) = \left\lvert \left[\mathbb{U}(l_1,l_0)\right]_{\beta\alpha} \right\rvert^2 .\]

In vacuum,

(2)\[\mathbb{H}^{\rm vac}(E) = \frac{1}{2E}\, \mathbb{R}\, \mathbb{M}^2\, \mathbb{R}^\dagger , \qquad \mathbb{M}^2 = {\rm diag}(0, \Delta m^2_{21}, \Delta m^2_{31}, \ldots) ,\]

with the mixing matrix \(\mathbb{R}\) a product of rotations. Each \(\mathbb{R}_{ij}\) rotates by \(\theta_{ij}\) in the \((i,j)\) plane, with \(\sin\theta_{ij}\,e^{-i\delta_{ij}}\) in entry \((i,j)\):

(3)\[\begin{split}\mathbb{R} = \begin{cases} \mathbb{R}_{12} , & n = 2 , \\ \mathbb{R}_{23}\, \mathbb{R}_{13}\, \mathbb{R}_{12} , & n = 3 , \\ \mathbb{R}_{34}\, \mathbb{R}_{24}\, \mathbb{R}_{14}\, \mathbb{R}_{23}\, \mathbb{R}_{13}\, \mathbb{R}_{12} , & n = 4 , \\ \mathbb{R}_{35}\, \mathbb{R}_{25}\, \mathbb{R}_{15}\, \mathbb{R}_{34}\, \mathbb{R}_{24}\, \mathbb{R}_{14}\, \mathbb{R}_{23}\, \mathbb{R}_{13}\, \mathbb{R}_{12} , & n = 5 . \end{cases}\end{split}\]

At three flavors, this is the PMNS matrix, and the phase of \(\mathbb{R}_{13}\) is \(\delta_{\rm CP}\). Of the sterile rotations, only \(\mathbb{R}_{14}\), \(\mathbb{R}_{24}\), \(\mathbb{R}_{15}\) and \(\mathbb{R}_{35}\) carry a phase.

In matter, the charged-current potential enters through a diagonal projector \(\mathbb{P}\):

(4)\[\mathbb{H}(E, l) = \mathbb{H}^{\rm vac}(E) + V_{\rm CC}(l)\, \mathbb{P}(l) , \qquad V_{\rm CC} = \sqrt{2}\, G_F\, n_e(l) .\]

At three flavors, \(\mathbb{P} = {\rm diag}(1, 0, 0)\). With sterile states, \(\mathbb{P} = {\rm diag}(1, 0, 0, r/2, \ldots)\), with \(r = n_n/n_p\) (ratio_number_neutrons_to_protons): the neutral-current potential, common to the active flavors, is subtracted from the whole diagonal, and what it leaves on a sterile state is \((r/2)\,V_{\rm CC}\). magnus.matter.matter_potential_projector() builds \(\mathbb{P}\) for every flavor count.

Ordering of the probabilities

Every osc_prob_* function returns the probability matrix indexed initial flavor first:

\[P[\nu_i][\nu_f] \;=\; P(\nu_i \to \nu_f) .\]

For example, P[1][0] is \(P(\nu_\mu \to \nu_e)\), not the reverse. Flavors are in the standard order \((e, \mu, \tau, s_1, s_2)\), so index 0 is always \(\nu_e\).

Each row sums to one — a neutrino that started as \(\nu_i\) ends as something. Each column also sums to one; both sums follow from unitarity. Passing nu_i and nu_f returns that one entry instead of the matrix.

For a batched call, the point index comes first: the shape is (n_points, d, d), so P[:, 1, 0] is \(P_{\mu e}\) along a scan.

Sign of the matter potential

The charged-current potential enters the electron-flavor diagonal entry,

\[\mathbb{H} \;=\; \mathbb{H}^{\rm vac} \;+\; \mathrm{diag}(V_{\rm CC},\, 0,\, \ldots) , \qquad V_{\rm CC} = +\sqrt{2}\, G_F n_e ,\]

and for antineutrinos it changes sign. That flip is applied once, inside magnus.matter.vcc_func_from_rho_func(), so a user passing nubar=True gets it automatically, and code downstream must not apply it again. Applied twice, it would give antineutrinos a positive potential, and the resulting probabilities would still look plausible.

Mass ordering

The ordering is carried by the sign of \(\Delta m^2_{31}\), not by a flag: positive is normal, negative is inverted. The default parameter set, OSC_PARAMS_PREDEFINED['OSC_PARAMS_DEFAULT'], is the normal ordering, with \(\Delta m^2_{31} = +2.511 \times 10^{-3}\) eV². It is NuFIT 6.1 with Super-Kamiokande atmospheric data, the release load_nufit_params() returns by default. To name a fit explicitly, magnus.globaldefs.OSC_PARAMS_PREDEFINED holds every NuFIT release from 1.0 on, in both orderings and, from 4.0 on, with and without that atmospheric data. Its names are generated from gd.NUFIT_GLOBAL_FITS, the table that load_nufit_params() reads.

For two flavors, the same rule applies to \(\Delta m^2\), which is what makes the two-flavor case easy to get backwards: flipping its sign moves the MSW resonance into the other channel, and the result still looks like an ordinary probability.

Mixing parameters

Angles are given as sines by default (angles='sin'): s12 is \(\sin\theta_{12}\), not the angle and not \(\sin^2\theta_{12}\). Setting angles='sin2', 'rad' or 'deg' selects the other forms. Under 'rad' and 'deg', an angle beyond \(\pm 90^\circ\) is refused, because the cosine is taken as \(+\sqrt{1 - s^2}\). Fits usually quote \(\sin^2\theta\); pass those with angles='sin2', or take the square root, e.g., s12 = np.sqrt(0.309). Phases are in radians; the default \(\delta_{\rm CP}\) is 3.7001 rad, i.e., 212°.

Two flavors take sth and Dm2 rather than s12 and D21. A mistake here cannot pass unnoticed: unrecognized keywords are refused by name, so a two-flavor call written with the three-flavor names raises an error instead of returning a probability computed from the defaults.

Units

Magνs uses natural units throughout: energies in eV, baselines and positions in eV^-1, so that \(\mathbb{H} L\) is dimensionless. magnus.globaldefs supplies the conversions — multiply by UNIT_KM, UNIT_MEV, UNIT_GEV, UNIT_G_PER_CM3 — and Units lists them.

Coming from GLoBES, Prob3++ or nuSQuIDS

The physics is the same; the bookkeeping differs. Each entry for the other codes is what the drivers behind Against other codes pass them (resources/benchmarks/external_drivers, in a source checkout).

	Magνs	GLoBES	Prob3++	nuSQuIDS
Energy	eV: multiply GeV by `gd.UNIT_GEV`	GeV	GeV	its own units: `E*units.GeV`, with `units = nuSQuIDS.Const()`
Baseline	eV^-1: multiply km by `gd.UNIT_KM`	km	km, or a zenith cosine through the Earth	its own units: `L*units.km`
Density	natural units, or g cm^-3 with `density_matter_is_in_g_per_cm3=True`	g cm^-3	g cm^-3	g cm^-3
\(Y_e\)	`electron_fraction`, 0.5 by default; per layer through the Earth	fixed at 0.5 inside the code	from the profile file on the Earth path	an argument of the body, e.g., `ConstantDensity(rho, 0.5)`
Flavor indices	0, 1, 2 = e, μ, τ; `P[initial][final]`	1, 2, 3; `(initial, final, …)`	1, 2, 3; `GetProb(initial, final)`	0, 1, 2; initial state set, then `EvalFlavor(final)`
Antineutrinos	`nubar=True`	`cp_sign = -1`	a negative neutrino type	`NeutrinoType.antineutrino`
Angles	sines by default; `angles='sin2'`, `'rad'` or `'deg'`	radians, in the order θ12, θ13, θ23	\(\sin^2\theta\) (`kSquared=true`) or \(\sin^2 2\theta\), in the order θ12, θ13, θ23	radians, `Set_MixingAngle(i, j, θ)`
Splittings and ordering	`D21`, `D31`; the sign of `D31` is the ordering	Δm²21, Δm²31	Δm²21 and Δm²32, not Δm²31	`Set_SquareMassDifference(1, Δm²21)`, `(2, Δm²31)`
\(\delta_{\rm CP}\)	radians (degrees under `angles='deg'`)	radians	radians	radians, `Set_CPPhase(0, 2, δ)`

Two differences matter at the level of the comparison itself:

The matter potential for a given density. Every code converts g cm^-3 to an electron density as \(\rho N_A Y_e\), but with its own rounding of the constant \(\sqrt{2} G_F N_A\): Prob3++’s value is 0.04% below Magνs’s, and the comparison drivers correct for the difference. The stored benchmark data predate the potential correction in version 1.2.0, which raised the Magνs potential by 0.8%; their rescaling factors (0.992 for GLoBES) include that difference.
Prob3++ takes Δm²32. Passing Δm²31 in its place changes probabilities by up to 0.26.