magnus.hamiltonians.hamiltonians2nu

hamiltonians2nu.py

Compute two-neutrino Hamiltonians for selected scenarios.

This module contains the routines to compute the two-neutrino Hamiltonians for the following scenarios: oscillations in vacuum, in matter of constant density, in matter with non-standard interactions (NSI), and in a CPT-odd Lorentz invariance-violating background (LIV).

Routine listings

  • mixing_matrix_2nu - Returns 2x2 rotation matrix

  • hamiltonian_2nu_vacuum_energy_independent - Returns H_vac (no 1/E)

  • hamiltonian_2nu_vacuum_energy_independent_td - Returns H_vac (no

    1/E), as a function of position

  • hamiltonian_2nu_vacuum - Returns H_vac

  • hamiltonian_2nu_vacuum_td - Returns H_vac, as a function of position

  • hamiltonian_2nu_matter - Returns H_matter

  • hamiltonian_2nu_matter_td - Returns H_matter, as a function of position

  • hamiltonian_2nu_nsi - Returns H_NSI

  • hamiltonian_2nu_nsi_td - Returns H_NSI, as a function of position

  • hamiltonian_2nu_liv - Returns H_LIV

  • hamiltonian_2nu_liv_energy_independent - Returns H_LIV (no energy

    dependence)

Functions

mixing_matrix_2nu(→ numpy.ndarray)

Returns the 2x2 rotation matrix.

hamiltonian_2nu_vacuum_energy_independent(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for vacuum oscillations.

hamiltonian_2nu_vacuum_energy_independent_td(...)

Returns the two-neutrino Hamiltonian for vacuum oscillations, as a function of distance,

hamiltonian_2nu_vacuum(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for vacuum oscillations.

hamiltonian_2nu_vacuum_td(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for vacuum oscillations, as a function of distance,

hamiltonian_2nu_matter(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for matter oscillations.

hamiltonian_2nu_matter_td(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for matter oscillations, as a function of distance.

hamiltonian_2nu_nsi(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for oscillations with NSI.

hamiltonian_2nu_nsi_td(→ numpy.ndarray)

Returns the two-neutrino NSI Hamiltonian as a function of position.

hamiltonian_2nu_liv(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for oscillations with LIV.

hamiltonian_2nu_liv_energy_independent(→ numpy.ndarray)

Returns the two-neutrino Hamiltonian for oscillations with LIV.

Module Contents

magnus.hamiltonians.hamiltonians2nu.mixing_matrix_2nu(sth: float) → numpy.ndarray[source]

Returns the 2x2 rotation matrix.

Computes and returns a 2x2 real rotation matrix parametrized by a single rotation angle theta.

Added in version 1.0.0.

Parameters:

sth (float) – Sine of the mixing angle \(\theta\).

Returns:

Rotation matrix [[cth, sth], [-sth, cth]], with cth = cos(theta) and sth = sin(theta).

Return type:

np.ndarray

Examples

import numpy as np

from magnus.hamiltonians import hamiltonians2nu

U = np.asarray(hamiltonians2nu.mixing_matrix_2nu(np.sqrt(0.308)))

print(np.round(U, 6))
print('unitary to %.1e' % np.max(np.abs(U.conj().T @ U - np.eye(2))))

[[ 0.831865  0.554977]
 [-0.554977  0.831865]]
unitary to 1.1e-16

Note the argument is \(\sin\theta\), not \(\sin^2\theta\); fits are usually quoted as the latter, hence the square root.

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_vacuum_energy_independent(sth: float, Dm2: float, compute_matrix_multiplication: bool | None = False) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for vacuum oscillations.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in vacuum, parametrized by a single mixing angle theta and a single mass-squared difference Dm2. The Hamiltonian is H = (1/2)*R.M2.R^dagger, with R the 2x2 rotation matrix and M2 the mass matrix. The multiplicative factor 1/E is not applied.

Added in version 1.0.0.

Parameters:

  • sth (float) – Sine of the mixing angle \(\theta\).

  • Dm2 (float) – Mass-squared difference \(\Delta m^2\).

  • compute_matrix_multiplication (bool, optional) – If False (default), use the pre-computed expressions; otherwise, multiply R.M2.R^dagger live.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

Examples

import numpy as np

import magnus.globaldefs as gd
from magnus.hamiltonians import hamiltonians2nu

h = np.asarray(hamiltonians2nu.hamiltonian_2nu_vacuum_energy_independent(
    np.sqrt(0.308), 7.49e-5))

print(np.round(h/1e-5, 6), ' [1e-5 eV^2]')
print('at 1 GeV, in eV: %.3e' % (h/(1.0*gd.UNIT_GEV))[0][0])

[[-0.71904   1.728941]
 [ 1.728941  0.71904 ]]  [1e-5 eV^2]
at 1 GeV, in eV: -7.190e-15

The energy is divided out separately, which is what makes this the piece to build once and reuse across a scan over energies.

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_vacuum_energy_independent_td(l: float, sth: float, Dm2: float, compute_matrix_multiplication: bool | None = False) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for vacuum oscillations, as a function of distance, even if it does not depend on it.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in vacuum, as a function of distance, parametrized by a single mixing angle theta and a single mass-squared difference Dm2. The Hamiltonian is H = (1/2)*R.M2.R^dagger, with R the 2x2 rotation matrix and M2 the mass matrix. The multiplicative factor 1/E is not applied. The vacuum Hamiltonian does not depend on distance in reality, but we include the dependence here as a way to validate the routine to compute probabilities for time-dependent Hamiltonians.

Added in version 1.0.0.

Parameters:

  • l (float) – Position at which the Hamiltonian is evaluated.

  • sth (float) – Sine of the mixing angle \(\theta\).

  • Dm2 (float) – Mass-squared difference \(\Delta m^2\).

  • compute_matrix_multiplication (bool, optional) – If False (default), use the pre-computed expressions; otherwise, multiply R.M2.R^dagger live.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_vacuum(energy: float, sth: float, Dm2: float, compute_matrix_multiplication: bool | None = False) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for vacuum oscillations.

Same as hamiltonian_2nu_vacuum_energy_independent(), but with the 1/E factor applied.

Added in version 1.0.0.

Parameters:

  • energy (float) – Neutrino energy.

  • sth (float) – Sine of the mixing angle \(\theta\).

  • Dm2 (float) – Mass-squared difference \(\Delta m^2\).

  • compute_matrix_multiplication (bool, optional) – If False (default), use the pre-computed expressions; otherwise, multiply R.M2.R^dagger live.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_vacuum_td(l: float, energy: float, sth: float, Dm2: float, compute_matrix_multiplication: bool | None = False) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for vacuum oscillations, as a function of distance, even if it does not depend on it.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in vacuum, as a function of distance, parametrized by a single mixing angle theta and a single mass-squared difference Dm2. The Hamiltonian is H = (1/2)*R.M2.R^dagger, with R the 2x2 rotation matrix and M2 the mass matrix. The multiplicative factor 1/E is not applied. The vacuum Hamiltonian does not depend on distance in reality, but we include the dependence here as a way to validate the routine to compute probabilities for time-dependent Hamiltonians.

Added in version 1.0.0.

Parameters:

  • l (float) – Position at which the Hamiltonian is evaluated.

  • energy (float) – Neutrino energy.

  • sth (float) – Sine of the mixing angle \(\theta\).

  • Dm2 (float) – Mass-squared difference \(\Delta m^2\).

  • compute_matrix_multiplication (bool, optional) – If False (default), use the pre-computed expressions; otherwise, multiply R.M2.R^dagger live.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_matter(VCC: float) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for matter oscillations.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in matter with constant density.

Added in version 1.0.0.

Parameters:

VCC (float) – Potential due to charged-current interactions of nu_e with electrons.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

Examples

import numpy as np

from magnus.hamiltonians import hamiltonians2nu

print(np.asarray(hamiltonians2nu.hamiltonian_2nu_matter(1.0e-13)))

[[1.e-13 0.e+00]
 [0.e+00 0.e+00]]

Only the electron-flavour entry is filled: matter is what the other flavours do not feel.

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_matter_td(l: float, VCC_func: Callable) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for matter oscillations, as a function of distance.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in matter with a given density as a function of position.

Added in version 1.0.0.

Parameters:

  • l (float) – Position at which the Hamiltonian is evaluated.

  • VCC_func (Callable) – Potential due to charged-current interactions of nu_e with electrons, as a function of position, l.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_nsi(VCC: float, eps_aa: float, eps_ab: complex) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for oscillations with NSI.

Computes and returns the 2x2 complex two-neutrino Hamiltonian for oscillations with non-standard interactions (NSI) in matter with constant density.

Two flavors admit only one physically meaningful diagonal NSI degree of freedom: an overall (flavor-universal) diagonal shift is proportional to the identity matrix, so it commutes with everything, contributes only an unobservable global phase to the evolution operator, and cannot affect any oscillation probability. eps_aa is therefore defined here as the non-universal (flavor-off-diagonal-difference) coupling, following the convention eps_mumu = 0, i.e., it parametrizes the coupling of \(\nu_e\) alone, relative to \(\nu_\mu\). [Earlier versions of this function placed eps_aa on both diagonal entries, making it a pure multiple of the identity and therefore a no-op on every oscillation probability – this was a bug, not a convention choice, confirmed by direct calculation.]

Added in version 1.0.0.

Parameters:

  • VCC (float) – Potential due to charged-current interactions of nu_e with electrons.

  • eps_aa (float) – Non-universal diagonal NSI coupling of nu_e (relative to nu_mu, whose diagonal coupling is fixed to 0 by this convention).

  • eps_ab (complex) – Flavor-off-diagonal (nu_e-nu_mu) NSI coupling.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_nsi_td(l: float, VCC_func: Callable, eps_aa: float, eps_ab: complex) → numpy.ndarray[source]

Returns the two-neutrino NSI Hamiltonian as a function of position.

Same as hamiltonian_2nu_nsi(), but evaluates the position-dependent matter potential VCC_func(l) first.

Added in version 1.0.0.

Parameters:

  • l (float) – Position at which the Hamiltonian is evaluated.

  • VCC_func (Callable) – Potential due to charged-current interactions of nu_e with electrons, as a function of position, l.

  • eps_aa (float) – Non-universal diagonal NSI coupling of nu_e; see hamiltonian_2nu_nsi().

  • eps_ab (complex) – Flavor-off-diagonal (nu_e-nu_mu) NSI coupling.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_liv(energy: float, sxi: float, b1: float, b2: float, Lambda: float, n_liv: int, nubar: bool | None = False) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for oscillations with LIV.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in a CPT-odd Lorentz invariance-violating background. Same as hamiltonian_2nu_liv_energy_independent(), but with the \(E^{n_{\rm liv}}\) energy dependence of the LIV operator applied.

Added in version 1.0.0.

Parameters:

  • energy (float) – Neutrino energy.

  • sxi (float) – Sin(xi), with xi the rotation angle between the space of the eigenvectors of B2 and the flavor states.

  • b1 (float) – Eigenvalue b1 of the LIV operator B2.

  • b2 (float) – Eigenvalue b2 of the LIV operator B2.

  • Lambda (float) – Energy scale of the LIV operator B2.

  • n_liv (int) – Power of the energy dependence of the LIV operator (dimension of the operator minus 3).

  • nubar (bool, optional) – Accepted for interface parity with magnus.hamiltonians.hamiltonians3nu.hamiltonian_3nu_liv() and its 4nu/5nu siblings, which conjugate their (complex) LIV mixing matrix for antineutrinos. The 2-flavor LIV rotation has no CP-violating phase (only the real angle sxi), so there is nothing to conjugate and this parameter currently has no effect. Default: False.

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray

magnus.hamiltonians.hamiltonians2nu.hamiltonian_2nu_liv_energy_independent(sxi: float, b1: float, b2: float, Lambda: float, n_liv: int) → numpy.ndarray[source]

Returns the two-neutrino Hamiltonian for oscillations with LIV.

Computes and returns the 2x2 real two-neutrino Hamiltonian for oscillations in a CPT-odd Lorentz invariance-violating background, without the energy-dependent prefactor.

Added in version 1.0.0.

Parameters:

  • sxi (float) – Sin(xi), with xi the rotation angle between the space of the eigenvectors of B2 and the flavor states.

  • b1 (float) – Eigenvalue b1 of the LIV operator B2.

  • b2 (float) – Eigenvalue b2 of the LIV operator B2.

  • Lambda (float) – Energy scale of the LIV operator B2.

  • n_liv (int) – Power of the energy dependence of the LIV operator (dimension of the operator minus 3).

Returns:

Hamiltonian 2x2 matrix.

Return type:

np.ndarray