ekore.anomalous_dimensions package

The Altarelli-Parisi splitting kernels.

Normalization is given by

\[\mathbf{P}(x) = \sum\limits_{j=0} a_s^{j+1} \mathbf P^{(j)}(x)\]

with \(a_s = \frac{\alpha_S(\mu^2)}{4\pi}\). The 3-loop references for the non-singlet [MVV04] and singlet [VMV04] case contain also the lower order results. The results are also determined in Mellin space in terms of the anomalous dimensions (note the additional sign!)

\[\gamma(N) = - \mathcal{M}[\mathbf{P}(x)](N)\]
ekore.anomalous_dimensions.exp_matrix_2D(gamma_S)[source]

Compute the exponential and the eigensystem of the singlet anomalous dimension matrix.

Parameters:

gamma_S (numpy.ndarray) – singlet anomalous dimension matrix

Returns:

  • exp (numpy.ndarray) – exponential of the singlet anomalous dimension matrix \(\gamma_{S}(N)\)

  • lambda_p (complex) – positive eigenvalue of the singlet anomalous dimension matrix \(\gamma_{S}(N)\)

  • lambda_m (complex) – negative eigenvalue of the singlet anomalous dimension matrix \(\gamma_{S}(N)\)

  • e_p (numpy.ndarray) – projector for the positive eigenvalue of the singlet anomalous dimension matrix \(\gamma_{S}(N)\)

  • e_m (numpy.ndarray) – projector for the negative eigenvalue of the singlet anomalous dimension matrix \(\gamma_{S}(N)\)

ekore.anomalous_dimensions.exp_matrix(gamma)[source]

Compute the exponential and the eigensystem of a matrix.

Parameters:

gamma (numpy.ndarray) – input matrix

Returns:

  • exp (numpy.ndarray) – exponential of the matrix gamma \(\gamma(N)\)

  • w (numpy.ndarray) – array of the eigenvalues of the matrix lambda

  • e (numpy.ndarray) – projectors on the eigenspaces of the matrix gamma \(\gamma(N)\)

Subpackages