eko.scale_variations package

Define possible scale variations schemes.

A Mathematica snippet to check the formulas is available in the extras folder.

class eko.scale_variations.Modes(value)

Bases: IntEnum

Enumerate scale Variation modes.

unvaried = 1

exponentiated = 2

expanded = 3

eko.scale_variations.sv_mode(s)

Return the scale variation mode.

Parameters:

s (str) – string representation

Returns:

enum representation

Return type:

enum.IntEnum

class eko.scale_variations.ModeMixin

Bases: object

Mixin to cast scale variation mode.

property sv_mode

Return the scale variation mode.

Submodules

eko.scale_variations.expanded module

Scale variation operator for the expanded scheme (ModSV=expanded).

The expressions can be obtained using Eqs. (3.33) and (3.38) of [AK+19]. Be aware that corresponding the signs of the ingredients there are a number of differences. However, the ultimate sign can be obtained by comparing to the exponentiated scheme in the trunctated solution.

eko.scale_variations.expanded.variation_as1(gamma, L)

Compute the NLO anomalous dimension variation.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

variation at NLO

Return type:

complex

eko.scale_variations.expanded.variation_as2(gamma, L, beta0, g0e2)

Compute the NNLO anomalous dimension variation.

These kernels are meant to be used with alpha_s evaluated at the factorization scale. If one expresses everything in terms of alpha_s evaluated at the process scale, the sign of beta*gamma0 flips.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• L (float) – logarithmic ratio of factorization and renormalization scale

• beta0 (float) – \(\beta_0\)

• g0e2 (complex or numpy.ndarray) – \(\left(\gamma^{(0)}\right)^2\)

Returns:

variation at NNLO

Return type:

complex

eko.scale_variations.expanded.variation_as3(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1)

Compute the N3LO anomalous dimension variation.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• L (float) – logarithmic ratio of factorization and renormalization scale

• beta0 (float) – \(\beta_0\)

• beta0 – \(\beta_1\)

• g0e2 (complex or numpy.ndarray) – \(\left(\gamma^{(0)}\right)^2\)

• g0e3 (complex or numpy.ndarray) – \(\left(\gamma^{(0)}\right)^3\)

• g1g0 (complex or numpy.ndarray) – \(\gamma^{(1)} \gamma^{(0)}\)

• g0g1 (complex or numpy.ndarray) – \(\gamma^{(0)} \gamma^{(1)}\)

Returns:

variation at N3LO

Return type:

complex

eko.scale_variations.expanded.non_singlet_variation(gamma, a_s, order, nf, L)

Non-singlet scale variation dispatcher.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• a_s (float) – target coupling value

• order (int) – perturbation order

• nf (int) – number of active flavors

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

scale variation kernel

Return type:

complex

eko.scale_variations.expanded.singlet_variation(gamma, a_s, order, nf, L, dim)

Singlet scale variation dispatcher.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• a_s (float) – target coupling value

• order (int) – perturbation order

• nf (int) – number of active flavors

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

scale variation kernel

Return type:

numpy.ndarray

eko.scale_variations.expanded.non_singlet_variation_qed(gamma, a_s, a_em, alphaem_running, order, nf, L)

Non-singlet scale variation dispatcher.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• a_s (float) – target coupling value

• order (int) – perturbation order

• nf (int) – number of active flavors

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

scale variation kernel

Return type:

complex

eko.scale_variations.expanded.singlet_variation_qed(gamma, a_s, a_em, alphaem_running, order, nf, L)

Singlet scale variation dispatcher.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• a_s (float) – target coupling value

• order (int) – perturbation order

• nf (int) – number of active flavors

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

scale variation kernel

Return type:

numpy.ndarray

eko.scale_variations.expanded.valence_variation_qed(gamma, a_s, a_em, alphaem_running, order, nf, L)

Singlet scale variation dispatcher.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• a_s (float) – target coupling value

• order (int) – perturbation order

• nf (int) – number of active flavors

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

scale variation kernel

Return type:

numpy.ndarray

eko.scale_variations.exponentiated module

Contains the scale variation for ModSV=exponentiated.

eko.scale_variations.exponentiated.gamma_variation(gamma, order, nf, L)

Adjust the anomalous dimensions with the scale variations.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• order (tuple(int,int)) – perturbation order

• nf (int) – number of active flavors

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

adjusted anomalous dimensions

Return type:

numpy.ndarray

eko.scale_variations.exponentiated.gamma_variation_qed(gamma, order, nf, nl, L, alphaem_running)

Adjust the anomalous dimensions with the scale variations.

Parameters:

• gamma (numpy.ndarray) – anomalous dimensions

• order (tuple(int,int)) – perturbation order

• nf (int) – number of active flavors

• nl (int) – number of leptons partecipating to alphaem running

• L (float) – logarithmic ratio of factorization and renormalization scale

Returns:

gamma – adjusted anomalous dimensions

Return type:

numpy.ndarray