ekore.anomalous_dimensions.polarized.space_like package

The polarized, space-like Altarelli-Parisi splitting kernels.

Normalization is given by

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

with \(a_s = \frac{\alpha_S(\mu^2)}{4\pi}\).

ekore.anomalous_dimensions.polarized.space_like.gamma_ns(order, mode, n, nf)[source]

Compute the tower of the non-singlet anomalous dimensions.

Parameters:

order (tuple(int,int)) – perturbative orders
mode (10201 | 10101 | 10200) – sector identifier
n (complex) – Mellin variable
nf (int) – Number of active flavors

Returns:

non-singlet anomalous dimensions

Return type:

numpy.ndarray

ekore.anomalous_dimensions.polarized.space_like.gamma_singlet(order, n, nf)[source]

Compute the tower of the singlet anomalous dimensions matrices.

Parameters:

order (tuple(int,int)) – perturbative orders
n (complex) – Mellin variable
nf (int) – Number of active flavors

Returns:

singlet anomalous dimensions matrices

Return type:

numpy.ndarray

Submodules

ekore.anomalous_dimensions.polarized.space_like.as1 module

The LO Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.polarized.space_like.as1.gamma_qg(N, nf)[source]

Compute the LO polarized quark-gluon anomalous dimension.

Implements A.1 from [GRSV96].

Parameters:

N (complex) – Mellin moment
nf (int) – Number of active flavors

Returns:

LO polarized quark-gluon anomalous dimension \(\gamma_{qg}^{(0)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as1.gamma_gq(N)[source]

Compute the LO polarized gluon-quark anomalous dimension.

Implements A.1 from [GRSV96].

Parameters:: N (complex) – Mellin moment
Returns:: LO gluon-quark anomalous dimension \(\gamma_{gq}^{(0)}(N)\)
Return type:: complex

ekore.anomalous_dimensions.polarized.space_like.as1.gamma_gg(N, cache, nf)[source]

Compute the LO polarized gluon-gluon anomalous dimension.

Implements A.1 from [GRSV96].

Parameters:

N (complex) – Mellin moment
cache (numpy.ndarray) – Harmonic sum cache
nf (int) – Number of active flavors

Returns:

LO gluon-gluon anomalous dimension \(\gamma_{gg}^{(0)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as1.gamma_singlet(N, cache, nf)[source]

Compute the LO polarized singlet anomalous dimension matrix.

\[\begin{split}\gamma_S^{(0)} = \left(\begin{array}{cc} \gamma_{qq}^{(0)} & \gamma_{qg}^{(0)}\\ \gamma_{gq}^{(0)} & \gamma_{gg}^{(0)} \end{array}\right)\end{split}\]

Parameters:

N (complex) – Mellin moment
cache (numpy.ndarray) – Harmonic sum cache
nf (int) – Number of active flavors

Returns:

LO singlet anomalous dimension matrix \(\gamma_{S}^{(0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.polarized.space_like.as2 module

The NLO Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.polarized.space_like.as2.gamma_ps(n, nf)[source]

Compute the NLO polarized pure-singlet quark-quark anomalous dimension [GRSV96] (eq A.3).

Parameters:

n (complex) – Mellin moment
nf (int) – number of active flavors

Returns:

NLO pure-singlet quark-quark anomalous dimension \(\\gamma_{ps}^{(1)}(n)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as2.gamma_qg(n, nf, cache)[source]

Compute the NLO polarized quark-gluon singlet anomalous dimension [GRSV96] (eq A.4).

Parameters:

n (complex) – Mellin moment
nf (int) – number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO quark-gluon singlet anomalous dimension \(\\gamma_{qg}^{(1)}(n)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as2.gamma_gq(n, nf, cache)[source]

Compute the NLO polarized gluon-quark singlet anomalous dimension [GRSV96] (eq A.5).

Parameters:

n (complex) – Mellin moment
nf (int) – number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO gluon-quark singlet anomalous dimension \(\\gamma_{gq}^{(1)}(n)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as2.gamma_gg(n, nf, cache)[source]

Compute the NLO polarized gluon-gluon singlet anomalous dimension [GRSV96] (eq A.6).

Parameters:

n (complex) – Mellin moment
nf (int) – number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO gluon-quark singlet anomalous dimension \(\\gamma_{gq}^{(1)}(n)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as2.gamma_singlet(n, nf, cache)[source]

Compute the NLO polarized singlet anomalous dimension matrix.

\[\begin{split}\gamma_S^{(1)} = \left(\begin{array}{cc} \gamma_{qq}^{(1)} & \gamma_{qg}^{(1)}\\ \gamma_{gq}^{(1)} & \gamma_{gg}^{(1)} \end{array}\right)\end{split}\]

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO singlet anomalous dimension matrix \(\gamma_{S}^{(1)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.polarized.space_like.as3 module

The NNLO polarized Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_gg(N, nf, cache)[source]

Compute the parametrized NNLO gluon-gluon polarized anomalous dimension.

Implement Eq. (4.12) of [MVV14].

Parameters:

N (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO gluon-gluon anomalous dimension \(\\gamma_{gg}^{(2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_qg(N, nf, cache)[source]

Compute the parametrized NNLO quark-gluon polarized anomalous dimension.

Implement Eq. (4.10) of [MVV14].

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO quark-gluon anomalous dimension \(\\gamma_{qg}^{(2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_gq(N, nf, cache)[source]

Compute the parametrized NNLO gluon-quark polarized anomalous dimension.

Implement Eq. (4.11) of [MVV14].

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO gluon-quark anomalous dimension \(\\gamma_{gq}^{(2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_ps(N, nf, cache)[source]

Compute the parametrized NNLO pure-singlet quark-quark polarized anomalous dimension.

Implement Eq. (4.9) of [MVV14].

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO pure-singlet quark-quark anomalous dimension \(\\gamma_{ps}^{(2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_nss(N, nf, cache)[source]

Compute the NNLO sea-like polarized non-singlet anomalous dimension.

Implement Eq. (24) of [MVV15].

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO valence non-singlet anomalous dimension \(\\gamma_{ns,s}^{(2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_nsv(N, nf, cache)[source]

Compute the NNLO valence polarized non-singlet anomalous dimension.

Implement Eq. (23) of [MVV15].

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO valence non-singlet anomalous dimension \(\\gamma_{ns,v}^{(2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.polarized.space_like.as3.gamma_singlet(N, nf, cache)[source]

Compute the NNLO polarized singlet anomalous dimension matrix.

\[\begin{split}\gamma_S^{(1)} = \left(\begin{array}{cc} \gamma_{qq}^{(2)} & \gamma_{qg}^{(2)}\\ \gamma_{gq}^{(2)} & \gamma_{gg}^{(2)} \end{array}\right)\end{split}\]

Parameters:

n (complex) – Mellin moment
nf (int) – Number of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NNLO singlet anomalous dimension matrix \(\gamma_{S}^{(2)}(N)\)

Return type:

numpy.ndarray