ekore.anomalous_dimensions.unpolarized.time_like package

The unpolarized, time-like Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.time_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.unpolarized.time_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.unpolarized.time_like.as1 module

The unpolarized, time-like LO Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.time_like.as1.gamma_qq(N, cache)[source]

Compute the LO quark-quark anomalous dimension.

Implements B.3 from [MM06].

Parameters:

N (complex) – Mellin moment
cache (numpy.ndarray) – Harmonic sum cache

Returns:

LO quark-quark anomalous dimension \(\gamma_{qq}^{(0)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as1.gamma_qg(N)[source]

Compute the LO quark-gluon anomalous dimension.

Implements B.4 from [MM06] and A1 from [GRV93].

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

ekore.anomalous_dimensions.unpolarized.time_like.as1.gamma_gq(N, nf)[source]

Compute the LO gluon-quark anomalous dimension.

Implements B.5 from [MM06] and A1 from [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as1.gamma_gg(N, nf, cache)[source]

Compute the LO gluon-gluon anomalous dimension.

Implements B.6 from [MM06].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as1.gamma_ns(N, cache)[source]

Compute the LO non-singlet anomalous dimension.

At LO, \(\gamma_{ns}^{(0)} = \gamma_{qq}^{(0)}\).

Parameters:

N (complex) – Mellin moment
cache (numpy.ndarray) – Harmonic sum cache

Returns:

LO quark-quark anomalous dimension \(\gamma_{ns}^{(0)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as1.gamma_singlet(N, nf, cache)[source]

Compute the LO singlet anomalous dimension matrix.

Implements 2.13 from [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.time_like.as2 module

The unpolarized, time-like NLO Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_nsp(N, nf, cache)[source]

Compute the NLO non-singlet positive anomalous dimension.

Implements A6 from [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO non-singlet positive anomalous dimension \(\gamma_{ns}^{(1)+}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_nsm(N, nf, cache)[source]

Compute the NLO non-singlet negative anomalous dimension.

Based on [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO non-singlet negative anomalous dimension \(\gamma_{ns}^{(1)-}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_qqs(N, nf)[source]

Compute the NLO quark-quark singlet anomalous dimension.

Implements B.9 from [MM06].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors

Returns:

NLO quark-quark singlet anomalous dimension \(\gamma_{qq}^{(1)s}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_qg(N, nf, cache)[source]

Compute the NLO quark-gluon anomalous dimension.

Implements B.10 from [MM06] and A1 from [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_gq(N, nf, cache)[source]

Compute the NLO gluon-quark anomalous dimension.

Implements B.11 from [MM06] and A1 from [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_gg(N, nf, cache)[source]

Compute the NLO gluon-gluon anomalous dimension.

Implements B.12 from [MM06].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

NLO gluon-gluon anomalous dimension \(\gamma_{gg}^{(1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as2.gamma_singlet(N, nf, cache)[source]

Compute the NLO singlet anomalous dimension matrix.

Implements 2.13 from [GRV93].

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.time_like.as3 module

The unpolarized, time-like NNLO Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.time_like.as3.gamma_nsp(N, nf, cache)[source]

Compute the NNLO non-singlet positive anomalous dimension.

Implements 15 from [MMV06] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
ache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as3.gamma_nsm(N, nf, cache)[source]

Compute the NNLO non-singlet negative anomalous dimension.

Implements 16 from [MMV06] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as3.gamma_nsv(N, nf, cache)[source]

Compute the NNLO non-singlet valence anomalous dimension.

Implements 16 from [MMV06] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as3.gamma_qq(N, nf, cache)[source]

Compute the NNLO quark-quark anomalous dimension.

Implements 11 from [MV08] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.time_like.as3.gamma_qg(N, nf, cache)[source]

Compute the NNLO quark-gluon anomalous dimension.

Implements 18 from [AMV12] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. 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.unpolarized.time_like.as3.gamma_gq(N, nf, cache)[source]

Compute the NNLO gluon-quark anomalous dimension.

Implements 19 from [AMV12] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. 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.unpolarized.time_like.as3.gamma_gg(N, nf, cache)[source]

Compute the NNLO gluon-gluon anomalous dimension.

Implements 12 from [MV08] via the N-space translation from A. Vogt.

Parameters:

N (complex) – Mellin moment
nf (int) – No. 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.unpolarized.time_like.as3.gamma_singlet(N, nf, cache)[source]

Compute the NNLO singlet anomalous dimension matrix.

Parameters:

N (complex) – Mellin moment
nf (int) – No. of active flavors
cache (numpy.ndarray) – Harmonic sum cache

Returns:

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

Return type:

numpy.ndarray