ekore.anomalous_dimensions.unpolarized.space_like package

The unpolarized, 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 \(a_s = \frac{\alpha_S(\mu^2)}{4\pi}\). 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.unpolarized.space_like.gamma_ns(order, mode, n, nf, n3lo_ad_variation, use_fhmruvv=False)[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

  • n3lo_ad_variation (tuple) – N3LO anomalous dimension variation (gg, gq, qg, qq, nsp, nsm, nsv)

  • use_fhmruvv (bool) – if True use the FHMRUVV N3LO anomalous dimensions

Returns:

non-singlet anomalous dimensions

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.gamma_singlet(order, n, nf, n3lo_ad_variation, use_fhmruvv=False)[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

  • n3lo_ad_variation (tuple) – N3LO anomalous dimension variation (gg, gq, qg, qq, nsp, nsm, nsv)

  • use_fhmruvv (bool) – if True use the FHMRUVV N3LO anomalous dimensions

Returns:

singlet anomalous dimensions matrices

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.gamma_ns_qed(order, mode, n, nf, n3lo_ad_variation, use_fhmruvv=False)[source]

Compute the grid of the QED non-singlet anomalous dimensions.

Parameters:

  • order (tuple(int,int)) – perturbative orders

  • mode (10102 | 10103 | 10202 | 10203) – sector identifier

  • n (complex) – Mellin variable

  • nf (int) – Number of active flavors

  • n3lo_ad_variation (tuple) – N3LO anomalous dimension variation (gg, gq, qg, qq, nsp, nsm, nsv)

  • use_fhmruvv (bool) – if True use the FHMRUVV N3LO anomalous dimensions

Returns:

gamma_ns – non-singlet QED anomalous dimensions

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.choose_ns_ad_aem1(mode, n, cache)[source]

Select the non-singlet anomalous dimension at O(aem1) with the correct charge factor.

Parameters:

  • mode (10102 | 10202 | 10103 | 10203) – sector identifier

  • n (complex) – Mellin variable

  • cache (numpy.ndarray) – Harmonic sum cache

Returns:

non-singlet anomalous dimensions

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.choose_ns_ad_as1aem1(mode, n, cache)[source]

Select the non-singlet anomalous dimension at O(as1aem1) with the correct charge factor.

Parameters:

  • mode (10102 | 10202 | 10103 | 10203) – sector identifier

  • n (complex) – Mellin variable

  • cache (numpy.ndarray) – Harmonic sum cache

Returns:

non-singlet anomalous dimensions

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.choose_ns_ad_aem2(mode, n, nf, cache)[source]

Select the non-singlet anomalous dimension at O(aem2) with the correct charge factor.

Parameters:

  • mode (10102 | 10202 | 10103 | 10203) – sector identifier

  • n (complex) – Mellin variable

  • nf (int) – Number of active flavors

  • cache (numpy.ndarray) – Harmonic sum cache

Returns:

non-singlet anomalous dimensions

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.gamma_singlet_qed(order, n, nf, n3lo_ad_variation, use_fhmruvv=False)[source]

Compute the grid of the QED singlet anomalous dimensions matrices.

Parameters:

  • order (tuple(int,int)) – perturbative orders

  • n (complex) – Mellin variable

  • nf (int) – Number of active flavors

  • n3lo_ad_variation (tuple) – N3LO anomalous dimension variation (gg, gq, qg, qq, nsp, nsm, nsv)

  • use_fhmruvv (bool) – if True use the FHMRUVV N3LO anomalous dimensions

Returns:

singlet anomalous dimensions matrices

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.gamma_valence_qed(order, n, nf, n3lo_ad_variation, use_fhmruvv=False)[source]

Compute the grid of the QED valence anomalous dimensions matrices.

Parameters:

  • order (tuple(int,int)) – perturbative orders

  • n (complex) – Mellin variable

  • nf (int) – Number of active flavors

  • n3lo_ad_variation (tuple) – N3LO anomalous dimension variation (gg, gq, qg, qq, nsp, nsm, nsv)

  • use_fhmruvv (bool) – if True use the FHMRUVV N3LO anomalous dimensions

Returns:

valence anomalous dimensions matrices

Return type:

numpy.ndarray

Subpackages

Submodules

ekore.anomalous_dimensions.unpolarized.space_like.aem1 module

The \(O(a_{em}^1)\) Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.space_like.aem1.gamma_phq(N)[source]

Compute the leading-order photon-quark anomalous dimension.

Implements Eq. (2.5) of [Car15].

Parameters:

N (complex) – Mellin moment

Returns:

Leading-order photon-quark anomalous dimension \(\\gamma_{\\gamma q}^{(0,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem1.gamma_qph(N, nf)[source]

Compute the leading-order quark-photon anomalous dimension.

Implements Eq. (2.5) of [Car15]. But adding the \(N_C\) and the \(2n_f\) factors from \(\\theta\) inside the definition of \(\\gamma_{q \\gamma}^{(0)}(N)\).

Parameters:

  • N (complex) – Mellin moment

  • nf (int) – Number of active flavors

Returns:

Leading-order quark-photon anomalous dimension \(\\gamma_{q \\gamma}^{(0,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem1.gamma_phph(nf)[source]

Compute the leading-order photon-photon anomalous dimension.

Implements Eq. (2.5) of [Car15].

Parameters:

nf (int) – Number of active flavors

Returns:

Leading-order phton-photon anomalous dimension \(\\gamma_{\\gamma \\gamma}^{(0,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem1.gamma_ns(N, cache)[source]

Compute the leading-order non-singlet QED anomalous dimension.

Implements Eq. (2.5) of [Car15].

Parameters:
Returns:

Leading-order non-singlet QED anomalous dimension \(\\gamma_{ns}^{(0,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem1.gamma_singlet(N, nf, cache)[source]

Compute the QED leading-order singlet anomalous dimension matrix.

\[\begin{split}\\gamma_S^{(0)} = \\left(\begin{array}{cc} 0 & 0 & 0 & 0 \\ 0 & \\gamma_{\\gamma \\gamma}^{(0,1)} & \\langle e^2 \rangle \\gamma_{\\gamma q}^{(0,1)} & \nu_u e^2_- \\gamma_{\\gamma q}^{(0,1)}\\ 0 & \\langle e^2 \rangle\\gamma_{q \\gamma}^{(0,1)} & \\langle e^2 \rangle \\gamma_{ns}^{(0,1)} & \nu_u e^2_- \\gamma_{ns}^{(0,1)}\\ 0 & \nu_d e^2_- \\gamma_{q \\gamma}^{(0,1)} & \nu_d e^2_- \\gamma_{ns}^{(0,1)} & e^2_\\Delta \\gamma_{ns}^{(0,1)} \\end{array}\right)\end{split}\]
Parameters:
Returns:

Leading-order singlet anomalous dimension matrix \(\\gamma_{S}^{(0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.aem1.gamma_valence(N, nf, cache)[source]

Compute the QED leading-order valence anomalous dimension matrix.

\[\begin{split}\\gamma_V^{(0,1)} = \\left(\begin{array}{cc} \\langle e^2 \rangle \\gamma_{ns}^{(0,1)} & \nu_u e^2_- \\gamma_{ns}^{(0,1)}\\ \nu_d e^2_- \\gamma_{ns}^{(0,1)} & e^2_\\Delta \\gamma_{ns}^{(0,1)} \\end{array}\right)\end{split}\]
Parameters:
Returns:

Leading-order singlet anomalous dimension matrix \(\\gamma_{S}^{(0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.aem2 module

The \(O(a_{em}^2)\) Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_phph(N, nf)[source]

Compute the \(O(a_{em}^2)\) photon-photon singlet anomalous dimension.

Implements Eq. (68) of [dFSR16b].

Parameters:

  • N (complex) – Mellin moment

  • nf (int) – Number of active flavors

Returns:

\(O(a_{em}^2)\) photon-photon singlet anomalous dimension \(\\gamma_{\\gamma \\gamma}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_uph(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) quark-photon anomalous dimension for up quarks.

Implements Eq. (55) of [dFSR16b] for q=u.

Parameters:
Returns:

\(O(a_{em}^2)\) quark-photon anomalous dimension \(\\gamma_{u \\gamma}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_dph(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) quark-photon anomalous dimension for down quarks.

Implements Eq. (55) of [dFSR16b] for q=d.

Parameters:
Returns:

\(O(a_{em}^2)\) quark-photon anomalous dimension \(\\gamma_{d \\gamma}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_phu(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) photon-quark anomalous dimension for up quarks.

Implements Eq. (56) of [dFSR16b] for q=u.

Parameters:
Returns:

\(O(a_{em}^2)\) photon-quark anomalous dimension \(\\gamma_{\\gamma u}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_phd(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) photon-quark anomalous dimension for down quarks.

Implements Eq. (56) of [dFSR16b] for q=d.

Parameters:
Returns:

\(O(a_{em}^2)\) photon-quark anomalous dimension \(\\gamma_{\\gamma d}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_nspu(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) singlet-like non-singlet anomalous dimension for up quarks.

Implements sum of Eqs. (57-58) of [dFSR16b] for q=u.

Parameters:
Returns:

\(O(a_{em}^2)\) singlet-like non-singlet anomalous dimension \(\\gamma_{ns,+,u}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_nspd(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) singlet-like non-singlet anomalous dimension for down quarks.

Implements sum of Eqs. (57-58) of [dFSR16b] for q=d.

Parameters:
Returns:

\(O(a_{em}^2)\) singlet-like non-singlet anomalous dimension \(\\gamma_{ns,+,d}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_nsmu(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) valence-like non-singlet anomalous dimension for up quarks.

Implements difference between Eqs. (57-58) of [dFSR16b] for q=u.

Parameters:
Returns:

\(O(a_{em}^2)\) valence-like non-singlet anomalous dimension \(\\gamma_{ns,-,u}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_nsmd(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) valence-like non-singlet anomalous dimension for down quarks.

Implements difference between Eqs. (57-58) of [dFSR16b] for q=d.

Parameters:
Returns:

\(O(a_{em}^2)\) valence-like non-singlet anomalous dimension \(\\gamma_{ns,-,d}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_ps(N, nf)[source]

Compute the \(O(a_{em}^2)\) pure-singlet quark-quark anomalous dimension.

Implements Eq. (59) of [dFSR16b].

Parameters:

  • N (complex) – Mellin moment

  • nf (int) – Number of active flavors

Returns:

\(O(a_{em}^2)\) pure-singlet quark-quark anomalous dimension \(\\gamma_{ps}^{(0,2)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_singlet(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) singlet sector.

Parameters:
Returns:

\(O(a_{em}^2)\) singlet anomalous dimension \(\\gamma_{S}^{(0,2)}\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.aem2.gamma_valence(N, nf, cache)[source]

Compute the \(O(a_{em}^2)\) valence sector.

Parameters:
Returns:

\(O(a_{em}^2)\) valence anomalous dimension \(\\gamma_{V}^{(0,2)}\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as1 module

Compute the leading-order Altarelli-Parisi splitting kernels.

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

Compute the leading-order non-singlet anomalous dimension.

Implements Eq. (3.4) of [MVV04].

Parameters:
Returns:

gamma_ns – Leading-order non-singlet anomalous dimension \(\\gamma_{ns}^{(0)}(N)\)

Return type:

complex

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

Compute the leading-order quark-gluon anomalous dimension.

Implements Eq. (3.5) of [VMV04].

Parameters:

  • N (complex) – Mellin moment

  • nf (int) – Number of active flavors

Returns:

gamma_qg – Leading-order quark-gluon anomalous dimension \(\\gamma_{qg}^{(0)}(N)\)

Return type:

complex

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

Compute the leading-order gluon-quark anomalous dimension.

Implements Eq. (3.5) of [VMV04].

Parameters:

N (complex) – Mellin moment

Returns:

gamma_gq – Leading-order gluon-quark anomalous dimension \(\\gamma_{gq}^{(0)}(N)\)

Return type:

complex

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

Compute the leading-order gluon-gluon anomalous dimension.

Implements Eq. (3.5) of [VMV04].

Parameters:
Returns:

gamma_gg – Leading-order gluon-gluon anomalous dimension \(\\gamma_{gg}^{(0)}(N)\)

Return type:

complex

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

Compute the leading-order 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:
Returns:

gamma_S_0 – Leading-order singlet anomalous dimension matrix \(\\gamma_{S}^{(0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as1.gamma_singlet_qed(N, cache, nf)[source]

Compute the leading-order singlet anomalous dimension matrix for the unified evolution basis.

\[\begin{split}\\gamma_S^{(1,0)} = \\left(\begin{array}{cccc} \\gamma_{gg}^{(1,0)} & 0 & \\gamma_{gq}^{(1,0)} & 0\\ 0 & 0 & 0 & 0 \\ \\gamma_{qg}^{(1,0)} & 0 & \\gamma_{qq}^{(1,0)} & 0 \\ 0 & 0 & 0 & \\gamma_{qq}^{(1,0)} \\ \\end{array}\right)\end{split}\]
Parameters:
Returns:

gamma_S – Leading-order singlet anomalous dimension matrix \(\\gamma_{S}^{(1,0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as1.gamma_valence_qed(N, cache)[source]

Compute the leading-order valence anomalous dimension matrix for the unified evolution basis.

\[\begin{split}\\gamma_V^{(1,0)} = \\left(\begin{array}{cc} \\gamma_{ns}^{(1,0)} & 0\\ 0 & \\gamma_{ns}^{(1,0)} \\end{array}\right)\end{split}\]
Parameters:
Returns:

gamma_V – Leading-order singlet anomalous dimension matrix \(\\gamma_{V}^{(1,0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1 module

The \(O(a_s^1a_{em}^1)\) Altarelli-Parisi splitting kernels.

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_phq(N, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) photon-quark anomalous dimension.

Implements Eq. (36) of [dFSR16a].

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) photon-quark anomalous dimension \(\\gamma_{\\gamma q}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_qph(N, nf, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) quark-photon anomalous dimension.

Implements Eq. (26) of [dFSR16a].

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) quark-photon anomalous dimension \(\\gamma_{q \\gamma}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_gph(N)[source]

Compute the \(O(a_s^1a_{em}^1)\) gluon-photon anomalous dimension.

Implements Eq. (27) of [dFSR16a].

Parameters:

N (complex) – Mellin moment

Returns:

\(O(a_s^1a_{em}^1)\) gluon-photon anomalous dimension \(\\gamma_{g \\gamma}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_phg(N)[source]

Compute the \(O(a_s^1a_{em}^1)\) photon-gluon anomalous dimension.

Implements Eq. (30) of [dFSR16a].

Parameters:

N (complex) – Mellin moment

Returns:

\(O(a_s^1a_{em}^1)\) photon-gluon anomalous dimension \(\\gamma_{\\gamma g}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_qg(N, nf, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) quark-gluon singlet anomalous dimension.

Implements Eq. (29) of [dFSR16a].

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) quark-gluon singlet anomalous dimension \(\\gamma_{qg}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_gq(N, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) gluon-quark singlet anomalous dimension.

Implements Eq. (35) of [dFSR16a].

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) gluon-quark singlet anomalous dimension \(\\gamma_{gq}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_phph(nf)[source]

Compute the \(O(a_s^1a_{em}^1)\) photon-photon singlet anomalous dimension.

Implements Eq. (28) of [dFSR16a].

Parameters:

nf (int) – Number of active flavors

Returns:

gamma_gg – \(O(a_s^1a_{em}^1)\) photon-photon singlet anomalous dimension \(\\gamma_{\\gamma \\gamma}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_gg()[source]

Compute the \(O(a_s^1a_{em}^1)\) gluon-gluon singlet anomalous dimension.

Implements Eq. (31) of [dFSR16a].

Returns:

\(O(a_s^1a_{em}^1)\) gluon-gluon singlet anomalous dimension \(\\gamma_{gg}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_nsp(N, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) singlet-like non-singlet anomalous dimension.

Implements sum of Eqs. (33-34) of [dFSR16a].

Parameters:
Returns:

gamma_nsp – \(O(a_s^1a_{em}^1)\) singlet-like non-singlet anomalous dimension \(\\gamma_{ns,+}^{(1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_nsm(N, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) valence-like non-singlet anomalous dimension.

Implements difference between Eqs. (33-34) of [dFSR16a].

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) singlet-like non-singlet anomalous dimension \(\\gamma_{ns,-}^{(1,1)}(N)\)

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_singlet(N, nf, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) singlet sector.

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) singlet anomalous dimension \(\\gamma_{S}^{(1,1)}(N,nf,cache)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as1aem1.gamma_valence(N, nf, cache)[source]

Compute the \(O(a_s^1a_{em}^1)\) valence sector.

Parameters:
Returns:

\(O(a_s^1a_{em}^1)\) valence anomalous dimension \(\\gamma_{V}^{(1,1)}(N,nf,cache)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as2 module

Compute the NLO Altarelli-Parisi splitting kernels.

These expression have been obtained using the procedure described in the wiki involving FormGet [HPS16].

ekore.anomalous_dimensions.unpolarized.space_like.as2.gamma_nsm(n, nf, cache)[source]

Compute the NLO valence-like non-singlet anomalous dimension.

Implements Eq. (3.5) of [MVV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as2.gamma_nsp(n, nf, cache)[source]

Compute the NLO singlet-like non-singlet anomalous dimension.

Implements Eq. (3.5) of [MVV04].

Parameters:
Returns:

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

Return type:

complex

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

Compute the NLO pure-singlet quark-quark anomalous dimension.

Implements Eq. (3.6) of [VMV04].

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.unpolarized.space_like.as2.gamma_qg(n, nf, cache)[source]

Compute the NLO quark-gluon singlet anomalous dimension.

Implements Eq. (3.7) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

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

Compute the NLO gluon-quark singlet anomalous dimension.

Implements Eq. (3.8) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

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

Compute the NLO gluon-gluon singlet anomalous dimension.

Implements Eq. (3.9) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

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

Compute the next-leading-order 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:
Returns:

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

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as2.gamma_singlet_qed(N, nf, cache)[source]

Compute the leading-order singlet anomalous dimension matrix for the unified evolution basis.

\[\begin{split}\\gamma_S^{(2,0)} = \\left(\begin{array}{cccc} \\gamma_{gg}^{(2,0)} & 0 & \\gamma_{gq}^{(2,0)} & 0\\ 0 & 0 & 0 & 0 \\ \\gamma_{qg}^{(2,0)} & 0 & \\gamma_{qq}^{(2,0)} & 0 \\ 0 & 0 & 0 & \\gamma_{qq}^{(2,0)} \\ \\end{array}\right)\end{split}\]
Parameters:
Returns:

Leading-order singlet anomalous dimension matrix \(\\gamma_{S}^{(2,0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as2.gamma_valence_qed(N, nf, cache)[source]

Compute the leading-order valence anomalous dimension matrix for the unified evolution basis.

\[\begin{split}\\gamma_V^{(2,0)} = \\left(\begin{array}{cc} \\gamma_{ns-}^{(2,0)} & 0\\ 0 & \\gamma_{ns-}^{(2,0)} \\end{array}\right)\end{split}\]
Parameters:
Returns:

Leading-order singlet anomalous dimension matrix \(\\gamma_{V}^{(2,0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as3 module

Compute the NNLO Altarelli-Parisi splitting kernels.

The expression have been obtained from [MVV04, Vog05].

Note that the QCD colour factors have been hard-wired in the parametrizations.

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_nsm(n, nf, cache)[source]

Compute the NNLO valence-like non-singlet anomalous dimension.

Implements Eq. (3.8) of [MVV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_nsp(n, nf, cache)[source]

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

Implements Eq. (3.7) of [MVV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_nsv(n, nf, cache)[source]

Compute the NNLO valence non-singlet anomalous dimension.

Implements Eq. (3.9) of [MVV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_ps(n, nf, cache)[source]

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

Implements Eq. (3.10) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_qg(n, nf, cache)[source]

Compute the NNLO quark-gluon singlet anomalous dimension.

Implements Eq. (3.11) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_gq(n, nf, cache)[source]

Compute the NNLO gluon-quark singlet anomalous dimension.

Implements Eq. (3.12) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_gg(n, nf, cache)[source]

Compute the NNLO gluon-gluon singlet anomalous dimension.

Implements Eq. (3.13) of [VMV04].

Parameters:
Returns:

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

Return type:

complex

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

Compute the NNLO singlet anomalous dimension matrix.

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

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

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_singlet_qed(N, nf, cache)[source]

Compute the leading-order singlet anomalous dimension matrix for the unified evolution basis.

\[\begin{split}\\gamma_S^{(3,0)} = \\left(\begin{array}{cccc} \\gamma_{gg}^{(3,0)} & 0 & \\gamma_{gq}^{(3,0)} & 0\\ 0 & 0 & 0 & 0 \\ \\gamma_{qg}^{(3,0)} & 0 & \\gamma_{qq}^{(3,0)} & 0 \\ 0 & 0 & 0 & \\gamma_{qq}^{(3,0)} \\ \\end{array}\right)\end{split}\]
Parameters:
Returns:

Leading-order singlet anomalous dimension matrix \(\\gamma_{S}^{(3,0)}(N)\)

Return type:

numpy.ndarray

ekore.anomalous_dimensions.unpolarized.space_like.as3.gamma_valence_qed(N, nf, cache)[source]

Compute the leading-order valence anomalous dimension matrix for the unified evolution basis.

\[\begin{split}\\gamma_V^{(3,0)} = \\left(\begin{array}{cc} \\gamma_{nsV}^{(3,0)} & 0\\ 0 & \\gamma_{ns-}^{(3,0)} \\end{array}\right)\end{split}\]
Parameters:
Returns:

Leading-order singlet anomalous dimension matrix \(\\gamma_{V}^{(3,0)}(N)\)

Return type:

numpy.ndarray