ekore.operator_matrix_elements.unpolarized.space_like package

The unpolarized, space-like OME.

ekore.operator_matrix_elements.unpolarized.space_like.A_singlet(matching_order, n, nf, L, is_msbar)

Compute the tower of the singlet OME.

Parameters:

• matching_order (tuple(int,int)) – perturbative matching order

• n (complex) – Mellin variable

• nf (int) – number of active flavor below threshold

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

• is_msbar (bool) – add the \(\overline{MS}\) contribution

Returns:

singlet OME

Return type:

numpy.ndarray

ekore.operator_matrix_elements.unpolarized.space_like.A_non_singlet(matching_order, n, nf, L)

Compute the tower of the non-singlet OME.

Parameters:

• matching_order (tuple(int,int)) – perturbative matching order

• n (complex) – Mellin variable

• nf (int) – number of active flavor below threshold

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

non-singlet OME

Return type:

numpy.ndarray

Subpackages

• ekore.operator_matrix_elements.unpolarized.space_like.as3 package
  • A_singlet()
  • A_ns()
  • Submodules
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.aHg module
    • A_Hg()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.aHg_param module
    • a_Hg3()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.aHq module
    • A_Hq()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.agg module
    • a_gg3()
    • A_gg()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.agq module
    • A_gq()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.aqg module
    • A_qg()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.aqqNS module
    • A_qqNS()
  • ekore.operator_matrix_elements.unpolarized.space_like.as3.aqqPS module
    • A_qqPS()

Submodules

ekore.operator_matrix_elements.unpolarized.space_like.as1 module

The unpolarized, spacelike NLO OME.

Heavy quark contribution for intrinsic evolution are taken from [BBB+16] and Mellin transformed with Mathematica. The other matching conditions for the VFNS at \(\mu_F^2 \neq m_H^2\) are provided in [BMSvN98].

ekore.operator_matrix_elements.unpolarized.space_like.as1.A_hh(n, cache, L)

NLO heavy-heavy OME \(A_{HH}^{(1)}\).

They are defined as the Mellin transform of \(K_{hh}\) given in 20a of [BBB+16].

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NLO heavy-heavy OME \(A_{HH}^{(1)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as1.A_gh(n, L)

NLO gluon-heavy OME \(A_{gH}^{(1)}\).

They are defined as the Mellin transform of \(K_{gh}\) given in 20b of [BBB+16].

Parameters:

• n (complex) – Mellin moment

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NLO gluon-heavy OME \(A_{gH}^{(1)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as1.A_hg(n, L)

NLO heavy-gluon OME \(A_{Hg}^{S,(1)}\).

They are defined as the Mellin transform of:eqref:B.2 from [BMSvN98].

Parameters:

• n (complex) – Mellin moment

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NLO heavy-gluon OME \(A_{Hg}^{S,(1)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as1.A_gg(L)

NLO gluon-gluon OME \(A_{gg,H}^{S,(1)}\).

They are defined as the Mellin transform of B.6 from [BMSvN98].

Parameters:

L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NLO gluon-gluon OME \(A_{gg,H}^{S,(1)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as1.A_singlet(n, cache, L)

Compute the NLO singlet OME.

\[\begin{split}A^{S,(1)} = \left(\begin{array}{cc} A_{gg,H}^{S,(1)} & 0 & A_{gH}^{(1)} \\ 0 & 0 & 0 \\ A_{hg}^{S,(1)} & 0 & A_{HH}^{(1)} \end{array}\right)\end{split}\]

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NLO singlet OME \(A^{S,(1)}\)

Return type:

numpy.ndarray

ekore.operator_matrix_elements.unpolarized.space_like.as1.A_ns(n, cache, L)

Compute the NLO non-singlet OME with intrinsic contributions.

\[\begin{split}A^{NS,(1)} = \left(\begin{array}{cc} 0 & 0 \\ 0 & A_{HH}^{(1)} \end{array}\right)\end{split}\]

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NLO non-singlet OME \(A^{S,(1)}\)

Return type:

numpy.ndarray

ekore.operator_matrix_elements.unpolarized.space_like.as2 module

The unpolarized, spacelike NNLO OME.

See, [BMSvN98] appendix B. The expression for \(\mu_F^2 = m_H^2\) are taken from [Vog05] directly in N space. While the parts proportional to \(\ln(\mu_F^2 / m_h^2)\) comes QCDNUM (https://github.com/N3PDF/external/blob/master/qcdnum/qcdnum/pij/ome.f) and Mellin transformed with Mathematica.

The expression for A_Hg_l0 comes form [BBK09b].

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_qq_ns(n, cache, L)

NNLO light-light non-singlet OME \(A_{qq,H}^{NS,(2)}\).

It is given in B.4 of [BMSvN98].

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NNLO light-light non-singlet OME \(A_{qq,H}^{NS,(2)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_hq_ps(n, cache, L)

NNLO heavy-light pure-singlet OME \(A_{Hq}^{PS,(2)}\).

It is given in B.1 of [BMSvN98].

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NNLO heavy-light pure-singlet OME \(A_{Hq}^{PS,(2)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_hg(n, cache, L)

NNLO heavy-gluon OME \(A_{Hg}^{S,(2)}\).

It is given in B.3 of [BMSvN98]. The expession for A_Hg_l0 comes form [BBK09b].

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NNLO heavy-gluon OME \(A_{Hg}^{S,(2)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_gq(n, cache, L)

NNLO gluon-quark OME \(A_{gq,H}^{S,(2)}\).

It is given in B.5 of [BMSvN98].

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NNLO gluon-quark OME \(A_{gq,H}^{S,(2)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_gg(n, cache, L)

NNLO gluon-gluon OME \(A_{gg,H}^{S,(2)}\).

It is given in B.7 of [BMSvN98].

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NNLO gluon-gluon OME \(A_{gg,H}^{S,(2)}\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_singlet(n, cache, L, is_msbar=False)

Compute the NNLO singlet OME.

\[\begin{split}A^{S,(2)} = \left(\begin{array}{cc} A_{gg, H}^{S,(2)} & A_{gq, H}^{S,(2)} & 0 \\ 0 & A_{qq,H}^{NS,(2)} & 0\\ A_{hg}^{S,(2)} & A_{hq}^{PS,(2)} & 0\\ \end{array}\right)\end{split}\]

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

• is_msbar (bool) – add the \(\overline{MS}\) contribution

Returns:

NNLO singlet OME \(A^{S,(2)}(N)\)

Return type:

numpy.ndarray

ekore.operator_matrix_elements.unpolarized.space_like.as2.A_ns(n, cache, L)

Compute the NNLO non-singlet OME.

\[\begin{split}A^{NS,(2)} = \left(\begin{array}{cc} A_{qq,H}^{NS,(2)} & 0 \\ 0 & 0 \\ \end{array}\right)\end{split}\]

Parameters:

• n (complex) – Mellin moment

• cache (numpy.ndarray) – Harmonic sum cache

• L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

NNLO non-singlet OME \(A^{NS,(2)}\)

Return type:

numpy.ndarray