"""The unpolarized, time-like |NLO| matching conditions."""
import numba as nb
import numpy as np
from eko.constants import CF
[docs]
@nb.njit(cache=True)
def A_hg(N, L):
r"""Compute the |NLO| heavy-gluon |OME|.
Implements :eqref:`27` from :cite:`Cacciari:2005ry`.
Parameters
----------
N : complex
Mellin moment
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
A_hg : complex
|NLO| heavy-gluon |OME|
:math:`A_{hg}^{S,(1)}`
"""
result = (
2
* CF
* (
(2 + N + N**2) / (N * (N**2 - 1)) * (L - 1)
+ 4 / (N - 1) ** 2
- 4 / N**2
+ 2 / (N + 1) ** 2
)
)
return result
[docs]
@nb.njit(cache=True)
def A_gg(L):
r"""Compute the |NLO| gluon-gluon |OME|.
Implements the Mellin transform of
:eqref:`24` from :cite:`Cacciari:2005ry`.
It is identical to the one in
:class:`~ekore.operator_matrix_elements.unpolarized.space_like.as1`.
Parameters
----------
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
A_gg : complex
|NLO| gluon-gluon |OME| :math:`A_{gg,H}^{S,(1)}`
"""
return -2.0 / 3.0 * L
[docs]
@nb.njit(cache=True)
def A_singlet(N, L):
r"""Compute the |NLO| singlet |OME|.
Parameters
----------
N : complex
Mellin moment
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
numpy.ndarray
|NLO| singlet |OME|
:math:`A^{S,(1)}`
"""
result = np.array(
[[A_gg(L), 0 + 0j, 0], [0 + 0j, 0, 0], [A_hg(N, L), 0, 0]], np.complex128
)
return result
[docs]
@nb.njit(cache=True)
def A_ns():
r"""Compute the |NLO| non-singlet |OME|.
Returns
-------
A_ns : numpy.ndarray
|NLO| non-singlet |OME|
:math:`A^{S,(1)}`
"""
result = np.array([[0, 0], [0, 0]], np.complex128)
return result