r"""The unpolarized, spacelike |NLO| |OME|.
Heavy quark contribution for intrinsic evolution are taken from :cite:`Ball_2016`
and Mellin transformed with Mathematica.
The other matching conditions for the |VFNS| at :math:`\mu_F^2 \neq m_H^2`
are provided in :cite:`Buza_1998`.
"""
import numba as nb
import numpy as np
from eko.constants import CF
from ....harmonics import cache as c
[docs]
@nb.njit(cache=True)
def A_hh(n, cache, L):
r"""|NLO| heavy-heavy |OME| :math:`A_{HH}^{(1)}`.
They are defined as the Mellin transform of :math:`K_{hh}`
given in :eqref:`20a` of :cite:`Ball_2016`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NLO| heavy-heavy |OME| :math:`A_{HH}^{(1)}`
"""
S1m = c.get(c.S1, cache, n) - 1 / n # harmonics.S1(n - 1)
S2m = c.get(c.S2, cache, n) - 1 / n**2 # harmonics.S2(n - 1)
ahh_l = (2 + n - 3 * n**2) / (n * (1 + n)) + 4 * S1m
ahh = 2 * (
2 + 5 * n + n**2 - 6 * n**3 - 2 * n**4 - 2 * n * (-1 - 2 * n + n**3) * S1m
) / (n * (1 + n)) ** 2 + 4 * (S1m**2 + S2m)
return -CF * (ahh_l * L + ahh)
[docs]
@nb.njit(cache=True)
def A_gh(n, L):
r"""|NLO| gluon-heavy |OME| :math:`A_{gH}^{(1)}`.
They are defined as the Mellin transform of :math:`K_{gh}`
given in :eqref:`20b` of :cite:`Ball_2016`.
Parameters
----------
n : complex
Mellin moment
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NLO| gluon-heavy |OME| :math:`A_{gH}^{(1)}`
"""
agh_l1 = (2 + n + n**2) / (n * (n**2 - 1))
agh_l0 = (-4 + 2 * n + n**2 * (15 + n * (3 + n - n**2))) / (n * (n**2 - 1)) ** 2
return 2 * CF * (agh_l0 + agh_l1 * L)
[docs]
@nb.njit(cache=True)
def A_hg(n, L):
r"""|NLO| heavy-gluon |OME| :math:`A_{Hg}^{S,(1)}`.
They are defined as the Mellin transform of:eqref:`B.2` from :cite:`Buza_1998`.
Parameters
----------
n : complex
Mellin moment
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NLO| heavy-gluon |OME| :math:`A_{Hg}^{S,(1)}`
"""
den = 1.0 / (n * (n + 1) * (2 + n))
num = 2 * (2 + n + n**2)
return num * den * L
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@nb.njit(cache=True)
def A_gg(L):
r"""|NLO| gluon-gluon |OME| :math:`A_{gg,H}^{S,(1)}`.
They are defined as the Mellin transform of :eqref:`B.6` from :cite:`Buza_1998`.
Parameters
----------
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
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, cache, L):
r"""Compute the |NLO| singlet |OME|.
.. math::
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)
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
numpy.ndarray
|NLO| singlet |OME| :math:`A^{S,(1)}`
"""
A_S = np.array(
[
[A_gg(L), 0.0, A_gh(n, L)],
[0 + 0j, 0 + 0j, 0 + 0j],
[A_hg(n, L), 0.0, A_hh(n, cache, L)],
],
np.complex128,
)
return A_S
[docs]
@nb.njit(cache=True)
def A_ns(n, cache, L):
r"""Compute the |NLO| non-singlet |OME| with intrinsic contributions.
.. math::
A^{NS,(1)} = \left(\begin{array}{cc}
0 & 0 \\
0 & A_{HH}^{(1)}
\end{array}\right)
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
numpy.ndarray
|NLO| non-singlet |OME| :math:`A^{S,(1)}`
"""
return np.array([[0 + 0j, 0 + 0j], [0 + 0j, A_hh(n, cache, L)]], np.complex128)