"""The |LO| Altarelli-Parisi splitting kernels."""
import numba as nb
import numpy as np
from eko import constants
from ....harmonics import cache as c
from ...unpolarized.space_like.as1 import gamma_ns
[docs]
@nb.njit(cache=True)
def gamma_qg(N, nf):
r"""Compute the |LO| polarized quark-gluon anomalous dimension.
Implements :eqref:`A.1` from :cite:`Gluck:1995yr`.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
Returns
-------
complex
|LO| polarized quark-gluon anomalous dimension :math:`\gamma_{qg}^{(0)}(N)`
"""
gamma = -(N - 1) / N / (N + 1)
result = 2.0 * constants.TR * 2.0 * nf * gamma
return result
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@nb.njit(cache=True)
def gamma_gq(N):
r"""Compute the |LO| polarized gluon-quark anomalous dimension.
Implements :eqref:`A.1` from :cite:`Gluck:1995yr`.
Parameters
----------
N : complex
Mellin moment
Returns
-------
complex
|LO| gluon-quark anomalous dimension :math:`\gamma_{gq}^{(0)}(N)`
"""
gamma = -(N + 2) / N / (N + 1)
result = 2.0 * constants.CF * gamma
return result
[docs]
@nb.njit(cache=True)
def gamma_gg(N, cache, nf):
r"""Compute the |LO| polarized gluon-gluon anomalous dimension.
Implements :eqref:`A.1` from :cite:`Gluck:1995yr`.
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
nf : int
Number of active flavors
Returns
-------
complex
|LO| gluon-gluon anomalous dimension :math:`\gamma_{gg}^{(0)}(N)`
"""
S1 = c.get(c.S1, cache, N)
gamma = -S1 + 2 / N / (N + 1)
result = constants.CA * (-4.0 * gamma - 11.0 / 3.0) + 4.0 / 3.0 * constants.TR * nf
return result
[docs]
@nb.njit(cache=True)
def gamma_singlet(N, cache, nf):
r"""Compute the |LO| polarized singlet anomalous dimension matrix.
.. math::
\gamma_S^{(0)} = \left(\begin{array}{cc}
\gamma_{qq}^{(0)} & \gamma_{qg}^{(0)}\\
\gamma_{gq}^{(0)} & \gamma_{gg}^{(0)}
\end{array}\right)
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
nf : int
Number of active flavors
Returns
-------
numpy.ndarray
|LO| singlet anomalous dimension matrix :math:`\gamma_{S}^{(0)}(N)`
"""
gamma_qq = gamma_ns(N, cache)
gamma_S_0 = np.array(
[[gamma_qq, gamma_qg(N, nf)], [gamma_gq(N), gamma_gg(N, cache, nf)]],
np.complex_,
)
return gamma_S_0