"""The unpolarized, time-like |LO| Altarelli-Parisi splitting kernels."""
import numba as nb
import numpy as np
from eko import constants
from ....harmonics import cache as c
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@nb.njit(cache=True)
def gamma_qq(N, cache):
r"""Compute the |LO| quark-quark anomalous dimension.
Implements :eqref:`B.3` from :cite:`Mitov:2006wy`.
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
LO quark-quark anomalous dimension
:math:`\gamma_{qq}^{(0)}(N)`
"""
S1 = c.get(c.S1, cache, N)
result = constants.CF * (-3.0 + (4.0 * S1) - 2.0 / (N * (N + 1.0)))
return result
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@nb.njit(cache=True)
def gamma_qg(N):
r"""Compute the |LO| quark-gluon anomalous dimension.
Implements :eqref:`B.4` from :cite:`Mitov:2006wy`
and :eqref:`A1` from :cite:`Gluck:1992zx`.
Parameters
----------
N : complex
Mellin moment
Returns
-------
complex
LO quark-gluon anomalous dimension
:math:`\gamma_{qg}^{(0)}(N)`
"""
result = -(N**2 + N + 2.0) / (N * (N + 1.0) * (N + 2.0))
return result
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@nb.njit(cache=True)
def gamma_gq(N, nf):
r"""Compute the |LO| gluon-quark anomalous dimension.
Implements :eqref:`B.5` from :cite:`Mitov:2006wy`
and :eqref:`A1` from :cite:`Gluck:1992zx`.
Parameters
----------
N : complex
Mellin moment
nf : int
No. of active flavors
Returns
-------
complex
LO quark-gluon anomalous dimension
:math:`\gamma_{gq}^{(0)}(N)`
"""
result = -4.0 * nf * constants.CF * (N**2 + N + 2.0) / (N * (N - 1.0) * (N + 1.0))
return result
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@nb.njit(cache=True)
def gamma_gg(N, nf, cache):
r"""Compute the |LO| gluon-gluon anomalous dimension.
Implements :eqref:`B.6` from :cite:`Mitov:2006wy`.
Parameters
----------
N : complex
Mellin moment
nf : int
No. of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
LO quark-quark anomalous dimension
:math:`\gamma_{gg}^{(0)}(N)`
"""
S1 = c.get(c.S1, cache, N)
result = (2.0 * nf - 11.0 * constants.CA) / 3.0 + 4.0 * constants.CA * (
S1 - 1.0 / (N * (N - 1.0)) - 1.0 / ((N + 1.0) * (N + 2.0))
)
return result
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@nb.njit(cache=True)
def gamma_ns(N, cache):
r"""Compute the |LO| non-singlet anomalous dimension.
At LO, :math:`\gamma_{ns}^{(0)} = \gamma_{qq}^{(0)}`.
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
LO quark-quark anomalous dimension
:math:`\gamma_{ns}^{(0)}(N)`
"""
return gamma_qq(N, cache)
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@nb.njit(cache=True)
def gamma_singlet(N, nf, cache):
r"""Compute the |LO| singlet anomalous dimension matrix.
Implements :eqref:`2.13` from :cite:`Gluck:1992zx`.
Parameters
----------
N : complex
Mellin moment
nf : int
No. of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
LO singlet anomalous dimension matrix
:math:`\gamma_{s}^{(0)}`
"""
result = np.array(
[
[gamma_qq(N, cache), gamma_gq(N, nf)],
[gamma_qg(N), gamma_gg(N, nf, cache)],
],
np.complex_,
)
return result