"""Compute the leading-order 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_ns(N, cache):
r"""Compute the leading-order non-singlet anomalous dimension.
Implements Eq. (3.4) of :cite:`Moch:2004pa`.
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
gamma_ns : complex
Leading-order non-singlet anomalous dimension :math:`\\gamma_{ns}^{(0)}(N)`
"""
S1 = c.get(c.S1, cache, N)
gamma = -(3.0 - 4.0 * S1 + 2.0 / N / (N + 1.0))
result = constants.CF * gamma
return result
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@nb.njit(cache=True)
def gamma_qg(N, nf):
r"""Compute the leading-order quark-gluon anomalous dimension.
Implements Eq. (3.5) of :cite:`Vogt:2004mw`.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
Returns
-------
gamma_qg : complex
Leading-order quark-gluon anomalous dimension :math:`\\gamma_{qg}^{(0)}(N)`
"""
gamma = -(N**2 + N + 2.0) / (N * (N + 1.0) * (N + 2.0))
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 leading-order gluon-quark anomalous dimension.
Implements Eq. (3.5) of :cite:`Vogt:2004mw`.
Parameters
----------
N : complex
Mellin moment
Returns
-------
gamma_gq : complex
Leading-order gluon-quark anomalous dimension :math:`\\gamma_{gq}^{(0)}(N)`
"""
gamma = -(N**2 + N + 2.0) / (N * (N + 1.0) * (N - 1.0))
result = 2.0 * constants.CF * gamma
return result
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@nb.njit(cache=True)
def gamma_gg(N, cache, nf):
r"""Compute the leading-order gluon-gluon anomalous dimension.
Implements Eq. (3.5) of :cite:`Vogt:2004mw`.
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
nf : int
Number of active flavors
Returns
-------
gamma_gg : complex
Leading-order gluon-gluon anomalous dimension :math:`\\gamma_{gg}^{(0)}(N)`
"""
S1 = c.get(c.S1, cache, N)
gamma = S1 - 1.0 / N / (N - 1.0) - 1.0 / (N + 1.0) / (N + 2.0)
result = constants.CA * (4.0 * gamma - 11.0 / 3.0) + 4.0 / 3.0 * constants.TR * nf
return result
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@nb.njit(cache=True)
def gamma_singlet(N, cache, nf):
r"""Compute the leading-order 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
-------
gamma_S_0 : numpy.ndarray
Leading-order 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
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@nb.njit(cache=True)
def gamma_singlet_qed(N, cache, nf):
r"""Compute the leading-order singlet anomalous dimension matrix for the unified evolution basis.
.. math::
\\gamma_S^{(1,0)} = \\left(\begin{array}{cccc}
\\gamma_{gg}^{(1,0)} & 0 & \\gamma_{gq}^{(1,0)} & 0\\
0 & 0 & 0 & 0 \\
\\gamma_{qg}^{(1,0)} & 0 & \\gamma_{qq}^{(1,0)} & 0 \\
0 & 0 & 0 & \\gamma_{qq}^{(1,0)} \\
\\end{array}\right)
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
nf : int
Number of active flavors
Returns
-------
gamma_S : numpy.ndarray
Leading-order singlet anomalous dimension matrix :math:`\\gamma_{S}^{(1,0)}(N)`
"""
gamma_qq = gamma_ns(N, cache)
gamma_S = np.array(
[
[gamma_gg(N, cache, nf), 0.0 + 0.0j, gamma_gq(N), 0.0 + 0.0j],
[0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j],
[gamma_qg(N, nf), 0.0 + 0.0j, gamma_qq, 0.0 + 0.0j],
[0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, gamma_qq],
],
np.complex_,
)
return gamma_S
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@nb.njit(cache=True)
def gamma_valence_qed(N, cache):
r"""Compute the leading-order valence anomalous dimension matrix for the unified evolution basis.
.. math::
\\gamma_V^{(1,0)} = \\left(\begin{array}{cc}
\\gamma_{ns}^{(1,0)} & 0\\
0 & \\gamma_{ns}^{(1,0)}
\\end{array}\right)
Parameters
----------
N : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
gamma_V : numpy.ndarray
Leading-order singlet anomalous dimension matrix :math:`\\gamma_{V}^{(1,0)}(N)`
"""
gamma_V = np.array(
[
[1.0, 0.0],
[0.0, 1.0],
],
np.complex_,
)
return gamma_V * gamma_ns(N, cache)