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as1.py
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as1.py
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"""This file contains the leading-order Altarelli-Parisi splitting kernels."""
import numba as nb
import numpy as np
from eko import constants
@nb.njit(cache=True)
def gamma_ns(N, s1):
"""
Computes the leading-order non-singlet anomalous dimension.
Implements Eq. (3.4) of :cite:`Moch:2004pa`.
Parameters
----------
N : complex
Mellin moment
s1 : complex
harmonic sum :math:`S_{1}`
Returns
-------
gamma_ns : complex
Leading-order non-singlet anomalous dimension :math:`\\gamma_{ns}^{(0)}(N)`
"""
gamma = -(3.0 - 4.0 * s1 + 2.0 / N / (N + 1.0))
result = constants.CF * gamma
return result
@nb.njit(cache=True)
def gamma_qg(N, nf):
"""
Computes 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
@nb.njit(cache=True)
def gamma_gq(N):
"""
Computes 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
@nb.njit(cache=True)
def gamma_gg(N, s1, nf):
"""
Computes the leading-order gluon-gluon anomalous dimension
Implements Eq. (3.5) of :cite:`Vogt:2004mw`.
Parameters
----------
N : complex
Mellin moment
s1 : complex
harmonic sum :math:`S_{1}`
nf : int
Number of active flavors
Returns
-------
gamma_gg : complex
Leading-order gluon-gluon anomalous dimension :math:`\\gamma_{gg}^{(0)}(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
@nb.njit(cache=True)
def gamma_singlet(N, s1, nf):
r"""
Computes 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
s1 : complex
harmonic sum :math:`S_{1}`
nf : int
Number of active flavors
Returns
-------
gamma_S_0 : numpy.ndarray
Leading-order singlet anomalous dimension matrix :math:`\gamma_{S}^{(0)}(N)`
See Also
--------
gamma_ns : :math:`\gamma_{qq}^{(0)}`
gamma_qg : :math:`\gamma_{qg}^{(0)}`
gamma_gq : :math:`\gamma_{gq}^{(0)}`
gamma_gg : :math:`\gamma_{gg}^{(0)}`
"""
gamma_qq = gamma_ns(N, s1)
gamma_S_0 = np.array(
[[gamma_qq, gamma_qg(N, nf)], [gamma_gq(N), gamma_gg(N, s1, nf)]], np.complex_
)
return gamma_S_0