r"""The unpolarized, space-like |N3LO| |OME|.
The expressions are based on:
- :cite:`Bierenbaum:2009mv`. Isabella Bierenbaum, Johannes Blümlein, and
Sebastian Klein. Mellin Moments of the O(alpha**3(s)) Heavy Flavor
Contributions to unpolarized Deep-Inelastic Scattering at Q**2 \ensuremath
>\ensuremath > m**2 and Anomalous Dimensions. Nucl. Phys. B, 820:417-482,
2009. arXiv:0904.3563, doi:10.1016/j.nuclphysb.2009.06.005.
- :cite:`Bl_mlein_2000`. Johannes Blümlein. Analytic continuation of mellin
transforms up to two-loop order. Computer Physics Communications,
133(1):76-104, Dec 2000. URL:
http://dx.doi.org/10.1016/S0010-4655(00)00156-9,
doi:10.1016/s0010-4655(00)00156-9.
- :cite:`Bierenbaum:2009zt`. Isabella Bierenbaum, Johannes Blümlein, and
Sebastian Klein. The Gluonic Operator Matrix Elements at O(alpha(s)**2) for
DIS Heavy Flavor Production. Phys. Lett. B, 672:401-406, 2009.
arXiv:0901.0669, doi:10.1016/j.physletb.2009.01.057.
- :cite:`Ablinger:2010ty`. J. Ablinger, J. Blümlein, S. Klein, C. Schneider,
and F. Wissbrock. The $O(\alpha _s^3)$ Massive Operator Matrix Elements of
$O(n_f)$ for the Structure Function $F_2(x,Q^2)$ and Transversity. Nucl.
Phys. B, 844:26-54, 2011. arXiv:1008.3347,
doi:10.1016/j.nuclphysb.2010.10.021.
- :cite:`Ablinger:2014vwa`. J. Ablinger, A. Behring, J. Blümlein, A. De
Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F.
Wißbrock. The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous
Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity. Nucl.
Phys. B, 886:733-823, 2014. arXiv:1406.4654,
doi:10.1016/j.nuclphysb.2014.07.010.
- :cite:`Ablinger:2014uka`. J. Ablinger, J. Blümlein, A. De Freitas, A.
Hasselhuhn, A. von Manteuffel, M. Round, and C. Schneider. The $O(\alpha
_s^3 T_F^2)$ Contributions to the Gluonic Operator Matrix Element. Nucl.
Phys. B, 885:280-317, 2014. arXiv:1405.4259,
doi:10.1016/j.nuclphysb.2014.05.028.
- :cite:`Behring:2014eya`. A. Behring, I. Bierenbaum, J. Blümlein, A. De
Freitas, S. Klein, and F. Wißbrock. The logarithmic contributions to the
$O(\alpha ^3_s)$ asymptotic massive Wilson coefficients and operator matrix
elements in deeply inelastic scattering. Eur. Phys. J. C, 74(9):3033, 2014.
arXiv:1403.6356, doi:10.1140/epjc/s10052-014-3033-x.
- :cite:`Blumlein:2017wxd`. Johannes Blümlein, Jakob Ablinger, Arnd Behring,
Abilio De Freitas, Andreas von Manteuffel, Carsten Schneider, and C.
Schneider. Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering:
Recent Results. PoS, QCDEV2017:031, 2017. arXiv:1711.07957,
doi:10.22323/1.308.0031.
- :cite:`Ablinger_2014`. J. Ablinger, J. Blümlein, A. De Freitas, A.
Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock. The
transition matrix element a_gq(n) of the variable flavor number scheme at
o(α_s^3). Nuclear Physics B, 882:263-288, May 2014. URL:
http://dx.doi.org/10.1016/j.nuclphysb.2014.02.007,
doi:10.1016/j.nuclphysb.2014.02.007.
- :cite:`Ablinger_2015`. J. Ablinger, A. Behring, J. Blümlein, A. De
Freitas, A. von Manteuffel, and C. Schneider. The 3-loop pure singlet heavy
flavor contributions to the structure function f2(x,q2) and the anomalous
dimension. Nuclear Physics B, 890:48-151, Jan 2015. URL:
http://dx.doi.org/10.1016/j.nuclphysb.2014.10.008,
doi:10.1016/j.nuclphysb.2014.10.008.
- :cite:`Ablinger:2022wbb`. J. Ablinger, J. and A. Behring, J. Blümlein, A. De
Freitas, C. Schneider, A. Goedicke, C. von Manteuffel and K. Schonwald.
The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator
Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$}. JHEP 12 (2022) 134,
doi:10.1007/JHEP12(2022)134,
- :cite:`Ablinger:2024xtt`. J. Ablinger, A. Behring, J. Blümlein, A. De
Freitas, A. von Manteuffel, C. Schneider, K. Schönwald.
The non-first-order-factorizable contributions to the three-loop single-mass
operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$.
"""
import numba as nb
import numpy as np
from .agg import A_gg
from .agq import A_gq
from .aHg import A_Hg
from .aHq import A_Hq
from .aqg import A_qg
from .aqqNS import A_qqNS
from .aqqPS import A_qqPS
[docs]
@nb.njit(cache=True)
def A_singlet(n, cache, nf, L):
r"""Compute the |N3LO| singlet |OME|.
.. math::
A^{S,(3)} = \left(\begin{array}{cc}
A_{gg, H}^{S,(3)} & A_{gq, H}^{S,(3)} & 0 \\
A_{qg, H}^{S,(3)} & A_{qq,H}^{NS,(3)} + A_{qq,H}^{PS,(3)} & 0\\
A_{Hg}^{S,(3)} & A_{Hq}^{PS,(3)} & 0\\
\end{array}\right)
When using the code, please cite the complete list of references
available at the top of this module :mod:`~ekore.operator_matrix_elements.unpolarized.space_like.as3`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
nf : int
number of active flavor below the threshold
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
A_S : numpy.ndarray
|N3LO| singlet |OME| :math:`A^{S,(3)}(N)`
"""
A_hq_3 = A_Hq(n, cache, nf, L)
A_hg_3 = A_Hg(n, cache, nf, L)
A_gq_3 = A_gq(n, cache, nf, L)
A_gg_3 = A_gg(n, cache, nf, L)
A_qq_ps_3 = A_qqPS(n, cache, nf, L)
A_qq_ns_3 = A_qqNS(n, cache, nf, L, 1)
A_qg_3 = A_qg(n, cache, nf, L)
A_S = np.array(
[
[A_gg_3, A_gq_3, 0.0],
[A_qg_3, A_qq_ps_3 + A_qq_ns_3, 0.0],
[A_hg_3, A_hq_3, 0.0],
],
np.complex_,
)
return A_S
[docs]
@nb.njit(cache=True)
def A_ns(n, cache, nf, L):
r"""Compute the |N3LO| non-singlet |OME|.
.. math::
A^{NS,(3)} = \left(\begin{array}{cc}
A_{qq,H}^{NS,(3)} & 0\\
0 & 0\\
\end{array}\right)
When using the code, please cite the complete list of references available
at the top of this module :mod:`~ekore.operator_matrix_elements.unpolarized.space_like.as3`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
nf : int
number of active flavor below the threshold
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
A_NS : numpy.ndarray
|N3LO| non-singlet |OME| :math:`A^{NS,(3)}`
"""
return np.array([[A_qqNS(n, cache, nf, L, -1), 0.0], [0 + 0j, 0 + 0j]], np.complex_)