Polarized Matching conditions

doc/source/refs.bib

@@ -777,6 +777,18 @@ @article{Albino:2000cp
     year = "2001"
 }
 
+@article{Bierenbaum_2023,
+	title = {$O(\alpha_{s}^2 Polarized Heavy Flavor Corrections to Deep-Inelastic Scattering at $Q^2 >> m^2$},
+    author = {I. Bierenbaum and J. Blümlein and A. De Freitas and A. Goedicke and S. Klein and K. Schönwald},
+	doi = {10.1016/j.nuclphysb.2023.116114},
+	url = {https://doi.org/10.1016%2Fj.nuclphysb.2023.116114},
+	year = 2023,
+	month = {mar},
+	publisher = {Elsevier {BV}},
+	volume = {988},
+	pages = {116114},
+	journal = {Nuclear Physics B}
+}
 @article{Gluck:1995yr,
     author = "Gluck, M. and Reya, E. and Stratmann, M. and Vogelsang, W.",
     title = "{Next-to-leading order radiative parton model analysis of polarized deep inelastic lepton - nucleon scattering}",
@@ -827,4 +839,4 @@ @article{Moch:2014sna
     volume = "889",
     pages = "351--400",
     year = "2014"
-}
+}

src/ekore/operator_matrix_elements/polarized/space_like/__init__.py

@@ -1,13 +1,55 @@
 r"""The polarized, space-like |OME|."""
 
 import numba as nb
+import numpy as np
+
+from . import as1
 
 
 @nb.njit(cache=True)
-def A_non_singlet(_matching_order, _n, _sx, _nf, _L):
-    raise NotImplementedError("Polarised, space-like is not yet implemented")
+def A_singlet(matching_order, n, sx, nf, L, is_msbar, sx_ns=None):
+    r"""
+    Computes the tower of the singlet |OME|.
+
+    Parameters
+    ----------
+        matching_order : tuple(int,int)
+            perturbative matching_order
+        n : complex
+            Mellin variable
+        sx : list
+            singlet like harmonic sums cache
+        nf: int
+            number of active flavor below threshold
+        L : float
+            :math:``\ln(\mu_F^2 / m_h^2)``
+        is_msbar: bool
+            add the |MSbar| contribution
+        sx_ns : list
+            non-singlet like harmonic sums cache
+
+    Returns
+    -------
+        A_singlet : numpy.ndarray
+            singlet |OME|
+
+    See Also
+    --------
+        ekore.matching_conditions.nlo.A_singlet_1 : :math:`A^{S,(1)}(N)`
+        ekore.matching_conditions.nlo.A_hh_1 : :math:`A_{HH}^{(1)}(N)`
+        ekore.matching_conditions.nlo.A_gh_1 : :math:`A_{gH}^{(1)}(N)`
+        ekore.matching_conditions.nnlo.A_singlet_2 : :math:`A_{S,(2)}(N)`
+    """
+    A_s = np.zeros((matching_order[0], 3, 3), np.complex_)
+    if matching_order[0] == 1:
+        A_s[0] = as1.A_singlet(n, L)
+    else:
+        raise NotImplementedError(
+            "Polarised, space-like is not yet implemented at this order"
+        )
+    return A_s
 
 
 @nb.njit(cache=True)
-def A_singlet(_matching_order, _n, _sx, _nf, _L, _is_msbar, _sx_ns=None):
+def A_non_singlet(_matching_order, _n, _sx, _nf, _L, _is_msbar, _sx_ns=None):
     raise NotImplementedError("Polarised, space-like is not yet implemented")

src/ekore/operator_matrix_elements/polarized/space_like/as1.py

@@ -0,0 +1,112 @@
+r"""
+This module contains the |NLO| |OME| (OMEs) in the polarized case for the matching conditions in the |VFNS|.
+Heavy quark contribution for intrinsic evolution are not considered for the polarized case.
+The matching conditions for the |VFNS| at :math:`\mu_F^2 \neq m_H^2` are provided in :cite:`Bierenbaum_2023`. In the paper, the fraction :math:`\mu_F^2 / m_H^2` inside the log is inverted, yielding an additional factor of (-1) wherever L has an odd power. Additionally, a different convention for the anomalous dimensions is used, yielding a factor 2 in the |OME|'s wherever they are present. The anomalous dimensions and beta function with the addition 'hat', have the form :math:`\gamma_hat = gamma(nf+1) - gamma(nf)`.
+"""
+import numba as nb
+import numpy as np
+
+from eko.constants import TR
+
+from ....anomalous_dimensions.polarized.space_like.as1 import gamma_qg as gamma0_qg
+from ...unpolarized.space_like.as1 import A_gg as A_gg_unpol
+
+
+@nb.njit(cache=True)
+def gamma0_qghat(n, nf):
+    r"""Compute the |LO| polarized quark-gluon anomalous dimension with the addition 'hat' and thus, discarding the part proportional 'nf'. The factor 2 is included due to convention mentioned above.
+
+    Parameters
+    ----------
+    N : complex
+      Mellin moment
+    nf : int
+      Number of active flavors
+
+    Returns
+    -------
+    complex
+      |LO| polarized quark-gluon anomalous dimension (hat) :math:`\\\hat{gamma_{qg}}^{(0)}(N)`
+
+    """
+    return 2 * gamma0_qg(n, nf) / nf
+
+
+@nb.njit(cache=True)
+def A_hg(n, L, nf):
+    r"""
+    |NLO| heavy-gluon |OME| :math:`A_{Hg}^{S,(1)}` given in Eq. (104) of :cite:`Bierenbaum_2023`.
+    Parameters
+    ----------
+        n : complex
+            Mellin moment
+        L : float
+            :math:`\ln(\mu_F^2 / m_h^2)`
+        nf : int
+            Number of active flavors
+
+    Returns
+    -------
+        A_hg : complex
+            |NLO| heavy-gluon |OME| :math:`A_{Hg}^{S,(1)}`
+    """
+    return (1 / 2) * gamma0_qghat(n, nf) * (-L)
+
+
+@nb.njit(cache=True)
+def A_gg(L):
+    r"""
+    |NLO| gluon-gluon |OME| :math:`A_{gg,H}^{S,(1)}` given in Eq. (186) of :cite:`Bierenbaum_2023`.
+
+    Parameters
+    ----------
+        L : float
+            :math:`\ln(\mu_F^2 / m_h^2)`
+
+    Returns
+    -------
+        A_gg : complex
+            |NLO| gluon-gluon |OME| :math:`A_{gg,H}^{S,(1)}`
+    """
+    return 2 * A_gg_unpol(L)
+
+
+@nb.njit(cache=True)
+def A_singlet(n, L, nf):
+    r"""
+    Computes the |NLO| singlet |OME|.
+    .. math::
+        A^{S,(1)} = \left(\begin{array}{cc}
+        A_{gg,H}^{S,(1)} & 0 & 0\\
+        0 & 0 & 0 \\
+        A_{hg}^{S,(1)} & 0 & 0
+        \end{array}\right)
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    L : float
+        :math:`\ln(\mu_F^2 / m_h^2)`
+    nf : int
+        Number of active flavors
+
+    Returns
+      -------
+    A_S : numpy.ndarray
+        |NLO| singlet |OME| :math:`A^{S,(1)}`
+
+      See Also
+      --------
+        A_gg : :math:`A_{gg,H}^{S,(1)}`
+        A_hg : :math:`A_{hg}^{S,(1)}`
+    """
+    A_S = np.array(
+        [
+            [A_gg(L), 0.0, 0.0],
+            [0 + 0j, 0 + 0j, 0 + 0j],
+            [A_hg(n, L, nf), 0.0, 0.0],
+        ],
+        np.complex_,
+    )
+    return A_S