Time-like Anomalous dimensions

src/ekore/anomalous_dimensions/unpolarized/time_like/__init__.py

@@ -1,21 +0,0 @@
-r"""The unpolarized, time-like Altarelli-Parisi splitting kernels.
-
-Normalization is given by
-
-.. math::
-    \mathbf{P}(x) = \sum\limits_{j=0} a_s^{j+1} \mathbf P^{(j)}(x)
-
-with :math:`a_s = \frac{\alpha_S(\mu^2)}{4\pi}`.
-"""
-
-import numba as nb
-
-
-@nb.njit(cache=True)
-def gamma_ns(_order, _mode, _n, _nf):
-    raise NotImplementedError("Polarised is not yet implemented")
-
-
-@nb.njit(cache=True)
-def gamma_singlet(_order, _n, _nf):
-    raise NotImplementedError("Polarised is not yet implemented")

src/ekore/anomalous_dimensions/unpolarized/time_like/as1.py

@@ -0,0 +1,117 @@
+# -*- coding: utf-8 -*-
+"""The following are the unpolarized time-like leading order Altarelli-Parisi splitting kernels."""
+
+import numba as nb
+import numpy as np
+from eko import constants
+
+
+
+@nb.njit(cache=True)
+def gamma_qq(N, s1):
+    """
+    Computes the LO quark-quark anomalous dimension
+    Implements Eqn. (B.3) from hep-ph/0604160
+
+    Input parameters
+    ----------------
+    N : Mellin moment (type: complex)
+    s1 : harmonic sum $S_{1}$ (type: complex)
+
+    Returns
+    -------
+    gamma_qq : LO quark-quark anomalous dimension $\gamma_{qq}^{(0)}(N)$ (type: complex)
+    """
+    result = constants.CF * (-3.0 + (4.0 * s1) - 2.0 / (N * (N + 1.0)))
+    return result
+
+@nb.njit(cache=True)
+def gamma_qg(N):
+    """
+    Computes the LO quark-gluon anomalous dimension
+    Implements Eqn. (B.4) from hep-ph/0604160 and Eqn. (A1) from PhysRevD.48.116
+
+    Input parameters
+    ----------------
+    N : Mellin moment (type: complex)
+
+    Returns
+    -------
+    gamma_qg : LO quark-gluon anomalous dimension $\gamma_{qg}^{(0)}(N)$ (type: complex)
+    """
+    result = - (N**2 + N + 2.0) / (N * (N + 1.0) * (N + 2.0))
+    return result
+
+@nb.njit(cache=True)
+def gamma_gq(N, nf):
+    """
+    Computes the LO gluon-quark anomalous dimension
+    Implements Eqn. (B.5) from hep-ph/0604160 and Eqn. (A1) from PhysRevD.48.116
+
+    Input parameters
+    ----------------
+    N : Mellin moment (type: complex)
+    nf : No. of active flavors (type: int)
+
+    Returns
+    -------
+    gamma_qg : LO quark-gluon anomalous dimension $\gamma_{gq}^{(0)}(N)$ (type: complex)
+    """
+    result = -4.0 * nf * constants.CF * (N**2 + N + 2.0) / (N * (N - 1.0) * (N + 1.0))
+    return result
+
+@nb.njit(cache=True)
+def gamma_gg(N, s1, nf):
+    """
+    Computes the LO gluon-gluon anomalous dimension
+    Implements Eqn. (B.6) from hep-ph/0604160
+
+    Input parameters
+    ----------------
+    N : Mellin moment (type: complex)
+    s1 : harmonic sum $S_{1}$ (type: complex)
+    nf : No. of active flavors (type: int)
+
+    Returns
+    -------
+    gamma_qq : LO quark-quark anomalous dimension $\gamma_{gg}^{(0)}(N)$ (type: complex)
+    """
+    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
+
+@nb.njit(cache=True)
+def gamma_ns(N, s1):
+    """
+    Computes the LO non-singlet anomalous dimension
+    At LO, $\gamma_{ns}^{(0)} = \gamma_{qq}^{(0)}$
+
+    Input parameters
+    ----------------
+    N : Mellin moment (type: complex)
+    s1 : harmonic sum $S_{1}$ (type: complex)
+
+    Returns
+    -------
+    gamma_ns : LO quark-quark anomalous dimension $\gamma_{ns}^{(0)}(N)$ (type: complex)
+    """
+    result = gamma_qq(N, s1)
+    return result
+
+@nb.njit(cache=True)
+def gamma_singlet(N, s1, nf):
+    """
+    Computes the LO singlet anomalous dimension matrix
+    Implements Eqn. (2.13) from PhysRevD.48.116
+
+    Input Parameters
+    ----------------
+    N : Mellin moment (type: complex)
+    s1 : harmonic sum $S_{1}$ (type: complex)
+    nf : No. of active flavors (type: int)
+
+    Returns
+    -------
+    gamma_singlet : LO singlet anomalous dimension matrix $\gamma_{s}^{(0)}$ (type: numpy.array)
+    """
+    result = np.array([[gamma_qq(N, s1), gamma_gq(N, nf)], [gamma_qg(N), gamma_gg(N, s1, nf)]], np.complex_)
+    return result