"""The |NLO| Altarelli-Parisi splitting kernels."""
import numba as nb
import numpy as np
from eko.constants import CA, CF, TR, zeta2, zeta3
from ....harmonics import cache as c
# Non Singlet sector is swapped
from ...unpolarized.space_like.as2 import gamma_nsm as gamma_nsp
from ...unpolarized.space_like.as2 import gamma_nsp as gamma_nsm # noqa
[docs]
@nb.njit(cache=True)
def gamma_ps(n, nf):
r"""Compute the |NLO| polarized pure-singlet quark-quark anomalous dimension
:cite:`Gluck:1995yr` (eq A.3).
Parameters
----------
n : complex
Mellin moment
nf : int
number of active flavors
Returns
-------
complex
|NLO| pure-singlet quark-quark anomalous dimension :math:`\\gamma_{ps}^{(1)}(n)`
"""
gqqps1_nfcf = (2 * (n + 2) * (1 + 2 * n + n**3)) / ((1 + n) ** 3 * n**3)
result = 4.0 * TR * nf * CF * gqqps1_nfcf
return result
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@nb.njit(cache=True)
def gamma_qg(n, nf, cache):
r"""Compute the |NLO| polarized quark-gluon singlet anomalous dimension
:cite:`Gluck:1995yr` (eq A.4).
Parameters
----------
n : complex
Mellin moment
nf : int
number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NLO| quark-gluon singlet anomalous dimension :math:`\\gamma_{qg}^{(1)}(n)`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
Sp2m = c.get(c.S2mh, cache, n)
gqg1_nfca = (
(S1**2 - S2 + Sp2m) * (n - 1) / (n * (n + 1))
- 4 * S1 / (n * (1 + n) ** 2)
- (-2 - 7 * n + 3 * n**2 - 4 * n**3 + n**4 + n**5) / (n**3 * (1 + n) ** 3)
) * 2.0
gqg1_nfcf = (
(-(S1**2) + S2 + 2 * S1 / n) * (n - 1) / (n * (n + 1))
- (n - 1)
* (1 + 3.5 * n + 4 * n**2 + 5 * n**3 + 2.5 * n**4)
/ (n**3 * (1 + n) ** 3)
+ 4 * (n - 1) / (n**2 * (1 + n) ** 2)
) * 2
result = 4.0 * TR * nf * (CA * gqg1_nfca + CF * gqg1_nfcf)
return result
[docs]
@nb.njit(cache=True)
def gamma_gq(n, nf, cache):
r"""Compute the |NLO| polarized gluon-quark singlet anomalous dimension
:cite:`Gluck:1995yr` (eq A.5).
Parameters
----------
n : complex
Mellin moment
nf : int
number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NLO| gluon-quark singlet anomalous dimension :math:`\\gamma_{gq}^{(1)}(n)`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
Sp2m = c.get(c.S2mh, cache, n)
ggq1_cfcf = (
(2 * (S1**2 + S2) * (n + 2)) / (n * (n + 1))
- (2 * S1 * (n + 2) * (1 + 3 * n)) / (n * (1 + n) ** 2)
- ((n + 2) * (2 + 15 * n + 8 * n**2 - 12.0 * n**3 - 9.0 * n**4))
/ (n**3 * (1 + n) ** 3)
+ 8 * (n + 2) / (n**2 * (1 + n) ** 2)
) * 0.5
ggq1_cfca = -(
-(-(S1**2) - S2 + Sp2m) * (n + 2) / (n * (n + 1))
- S1 * (12 + 22 * n + 11 * n**2) / (3 * n**2 * (n + 1))
+ (36 + 72 * n + 41 * n**2 + 254 * n**3 + 271 * n**4 + 76 * n**5)
/ (9 * n**3 * (1 + n) ** 3)
)
ggq1_cfnf = (-S1 * (n + 2)) / (3 * n * (n + 1)) + ((n + 2) * (2 + 5 * n)) / (
9 * n * (1 + n) ** 2
)
result = 4 * CF * (CA * ggq1_cfca + CF * ggq1_cfcf + 4.0 * TR * nf * ggq1_cfnf)
return result
[docs]
@nb.njit(cache=True)
def gamma_gg(n, nf, cache):
r"""Compute the |NLO| polarized gluon-gluon singlet anomalous dimension
:cite:`Gluck:1995yr` (eq A.6).
Parameters
----------
n : complex
Mellin moment
nf : int
number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NLO| gluon-quark singlet anomalous dimension :math:`\\gamma_{gq}^{(1)}(n)`
"""
S1 = c.get(c.S1, cache, n)
Sp1m = c.get(c.S1mh, cache, n)
Sp2m = c.get(c.S2mh, cache, n)
Sp3m = c.get(c.S3mh, cache, n)
S1h = c.get(c.S1h, cache, n)
g3 = c.get(c.g3, cache, n)
SSCHLM = zeta2 / 2 * (+Sp1m - S1h + 2 / n) - S1 / n**2 - g3 - 5 * zeta3 / 8
ggg1_caca = (
-4 * S1 * Sp2m
- Sp3m
+ 8 * SSCHLM
+ 8 * Sp2m / (n * (n + 1))
+ 2.0
* S1
* (72 + 144 * n + 67 * n**2 + 134 * n**3 + 67 * n**4)
/ (9 * n**2 * (n + 1) ** 2)
- (144 + 258 * n + 7 * n**2 + 698 * n**3 + 469 * n**4 + 144 * n**5 + 48 * n**6)
/ (9 * n**3 * (1 + n) ** 3)
) * 0.5
ggg1_canf = (
-5 * S1 / 9
+ (-3 + 13 * n + 16 * n**2 + 6 * n**3 + 3 * n**4) / (9 * n**2 * (1 + n) ** 2)
) * 4
ggg1_cfnf = (4 + 2 * n - 8 * n**2 + n**3 + 5 * n**4 + 3 * n**5 + n**6) / (
n**3 * (1 + n) ** 3
)
result = 4 * (CA**2 * ggg1_caca + TR * nf * (CA * ggg1_canf + CF * ggg1_cfnf))
return result
[docs]
@nb.njit(cache=True)
def gamma_singlet(n, nf, cache):
r"""Compute the |NLO| polarized singlet anomalous dimension matrix.
.. math::
\gamma_S^{(1)} = \left(\begin{array}{cc}
\gamma_{qq}^{(1)} & \gamma_{qg}^{(1)}\\
\gamma_{gq}^{(1)} & \gamma_{gg}^{(1)}
\end{array}\right)
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
|NLO| singlet anomalous dimension matrix :math:`\gamma_{S}^{(1)}(N)`
"""
gamma_qq = gamma_nsp(n, nf, cache) + gamma_ps(n, nf)
gamma_S_0 = np.array(
[
[gamma_qq, gamma_qg(n, nf, cache)],
[gamma_gq(n, nf, cache), gamma_gg(n, nf, cache)],
],
np.complex_,
)
return gamma_S_0