"""The |NNLO| polarized Altarelli-Parisi splitting kernels."""
import numba as nb
import numpy as np
from eko.constants import zeta2
from ....harmonics import cache as c
# Non Singlet sector is swapped
from ...unpolarized.space_like.as3 import gamma_nsm as gamma_nsp
from ...unpolarized.space_like.as3 import gamma_nsp as gamma_nsm
[docs]
@nb.njit(cache=True)
def gamma_gg(N, nf, cache):
r"""Compute the parametrized |NNLO| gluon-gluon polarized anomalous
dimension.
Implement Eq. (4.12) of :cite:`Moch:2014sna`.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| gluon-gluon anomalous dimension :math:`\\gamma_{gg}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
E1 = S1 / N**2 + (-zeta2 + S2) / N
gg_nf0 = (
+4427.76
- 12292 * E1
+ 12096 / N**5
- 22665.0 / N**4
+ 21804 / N**3
- 23091 / N**2
+ 33631.5 / N
- 7002 / (1 + N) ** 4
- 1726 / (1 + N) ** 3
- 13247 / (1 + N) ** 2
- 39925 / (1 + N)
+ 13447 / (2 + N)
- 4576 / (3 + N)
- 2643.52 * S1
+ (9446 * S1) / N
- (13247 * S1) / (1 + N)
)
gg_nf1 = (
-528.536
- 7932 * E1
- 6128 / (9 * N**5)
+ 2146.79 / N**4
- 3754.4 / N**3
+ 3524 / N**2
- 1585.67 / N
- 786.0 / (1 + N) ** 4
+ 1226.2 / (1 + N) ** 3
- 6746 / (1 + N) ** 2
+ 2648.6 / (1 + N)
- 2160.8 / (2 + N)
+ 1251.7 / (3 + N)
+ 412.172 * S1
+ (7041.7 * S1) / N
- (6746 * S1) / (1 + N)
)
gg_nf2 = (
6.4607
- 16.944 * E1
+ 7.0854 / N**4
- 13.358 / N**3
+ 13.29 / N**2
- 18.3838 / N
+ 31.528 / (1 + N) ** 3
+ 32.905 / (1 + N)
- 18.3 / (2 + N)
+ 2.637 / (3 + N)
+ (16 * S1) / 9
+ (0.21 * S1) / N
)
return -(gg_nf0 + gg_nf1 * nf + gg_nf2 * nf**2)
[docs]
@nb.njit(cache=True)
def gamma_qg(N, nf, cache):
r"""Compute the parametrized |NNLO| quark-gluon polarized anomalous
dimension.
Implement Eq. (4.10) of :cite:`Moch:2014sna`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| quark-gluon anomalous dimension :math:`\\gamma_{qg}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
S3 = c.get(c.S3, cache, N)
S4 = c.get(c.S4, cache, N)
B3 = (-(S1**3) - 3 * S1 * S2 - 2 * S3) / N
B4 = (S1**4 + 6 * S1**2 * S2 + 3 * S2**2 + 8 * S1 * S3 + 6 * S4) / N
E1 = S1 / N**2 + (-zeta2 + S2) / N
qg_nf1 = (
-5.3 * B3
+ 3.784 * B4
+ 825.4 * E1
- 1208 / N**5
+ 2313.84 / N**4
- 1789.6 / N**3
+ 1461.2 / N**2
- 2972.4 / N
+ 439.8 / (1 + N) ** 4
+ 2290.6 / (1 + N) ** 3
+ 4672 / (1 + N)
- 1221.6 / (2 + N)
- 18 / (3 + N)
- (278.32 * S1) / N
- (90.26 * S1**2) / N
- (90.26 * S2) / N
)
qg_nf2 = (
0.7374 * B3
- 47.3 * E1
+ 128 / (3 * N**5)
- 184.434 / N**4
+ 393.92 / N**3
- 526.3 / N**2
+ 499.65 / N
- 61.116 / (1 + N) ** 4
+ 358.2 / (1 + N) ** 3
- 432.18 / (1 + N)
- 141.63 / (2 + N)
- 11.34 / (3 + N)
+ (6.256 * S1) / N
+ (7.32 * S1**2) / N
+ (7.32 * S2) / N
)
return -(qg_nf1 * nf + qg_nf2 * nf**2)
[docs]
@nb.njit(cache=True)
def gamma_gq(N, nf, cache):
r"""Compute the parametrized |NNLO| gluon-quark polarized anomalous
dimension.
Implement Eq. (4.11) of :cite:`Moch:2014sna`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| gluon-quark anomalous dimension :math:`\\gamma_{gq}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
S3 = c.get(c.S3, cache, N)
S4 = c.get(c.S4, cache, N)
B3 = (-(S1**3) - 3 * S1 * S2 - 2 * S3) / N
B4 = (S1**4 + 6 * S1**2 * S2 + 3 * S2**2 + 8 * S1 * S3 + 6 * S4) / N
E1 = S1 / N**2 + (-zeta2 + S2) / N
gq_nf0 = (
59.3 * B3
+ 5.143 * B4
- 1424.8 * E1
+ 92096 / (27 * N**5)
- 5328.02 / N**4
+ 4280 / N**3
- 4046.6 / N**2
+ 6159 / N
- 1050.6 / (1 + N) ** 4
- 1701.4 / (1 + N) ** 3
- 3825.9 / (1 + N)
+ 1942.0 / (2 + N)
- 742.1 / (3 + N)
- (1843.7 * S1) / N
+ (451.55 * S1**2) / N
+ (451.55 * S2) / N
)
gq_nf1 = (
-4.963 * B3
- 16.18 * E1
- 1024 / (9 * N**5)
+ 236.323 / N**4
- 404.92 / N**3
+ 308.98 / N**2
- 301.07 / N
+ 180.138 / (1 + N) ** 4
- 253.06 / (1 + N) ** 3
- 296 / (1 + N)
+ 406.13 / (2 + N)
- 101.62 / (3 + N)
+ (171.78 * S1) / N
- (47.86 * S1**2) / N
- (47.86 * S2) / N
)
gq_nf2 = (
-(64 / (9 * N))
- 32 / (9 * (1 + N) ** 3)
- 32 / (27 * (1 + N) ** 2)
+ 160 / (27 * (1 + N))
- (128 * S1) / (27 * N)
- (32 * S1) / (9 * (1 + N) ** 2)
- (32 * S1) / (27 * (1 + N))
+ (32 * S1**2) / (9 * N)
- (16 * S1**2) / (9 * (1 + N))
+ (32 * S2) / (9 * N)
- (16 * S2) / (9 * (1 + N))
)
return -(gq_nf0 + gq_nf1 * nf + gq_nf2 * nf**2)
[docs]
@nb.njit(cache=True)
def gamma_ps(N, nf, cache):
r"""Compute the parametrized |NNLO| pure-singlet quark-quark polarized
anomalous dimension.
Implement Eq. (4.9) of :cite:`Moch:2014sna`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| pure-singlet quark-quark anomalous dimension :math:`\\gamma_{ps}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
S3 = c.get(c.S3, cache, N)
B3 = (-(S1**3) - 3 * S1 * S2 - 2 * S3) / N
B31 = (
-((1 / (1 + N) + S1) ** 3)
- 3 * (1 / (1 + N) + S1) * (1 / (1 + N) ** 2 + S2)
- 2 * (1 / (1 + N) ** 3 + S3)
) / (1 + N)
E1 = S1 / N**2 + (-zeta2 + S2) / N
E11 = (1 / (1 + N) + S1) / (1 + N) ** 2 + (1 / (1 + N) ** 2 - zeta2 + S2) / (1 + N)
ps_nf1 = (
1
/ 9
* (
-58.869 * B3
+ 58.869 * B31
+ 2093.13 * E1
- 2093.13 * E11
- 2752 / N**5
+ 4909.67 / N**4
- 6634.8 / N**3
+ 6651.0 / N**2
- 12263.4 / N
+ 2752 / (1 + N) ** 5
- 508.669 / (1 + N) ** 4
+ 13160.0 / (1 + N) ** 3
- 8493.84 / (1 + N) ** 2
+ 26820.0 / (1 + N)
- 4401.0 / (2 + N) ** 4
- 6298.2 / (2 + N) ** 3
- 20629.8 / (2 + N)
+ 7579.89 / (3 + N)
- 1506.69 / (4 + N)
+ (1842.84 * S1) / N
+ (226.98 * S1) / (1 + N) ** 2
- (1842.84 * S1) / (1 + N)
- (113.49 * S1**2) / N
+ (113.49 * S1**2) / (1 + N)
- (113.49 * S2) / N
+ (113.49 * S2) / (1 + N)
)
)
ps_nf2 = (
1
/ 9
* (
-(63.4014 / N**4)
+ 239.166 / N**3
- 409.338 / N**2
+ 442.17 / N
+ 107.968 / (1 + N) ** 4
- 79.389 / (1 + N) ** 3
+ 494.991 / (1 + N) ** 2
- 719.1 / (1 + N)
- 44.5662 / (2 + N) ** 4
- 191.826 / (2 + N) ** 3
+ 238.167 / (2 + N)
+ 34.1784 / (3 + N)
+ 4.5846 / (4 + N)
- (85.653 * S1) / N
- (32.049 * S1) / (1 + N) ** 2
+ (85.653 * S1) / (1 + N)
+ (16.0245 * S1**2) / N
- (16.0245 * S1**2) / (1 + N)
+ (16.0245 * S2) / N
- (16.0245 * S2) / (1 + N)
)
)
return -(ps_nf1 * nf + ps_nf2 * nf**2)
[docs]
@nb.njit(cache=True)
def gamma_nss(N, nf, cache):
r"""Compute the |NNLO| sea-like polarized non-singlet anomalous dimension.
Implement Eq. (24) of :cite:`Moch:2015usa`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| valence non-singlet anomalous dimension
:math:`\\gamma_{ns,s}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S3 = c.get(c.S3, cache, N)
Sm2 = c.get(c.Sm2, cache, N, is_singlet=True)
Sm3 = c.get(c.Sm3, cache, N, is_singlet=True)
Sm21 = c.get(c.Sm21, cache, N, is_singlet=True)
nss_nf1 = (
40
/ 9
* (
1 / N**2
+ (
-(2 / N**2)
- 12 / (N**4 * (1 + N) ** 4)
- 42 / (N**3 * (1 + N) ** 3)
- 14 / (N**2 * (1 + N) ** 2)
+ 8 / (N * (1 + N))
)
* S1
+ (4 / (N**2 * (1 + N) ** 2) + 6 / (N * (1 + N))) * S3
+ (
-(4 / N**2)
+ 8 / (N**3 * (1 + N) ** 3)
+ 20 / (N**2 * (1 + N) ** 2)
+ 8 / (N * (1 + N))
)
* Sm2
+ (32 * Sm21) / (N * (1 + N))
+ (8 / (N**2 * (1 + N) ** 2) - 20 / (N * (1 + N))) * Sm3
+ (-(16 / (N**2 * (1 + N) ** 2)) + 8 / (N * (1 + N)))
* (S1 * Sm2 - Sm21 + Sm3)
)
)
return -(nss_nf1 * nf)
[docs]
@nb.njit(cache=True)
def gamma_nsv(N, nf, cache):
r"""Compute the |NNLO| valence polarized non-singlet anomalous dimension.
Implement Eq. (23) of :cite:`Moch:2015usa`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| valence non-singlet anomalous dimension
:math:`\\gamma_{ns,v}^{(2)}(N)`
"""
return gamma_nsm(N, nf, cache) + gamma_nss(N, nf, cache)
[docs]
@nb.njit(cache=True)
def gamma_singlet(N, nf, cache):
r"""Compute the |NNLO| polarized singlet anomalous dimension matrix.
.. math::
\gamma_S^{(1)} = \left(\begin{array}{cc}
\gamma_{qq}^{(2)} & \gamma_{qg}^{(2)}\\
\gamma_{gq}^{(2)} & \gamma_{gg}^{(2)}
\end{array}\right)
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
|NNLO| singlet anomalous dimension matrix :math:`\gamma_{S}^{(2)}(N)`
"""
gamma_qq = gamma_nsp(N, nf, cache) + gamma_ps(N, nf, cache)
gamma_S_0 = np.array(
[
[gamma_qq, gamma_qg(N, nf, cache)],
[gamma_gq(N, nf, cache), gamma_gg(N, nf, cache)],
],
np.complex128,
)
return gamma_S_0