r"""The unpolarized, spacelike |NNLO| |OME|.
See, :cite:`Buza_1998` appendix B.
The expression for :math:`\mu_F^2 = m_H^2` are taken from :cite:`Vogt:2004ns` directly in N space.
While the parts proportional to :math:`\ln(\mu_F^2 / m_h^2)` comes |QCDNUM|
(https://github.com/N3PDF/external/blob/master/qcdnum/qcdnum/pij/ome.f)
and Mellin transformed with Mathematica.
The expression for ``A_Hg_l0`` comes form :cite:`Bierenbaum:2009zt`.
"""
import numba as nb
import numpy as np
from eko import constants
from eko.constants import zeta2, zeta3
from ....harmonics import cache as c
from .as1 import A_gg as A_gg_1
from .as1 import A_hg as A_hg_1
[docs]
@nb.njit(cache=True)
def A_qq_ns(n, cache, L):
r"""|NNLO| light-light non-singlet |OME| :math:`A_{qq,H}^{NS,(2)}`.
It is given in :eqref:`B.4` of :cite:`Buza_1998`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NNLO| light-light non-singlet |OME| :math:`A_{qq,H}^{NS,(2)}`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
S3 = c.get(c.S3, cache, n)
S1m = S1 - 1 / n # harmonic_S1(n - 1)
S2m = S2 - 1 / n**2 # harmonic_S2(n - 1)
a_qq_l0 = (
-224.0 / 27.0 * (S1 - 1.0 / n)
- 8.0 / 3.0 * zeta3
+ 40 / 9.0 * zeta2
+ 73.0 / 18.0
+ 44.0 / 27.0 / n
- 268.0 / 27.0 / (n + 1.0)
+ 8.0 / 3.0 * (-1.0 / n**2 + 1.0 / (n + 1.0) ** 2)
+ 20.0 / 9.0 * (S2 - 1.0 / n**2 - zeta2 + S2 + 1.0 / (n + 1.0) ** 2 - zeta2)
+ 2.0
/ 3.0
* (-2.0 * (S3 - 1.0 / n**3 - zeta3) - 2.0 * (S3 + 1.0 / (n + 1.0) ** 3 - zeta3))
)
a_qq_l1 = (
2 * (-12 - 28 * n + 9 * n**2 + 34 * n**3 - 3 * n**4) / (9 * (n * (n + 1)) ** 2)
+ 80 / 9 * S1m
- 16 / 3 * S2m
)
a_qq_l2 = -2 * ((2 + n - 3 * n**2) / (3 * n * (n + 1)) + 4 / 3 * S1m)
return constants.CF * constants.TR * (a_qq_l2 * L**2 + a_qq_l1 * L + a_qq_l0)
[docs]
@nb.njit(cache=True)
def A_hq_ps(n, cache, L):
r"""|NNLO| heavy-light pure-singlet |OME| :math:`A_{Hq}^{PS,(2)}`.
It is given in :eqref:`B.1` of :cite:`Buza_1998`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NNLO| heavy-light pure-singlet |OME| :math:`A_{Hq}^{PS,(2)}`
"""
S2 = c.get(c.S2, cache, n)
F1M = 1.0 / (n - 1.0) * (zeta2 - (S2 - 1.0 / n**2))
F11 = 1.0 / (n + 1.0) * (zeta2 - (S2 + 1.0 / (n + 1.0) ** 2))
F12 = 1.0 / (n + 2.0) * (zeta2 - (S2 + 1.0 / (n + 1.0) ** 2 + 1.0 / (n + 2.0) ** 2))
F21 = -F11 / (n + 1.0)
a_hq_l0 = (
-(
32.0 / 3.0 / (n - 1.0)
+ 8.0 * (1.0 / n - 1.0 / (n + 1.0))
- 32.0 / 3.0 * 1.0 / (n + 2.0)
)
* zeta2
- 448.0 / 27.0 / (n - 1.0)
- 4.0 / 3.0 / n
- 124.0 / 3.0 * 1.0 / (n + 1.0)
+ 1600.0 / 27.0 / (n + 2.0)
- 4.0 / 3.0 * (-6.0 / n**4 - 6.0 / (n + 1.0) ** 4)
+ 2.0 * 2.0 / n**3
+ 10.0 * 2.0 / (n + 1.0) ** 3
+ 16.0 / 3.0 * 2.0 / (n + 2.0) ** 3
- 16.0 * zeta2 * (-1.0 / n**2 - 1.0 / (n + 1.0) ** 2)
+ 56.0 / 3.0 / n**2
+ 88.0 / 3.0 / (n + 1.0) ** 2
+ 448.0 / 9.0 / (n + 2.0) ** 2
+ 32.0 / 3.0 * F1M
+ 8.0 * ((zeta2 - S2) / n - F11)
- 32.0 / 3.0 * F12
+ 16.0 * (-(zeta2 - S2) / n**2 + F21)
)
a_hq_l1 = (
8
* (2 + n * (5 + n))
* (4 + n * (4 + n * (7 + 5 * n)))
/ ((n - 1) * (n + 2) ** 2 * (n * (n + 1)) ** 3)
)
a_hq_l2 = -4 * (2 + n + n**2) ** 2 / ((n - 1) * (n + 2) * (n * (n + 1)) ** 2)
return constants.CF * constants.TR * (a_hq_l2 * L**2 + a_hq_l1 * L + a_hq_l0)
[docs]
@nb.njit(cache=True)
def A_hg(n, cache, L):
r"""|NNLO| heavy-gluon |OME| :math:`A_{Hg}^{S,(2)}`.
It is given in :eqref:`B.3` of :cite:`Buza_1998`.
The expession for ``A_Hg_l0`` comes form :cite:`Bierenbaum:2009zt`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NNLO| heavy-gluon |OME| :math:`A_{Hg}^{S,(2)}`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
Sm2 = c.get(c.Sm2, cache, n, is_singlet=True)
S3 = c.get(c.S3, cache, n)
Sm21 = c.get(c.Sm21, cache, n, is_singlet=True)
Sm3 = c.get(c.Sm3, cache, n, is_singlet=True)
S1m = S1 - 1 / n
S2m = S2 - 1 / n**2
a_hg_l0 = (
-(
3084
+ 192 / n**4
+ 1056 / n**3
+ 2496 / n**2
+ 2928 / n
+ 2970 * n
+ 1782 * n**2
+ 6 * n**3
- 1194 * n**4
- 1152 * n**5
- 516 * n**6
- 120 * n**7
- 12 * n**8
)
/ ((n - 1) * ((1 + n) * (2 + n)) ** 4)
+ (
764
- 16 / n**4
- 80 / n**3
- 100 / n**2
+ 3 * 72 / n
+ 208 * n**3
+ 3 * (288 * n + 176 * n**2 + 16 * n**4)
)
/ (3 * (1 + n) ** 4 * (2 + n))
+ 12 * Sm3 * (2 + n + n**2) / (n * (1 + n) * (2 + n))
- 24 * Sm2 * (4 + n - n**2) / ((1 + n) * (2 + n)) ** 2
- S1
* (48 / n + 432 + 564 * n + 324 * n**2 + 138 * n**3 + 48 * n**4 + 6 * n**5)
/ ((1 + n) * (2 + n)) ** 3
+ S1
* (-160 - 32 / n**2 - 80 / n + 8 * n * (n - 1))
/ (3 * (1 + n) ** 2 * (2 + n))
- 6 * S1**2 * (11 + 8 * n + n**2 + 2 / n) / ((1 + n) * (2 + n)) ** 2
+ 8 * S1**2 * (2 / (3 * n) + 1) / (n * (2 + n))
- 2
* S2
* (63 + 48 / n**2 + 54 / n + 39 * n + 63 * n**2 + 21 * n**3)
/ ((n - 1) * (1 + n) ** 2 * (2 + n) ** 2)
+ 8 * S2 * (17 - 2 / n**2 - 5 / n + n * (17 + n)) / (3 * (1 + n) ** 2 * (2 + n))
+ (1 + 2 / n + n)
/ ((1 + n) * (2 + n))
* (24 * Sm2 * S1 + 10 * S1**3 / 9 + 46 * S1 * S2 / 3 + 176 * S3 / 9 - 24 * Sm21)
)
a_hg_l1 = (
2
* (
+640
+ 2192 * n
+ 2072 * n**2
+ 868 * n**3
+ 518 * n**4
+ 736 * n**5
+ 806 * n**6
+ 542 * n**7
+ 228 * n**8
+ 38 * n**9
)
/ (3 * (n * (n + 1) * (n + 2)) ** 3 * (n - 1))
)
a_hg_l1 -= (
2
* (
n
* (n**2 - 1)
* (n + 2)
* (
4 * (36 + n * (88 + n * (33 + n * (8 + 9 * n)))) * S1m
+ n
* (n + 1)
* (n + 2)
* (2 + n + n**2)
* (10 * S1m**2 + 18 * (2 * Sm2 + zeta2) + 26 * S2m)
)
)
/ (3 * (n * (n + 1) * (n + 2)) ** 3 * (n - 1))
)
a_hg_l1 += 12 * zeta2 * (-2 + n + n**3) / (n * (n**2 - 1) * (n + 2))
a_hg_l2 = (
4
* (2 + n + n**2)
* (2 * (-11 + n + n**2) * (1 + n + n**2) / (n - 1))
/ (3 * (n * (n + 1) * (n + 2)) ** 2)
) + 20 * (2 + n + n**2) * S1 / (3 * n * (n + 1) * (n + 2))
return a_hg_l2 * L**2 + a_hg_l1 * L + a_hg_l0
[docs]
@nb.njit(cache=True)
def A_gq(n, cache, L):
r"""|NNLO| gluon-quark |OME| :math:`A_{gq,H}^{S,(2)}`.
It is given in :eqref:`B.5` of :cite:`Buza_1998`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NNLO| gluon-quark |OME| :math:`A_{gq,H}^{S,(2)}`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
S1m = S1 - 1 / n # harmonic_S1(n - 1)
B2M = ((S1 - 1.0 / n) ** 2 + S2 - 1.0 / n**2) / (n - 1.0)
B21 = ((S1 + 1.0 / (n + 1.0)) ** 2 + S2 + 1.0 / (n + 1.0) ** 2) / (n + 1.0)
a_gq_l0 = (
4.0 / 3.0 * (2.0 * B2M - 2.0 * (S1**2 + S2) / n + B21)
+ 8.0
/ 9.0
* (
-10.0 * (S1 - 1.0 / n) / (n - 1.0)
+ 10.0 * S1 / n
- 8.0 * (S1 + 1.0 / (n + 1.0)) / (n + 1.0)
)
+ 1.0 / 27.0 * (448.0 * (1.0 / (n - 1.0) - 1.0 / n) + 344.0 / (n + 1.0))
)
a_gq_l1 = -(
-96
+ 16 * n * (7 + n * (21 + 10 * n + 8 * n**2))
- 48 * n * (1 + n) * (2 + n + n**2) * S1m
) / (9 * (n - 1) * (n * (1 + n)) ** 2)
a_gq_l2 = 8 * (2 + n + n**2) / (3 * n * (n**2 - 1))
return constants.CF * constants.TR * (a_gq_l2 * L**2 + a_gq_l1 * L + a_gq_l0)
[docs]
@nb.njit(cache=True)
def A_gg(n, cache, L):
r"""|NNLO| gluon-gluon |OME| :math:`A_{gg,H}^{S,(2)}`.
It is given in :eqref:`B.7` of :cite:`Buza_1998`.
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
complex
|NNLO| gluon-gluon |OME| :math:`A_{gg,H}^{S,(2)}`
"""
S1 = c.get(c.S1, cache, n)
S1m = S1 - 1 / n # harmonic_S1(n - 1)
D1 = -1.0 / n**2
D11 = -1.0 / (n + 1.0) ** 2
D2 = 2.0 / n**3
D21 = 2.0 / (n + 1.0) ** 3
a_gg_f = (
-15.0
- 8.0 / (n - 1.0)
+ 80.0 / n
- 48.0 / (n + 1.0)
- 24.0 / (n + 2.0)
+ 4.0 / 3.0 * (-6.0 / n**4 - 6.0 / (n + 1.0) ** 4)
+ 6.0 * D2
+ 10.0 * D21
+ 32.0 * D1
+ 48.0 * D11
)
a_gg_a = (
-224.0 / 27.0 * (S1 - 1.0 / n)
+ 10.0 / 9.0
+ 4.0 / 3.0 * (S1 + 1.0 / (n + 1.0)) / (n + 1.0)
+ 1.0
/ 27.0
* (556.0 / (n - 1.0) - 628.0 / n + 548.0 / (n + 1.0) - 700.0 / (n + 2.0))
+ 4.0 / 3.0 * (D2 + D21)
+ 1.0 / 9.0 * (52.0 * D1 + 88.0 * D11)
)
a_gg_l0 = constants.TR * (constants.CF * a_gg_f + constants.CA * a_gg_a)
a_gg_l1 = (
8
/ 3
* (
(
8
+ 2 * n
- 34 * n**2
- 72 * n**3
- 77 * n**4
- 37 * n**5
- 19 * n**6
- 11 * n**7
- 4 * n**8
)
/ ((n * (n + 1)) ** 3 * (-2 + n + n**2))
+ 5 * S1m
)
)
a_gg_l2 = (
4
/ 9
* (
1
+ 6 * (2 + n + n**2) ** 2 / ((n * (n + 1)) ** 2 * (-2 + n + n**2))
- 9 * (-4 - 3 * n + n**3) / (n * (n + 1) * (-2 + n + n**2))
)
- 4 * S1m
)
return a_gg_l2 * L**2 + a_gg_l1 * L + a_gg_l0
[docs]
@nb.njit(cache=True)
def A_singlet(n, cache, L, is_msbar=False):
r"""Compute the |NNLO| singlet |OME|.
.. math::
A^{S,(2)} = \left(\begin{array}{cc}
A_{gg, H}^{S,(2)} & A_{gq, H}^{S,(2)} & 0 \\
0 & A_{qq,H}^{NS,(2)} & 0\\
A_{hg}^{S,(2)} & A_{hq}^{PS,(2)} & 0\\
\end{array}\right)
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
is_msbar: bool
add the |MSbar| contribution
Returns
-------
numpy.ndarray
|NNLO| singlet |OME| :math:`A^{S,(2)}(N)`
"""
A_hq_2 = A_hq_ps(n, cache, L)
A_qq_2 = A_qq_ns(n, cache, L)
A_hg_2 = A_hg(n, cache, L)
A_gq_2 = A_gq(n, cache, L)
A_gg_2 = A_gg(n, cache, L)
if is_msbar:
A_hg_2 -= 2.0 * 4.0 * constants.CF * A_hg_1(n, L=1.0)
A_gg_2 -= 2.0 * 4.0 * constants.CF * A_gg_1(L=1.0)
A_S = np.array(
[[A_gg_2, A_gq_2, 0.0], [0.0, A_qq_2, 0.0], [A_hg_2, A_hq_2, 0.0]], np.complex_
)
return A_S
[docs]
@nb.njit(cache=True)
def A_ns(n, cache, L):
r"""Compute the |NNLO| non-singlet |OME|.
.. math::
A^{NS,(2)} = \left(\begin{array}{cc}
A_{qq,H}^{NS,(2)} & 0 \\
0 & 0 \\
\end{array}\right)
Parameters
----------
n : complex
Mellin moment
cache: numpy.ndarray
Harmonic sum cache
L : float
:math:`\ln(\mu_F^2 / m_h^2)`
Returns
-------
numpy.ndarray
|NNLO| non-singlet |OME| :math:`A^{NS,(2)}`
"""
return np.array([[A_qq_ns(n, cache, L), 0.0], [0 + 0j, 0 + 0j]], np.complex_)