"""Compute the |NNLO| Altarelli-Parisi splitting kernels.
The expression have been obtained from :cite:`Moch:2004pa,Vogt:2004ns`.
Note that the QCD colour factors have been hard-wired in the
parametrizations.
"""
import numba as nb
import numpy as np
from eko.constants import zeta2, zeta3
from ....harmonics import cache as c
[docs]
@nb.njit(cache=True)
def gamma_nsm(n, nf, cache):
r"""Compute the |NNLO| valence-like non-singlet anomalous dimension.
Implements Eq. (3.8) of :cite:`Moch:2004pa`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| valence-like non-singlet anomalous dimension
:math:`\\gamma_{ns,-}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
S3 = c.get(c.S3, cache, n)
E1 = S1 / n**2 + (S2 - zeta2) / n
E2 = 2.0 * (-S1 / n**3 + (zeta2 - S2) / n**2 - (S3 - zeta3) / n)
pm2 = (
-1174.898 * (S1 - 1.0 / n)
+ 1295.470
- 714.1 * S1 / n
- 433.2 / (n + 3.0)
+ 297.0 / (n + 2.0)
- 3505.0 / (n + 1.0)
+ 1860.2 / n
- 1465.2 / n**2
+ 399.2 * 2.0 / n**3
- 320.0 / 9.0 * 6.0 / n**4
+ 116.0 / 81.0 * 24.0 / n**5
+ 684.0 * E1
+ 251.2 * E2
)
pm2_nf = (
+183.187 * (S1 - 1.0 / n)
- 173.933
+ 5120 / 81.0 * S1 / n
+ 34.76 / (n + 3.0)
+ 77.89 / (n + 2.0)
+ 406.5 / (n + 1.0)
- 216.62 / n
+ 172.69 / n**2
- 3216.0 / 81.0 * 2.0 / n**3
+ 256.0 / 81.0 * 6.0 / n**4
- 65.43 * E1
+ 1.136 * 6.0 / (n + 1.0) ** 4
)
pf2_nfnf = (
-(
17.0 / 72.0
- 2.0 / 27.0 * S1
- 10.0 / 27.0 * S2
+ 2.0 / 9.0 * S3
- (12.0 * n**4 + 2.0 * n**3 - 12.0 * n**2 - 2.0 * n + 3.0)
/ (27.0 * n**3 * (n + 1.0) ** 3)
)
* 32.0
/ 3.0
)
result = pm2 + nf * pm2_nf + nf**2 * pf2_nfnf
return -result
[docs]
@nb.njit(cache=True)
def gamma_nsp(n, nf, cache):
r"""Compute the |NNLO| singlet-like non-singlet anomalous dimension.
Implements Eq. (3.7) of :cite:`Moch:2004pa`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| singlet-like non-singlet anomalous dimension
:math:`\\gamma_{ns,+}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
S3 = c.get(c.S3, cache, n)
E1 = S1 / n**2 + (S2 - zeta2) / n
E2 = 2.0 * (-S1 / n**3 + (zeta2 - S2) / n**2 - (S3 - zeta3) / n)
pp2 = (
-1174.898 * (S1 - 1.0 / n)
+ 1295.384
- 714.1 * S1 / n
- 522.1 / (n + 3.0)
+ 243.6 / (n + 2.0)
- 3135.0 / (n + 1.0)
+ 1641.1 / n
- 1258.0 / n**2
+ 294.9 * 2.0 / n**3
- 800 / 27.0 * 6.0 / n**4
+ 128 / 81.0 * 24.0 / n**5
+ 563.9 * E1
+ 256.8 * E2
)
pp2_nf = (
+183.187 * (S1 - 1.0 / n)
- 173.924
+ 5120 / 81.0 * S1 / n
+ 44.79 / (n + 3.0)
+ 72.94 / (n + 2.0)
+ 381.1 / (n + 1.0)
- 197.0 / n
+ 152.6 / n**2
- 2608.0 / 81.0 * 2.0 / n**3
+ 192.0 / 81.0 * 6.0 / n**4
- 56.66 * E1
+ 1.497 * 6.0 / (n + 1.0) ** 4
)
pf2_nfnf = (
-(
17.0 / 72.0
- 2.0 / 27.0 * S1
- 10.0 / 27.0 * S2
+ 2.0 / 9.0 * S3
- (12.0 * n**4 + 2.0 * n**3 - 12.0 * n**2 - 2.0 * n + 3.0)
/ (27.0 * n**3 * (n + 1.0) ** 3)
)
* 32.0
/ 3.0
)
result = pp2 + nf * pp2_nf + nf**2 * pf2_nfnf
return -result
[docs]
@nb.njit(cache=True)
def gamma_nsv(n, nf, cache):
r"""Compute the |NNLO| valence non-singlet anomalous dimension.
Implements Eq. (3.9) of :cite:`Moch:2004pa`.
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)`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
S3 = c.get(c.S3, cache, n)
E1 = S1 / n**2 + (S2 - zeta2) / n
E2 = 2.0 * (-S1 / n**3 + (zeta2 - S2) / n**2 - (S3 - zeta3) / n)
B11 = -(S1 + 1.0 / (n + 1.0)) / (n + 1.0)
B12 = -(S1 + 1.0 / (n + 1.0) + 1.0 / (n + 2.0)) / (n + 2.0)
# with special care for the first moment of x^-1 ln(1-x)
if abs(n.imag) < 1.0e-5 and abs(n - 1.0) < 1.0e-5:
B1M = -zeta2
else:
B1M = -(S1 - 1.0 / n) / (n - 1.0)
ps2 = -(
-163.9 * (B1M + S1 / n)
- 7.208 * (B11 - B12)
+ 4.82 * (1.0 / (n + 3.0) - 1.0 / (n + 4.0))
- 43.12 * (1.0 / (n + 2.0) - 1.0 / (n + 3.0))
+ 44.51 * (1.0 / (n + 1.0) - 1.0 / (n + 2.0))
+ 151.49 * (1.0 / n - 1.0 / (n + 1.0))
- 178.04 / n**2
+ 6.892 * 2.0 / n**3
- 40.0 / 27.0 * (-2.0 * 6.0 / n**4 - 24.0 / n**5)
- 173.1 * E1
+ 46.18 * E2
)
result = gamma_nsm(n, nf, cache) + nf * ps2
return result
[docs]
@nb.njit(cache=True)
def gamma_ps(n, nf, cache):
r"""Compute the |NNLO| pure-singlet quark-quark anomalous dimension.
Implements Eq. (3.10) of :cite:`Vogt:2004mw`.
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)
E1 = S1 / n**2 + (S2 - zeta2) / n
E11 = (S1 + 1.0 / (n + 1.0)) / (n + 1.0) ** 2 + (
S2 + 1.0 / (n + 1.0) ** 2 - zeta2
) / (n + 1.0)
B21 = ((S1 + 1.0 / (n + 1.0)) ** 2 + S2 + 1.0 / (n + 1.0) ** 2) / (n + 1.0)
B3 = -(S1**3 + 3.0 * S1 * S2 + 2.0 * S3) / n
B31 = -(
(S1 + 1.0 / (n + 1.0)) ** 3
+ 3.0 * (S1 + 1.0 / (n + 1.0)) * (S2 + 1.0 / (n + 1.0) ** 2)
+ 2.0 * (S3 + 1.0 / (n + 1.0) ** 3)
) / (n + 1.0)
ps1 = (
-3584.0 / 27.0 * (-1.0 / (n - 1.0) ** 2 + 1.0 / n**2)
- 506.0 * (1.0 / (n - 1.0) - 1.0 / n)
+ 160.0 / 27.0 * (24.0 / n**5 - 24.0 / (n + 1.0) ** 5)
- 400.0 / 9.0 * (-6.0 / n**4 + 6.0 / (n + 1.0) ** 4)
+ 131.4 * (2.0 / n**3 - 2.0 / (n + 1.0) ** 3)
- 661.6 * (-1.0 / n**2 + 1.0 / (n + 1.0) ** 2)
- 5.926 * (B3 - B31)
- 9.751 * ((S1**2 + S2) / n - B21)
- 72.11 * (-S1 / n + (S1 + 1.0 / (n + 1.0)) / (n + 1.0))
+ 177.4 * (1.0 / n - 1.0 / (n + 1.0))
+ 392.9 * (1.0 / (n + 1.0) - 1.0 / (n + 2.0))
- 101.4 * (1.0 / (n + 2.0) - 1.0 / (n + 3.0))
- 57.04 * (E1 - E11)
)
ps2 = (
256.0 / 81.0 * (1.0 / (n - 1.0) - 1.0 / n)
+ 32.0 / 27.0 * (-6.0 / n**4 + 6.0 / (n + 1.0) ** 4)
+ 17.89 * (2.0 / n**3 - 2.0 / (n + 1.0) ** 3)
+ 61.75 * (-1.0 / n**2 + 1.0 / (n + 1.0) ** 2)
+ 1.778 * ((S1**2 + S2) / n - B21)
+ 5.944 * (-S1 / n + (S1 + 1.0 / (n + 1.0)) / (n + 1.0))
+ 100.1 * (1.0 / n - 1.0 / (n + 1.0))
- 125.2 * (1.0 / (n + 1.0) - 1.0 / (n + 2.0))
+ 49.26 * (1.0 / (n + 2.0) - 1.0 / (n + 3.0))
- 12.59 * (1.0 / (n + 3.0) - 1.0 / (n + 4.0))
- 1.889 * (E1 - E11)
)
result = nf * ps1 + nf**2 * ps2
return -result
[docs]
@nb.njit(cache=True)
def gamma_qg(n, nf, cache):
r"""Compute the |NNLO| quark-gluon singlet anomalous dimension.
Implements Eq. (3.11) of :cite:`Vogt:2004mw`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| quark-gluon singlet 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)
E1 = S1 / n**2 + (S2 - zeta2) / n
E2 = 2.0 * (-S1 / n**3 + (zeta2 - S2) / n**2 - (S3 - zeta3) / n)
B3 = -(S1**3 + 3.0 * S1 * S2 + 2.0 * S3) / n
B4 = (S1**4 + 6.0 * S1**2 * S2 + 8.0 * S1 * S3 + 3.0 * S2**2 + 6.0 * S4) / n
qg1 = (
+896.0 / 3.0 / (n - 1.0) ** 2
- 1268.3 / (n - 1.0)
+ 536.0 / 27.0 * 24.0 / n**5
+ 44.0 / 3.0 * 6.0 / n**4
+ 881.5 * 2.0 / n**3
- 424.9 / n**2
+ 100.0 / 27.0 * B4
- 70.0 / 9.0 * B3
- 120.5 * (S1**2 + S2) / n
- 104.42 * S1 / n
+ 2522.0 / n
- 3316.0 / (n + 1.0)
+ 2126.0 / (n + 2.0)
+ 1823.0 * E1
- 25.22 * E2
+ 252.5 * 6.0 / (n + 1.0) ** 4
)
qg2 = (
1112.0 / 243.0 / (n - 1.0)
- 16.0 / 9.0 * 24.0 / n**5
+ 376.0 / 27.0 * 6.0 / n**4
- 90.8 * 2.0 / n**3
+ 254.0 / n**2
+ 20.0 / 27.0 * B3
+ 200.0 / 27.0 * (S1**2 + S2) / n
+ 5.496 * S1 / n
- 252.0 / n
+ 158.0 / (n + 1.0)
+ 145.4 / (n + 2.0)
- 139.28 / (n + 3.0)
- 53.09 * E1
- 80.616 * E2
- 98.07 * 2.0 / (n + 1.0) ** 3
- 11.70 * 6.0 / (n + 1.0) ** 4
)
result = nf * qg1 + nf**2 * qg2
return -result
[docs]
@nb.njit(cache=True)
def gamma_gq(n, nf, cache):
r"""Compute the |NNLO| gluon-quark singlet anomalous dimension.
Implements Eq. (3.12) of :cite:`Vogt:2004mw`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| gluon-quark singlet 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)
E1 = S1 / n**2 + (S2 - zeta2) / n
E2 = 2.0 * (-S1 / n**3 + (zeta2 - S2) / n**2 - (S3 - zeta3) / n)
B21 = ((S1 + 1.0 / (n + 1.0)) ** 2 + S2 + 1.0 / (n + 1.0) ** 2) / (n + 1.0)
B3 = -(S1**3 + 3.0 * S1 * S2 + 2.0 * S3) / n
B4 = (S1**4 + 6.0 * S1**2 * S2 + 8.0 * S1 * S3 + 3.0 * S2**2 + 6.0 * S4) / n
gq0 = (
-1189.3 * 1.0 / (n - 1.0) ** 2
+ 6163.1 / (n - 1.0)
- 4288.0 / 81.0 * 24.0 / n**5
- 1568.0 / 9.0 * 6.0 / n**4
- 1794.0 * 2.0 / n**3
- 4033.0 * 1.0 / n**2
+ 400.0 / 81.0 * B4
+ 2200.0 / 27.0 * B3
+ 606.3 * (S1**2 + S2) / n
- 2193.0 * S1 / n
- 4307.0 / n
+ 489.3 / (n + 1.0)
+ 1452.0 / (n + 2.0)
+ 146.0 / (n + 3.0)
- 447.3 * E2
- 972.9 * 2.0 / (n + 1.0) ** 3
)
gq1 = (
-71.082 / (n - 1.0) ** 2
- 46.41 / (n - 1.0)
+ 128.0 / 27.0 * 24.0 / n**5
- 704 / 81.0 * 6.0 / n**4
+ 20.39 * 2.0 / n**3
- 174.8 * 1.0 / n**2
- 400.0 / 81.0 * B3
- 68.069 * (S1**2 + S2) / n
+ 296.7 * S1 / n
- 183.8 / n
+ 33.35 / (n + 1.0)
- 277.9 / (n + 2.0)
+ 108.6 * 2.0 / (n + 1.0) ** 3
- 49.68 * E1
)
gq2 = (
64.0 * (-1.0 / (n - 1.0) + 1.0 / n + 2.0 / (n + 1.0))
+ 320.0
* (
-(S1 - 1.0 / n) / (n - 1.0)
+ S1 / n
- 0.8 * (S1 + 1.0 / (n + 1.0)) / (n + 1.0)
)
+ 96.0
* (
((S1 - 1.0 / n) ** 2 + S2 - 1.0 / n**2) / (n - 1.0)
- (S1**2 + S2) / n
+ 0.5 * B21
)
) / 27.0
result = gq0 + nf * gq1 + nf**2 * gq2
return -result
[docs]
@nb.njit(cache=True)
def gamma_gg(n, nf, cache):
r"""Compute the |NNLO| gluon-gluon singlet anomalous dimension.
Implements Eq. (3.13) of :cite:`Vogt:2004mw`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
|NNLO| gluon-gluon singlet anomalous dimension
:math:`\\gamma_{gg}^{(2)}(N)`
"""
S1 = c.get(c.S1, cache, n)
S2 = c.get(c.S2, cache, n)
S3 = c.get(c.S3, cache, n)
E1 = S1 / n**2 + (S2 - zeta2) / n
E11 = (S1 + 1.0 / (n + 1.0)) / (n + 1.0) ** 2 + (
S2 + 1.0 / (n + 1.0) ** 2 - zeta2
) / (n + 1.0)
E2 = 2.0 * (-S1 / n**3 + (zeta2 - S2) / n**2 - (S3 - zeta3) / n)
gg0 = (
-2675.8 / (n - 1.0) ** 2
+ 14214.0 / (n - 1.0)
- 144.0 * 24.0 / n**5
- 72.0 * 6.0 / n**4
- 7471.0 * 2.0 / n**3
- 274.4 / n**2
- 20852.0 / n
+ 3968.0 / (n + 1.0)
- 3363.0 / (n + 2.0)
+ 4848.0 / (n + 3.0)
+ 7305.0 * E1
+ 8757.0 * E2
- 3589.0 * S1 / n
+ 4425.894
- 2643.521 * (S1 - 1.0 / n)
)
gg1 = (
-157.27 / (n - 1.0) ** 2
+ 182.96 / (n - 1.0)
+ 512.0 / 27.0 * 24.0 / n**5
- 832.0 / 9.0 * 6.0 / n**4
+ 491.3 * 2.0 / n**3
- 1541.0 / n**2
- 350.2 / n
+ 755.7 / (n + 1.0)
- 713.8 / (n + 2.0)
+ 559.3 / (n + 3.0)
+ 26.15 * E1
- 808.7 * E2
+ 320.0 * S1 / n
- 528.723
+ 412.172 * (S1 - 1.0 / n)
)
gg2 = (
-680.0 / 243.0 / (n - 1.0)
+ 32.0 / 27.0 * 6.0 / n**4
+ 9.680 * 2.0 / n**3
+ 3.422 / n**2
- 13.878 / n
+ 153.4 / (n + 1.0)
- 187.7 / (n + 2.0)
+ 52.75 / (n + 3.0)
- 115.6 * E1
+ 85.25 * E11
- 63.23 * E2
+ 6.4630
+ 16.0 / 9.0 * (S1 - 1.0 / n)
)
result = gg0 + nf * gg1 + nf**2 * gg2
return -result
[docs]
@nb.njit(cache=True)
def gamma_singlet(N, nf, cache):
r"""Compute the |NNLO| singlet anomalous dimension matrix.
.. math::
\\gamma_S^{(2)} = \\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
[docs]
@nb.njit(cache=True)
def gamma_singlet_qed(N, nf, cache):
r"""Compute the leading-order singlet anomalous dimension matrix for the
unified evolution basis.
.. math::
\\gamma_S^{(3,0)} = \\left(\begin{array}{cccc}
\\gamma_{gg}^{(3,0)} & 0 & \\gamma_{gq}^{(3,0)} & 0\\
0 & 0 & 0 & 0 \\
\\gamma_{qg}^{(3,0)} & 0 & \\gamma_{qq}^{(3,0)} & 0 \\
0 & 0 & 0 & \\gamma_{qq}^{(3,0)} \\
\\end{array}\right)
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
Leading-order singlet anomalous dimension matrix :math:`\\gamma_{S}^{(3,0)}(N)`
"""
gamma_np_p = gamma_nsp(N, nf, cache)
gamma_qq = gamma_np_p + gamma_ps(N, nf, cache)
gamma_S = np.array(
[
[gamma_gg(N, nf, cache), 0.0 + 0.0j, gamma_gq(N, nf, cache), 0.0 + 0.0j],
[0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j],
[gamma_qg(N, nf, cache), 0.0 + 0.0j, gamma_qq, 0.0 + 0.0j],
[0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, gamma_np_p],
],
np.complex128,
)
return gamma_S
[docs]
@nb.njit(cache=True)
def gamma_valence_qed(N, nf, cache):
r"""Compute the leading-order valence anomalous dimension matrix for the
unified evolution basis.
.. math::
\\gamma_V^{(3,0)} = \\left(\begin{array}{cc}
\\gamma_{nsV}^{(3,0)} & 0\\
0 & \\gamma_{ns-}^{(3,0)}
\\end{array}\right)
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
Leading-order singlet anomalous dimension matrix :math:`\\gamma_{V}^{(3,0)}(N)`
"""
gamma_V = np.array(
[
[gamma_nsv(N, nf, cache), 0.0],
[0.0, gamma_nsm(N, nf, cache)],
],
np.complex128,
)
return gamma_V