r"""The unpolarized, space-like anomalous dimension
:math:`\gamma_{gq}^{(3)}`."""
import numba as nb
from eko.constants import zeta2
from ......harmonics import cache as c
from ......harmonics.log_functions import (
lm11,
lm12,
lm12m1,
lm13,
lm14,
lm14m1,
lm15,
lm15m1,
)
[docs]
@nb.njit(cache=True)
def gamma_gq(n, nf, cache, variation):
r"""Compute the |N3LO| gluon-quark singlet anomalous dimension.
The routine is taken from :cite:`Falcioni:2024xyt`.
A previous version based only on the lowest 10 moments
was given in :cite:`Moch:2023tdj`.
Parameters
----------
n : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
variation : int
|N3LO| anomalous dimension variation
Returns
-------
complex
|N3LO| gluon-quark singlet anomalous dimension
:math:`\gamma_{gq}^{(3)}(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)
S5 = c.get(c.S5, cache, n)
nf2 = nf * nf
# Known large-x coefficients
x1L5cff = 1.3443073 * 10 - 5.4869684 * 0.1 * nf
x1L4cff = 3.7539831 * 10**2 - 3.4494742 * 10 * nf + 8.7791495 * 0.1 * nf2
y1L5cff = 2.2222222 * 10 - 5.4869684 * 0.1 * nf
y1L4cff = 6.6242163 * 10**2 - 4.7992684 * 10 * nf + 8.7791495 * 0.1 * nf2
# Small-x, Casimir scaled from P_gg (approx. for bfkl1)
bfkl0 = -8.3086173 * 10**3 / 2.25
bfkl1 = (-1.0691199 * 10**5 - nf * 9.9638304 * 10**2) / 2.25
# Small-x double-logs with x^0
x0L6cff = 5.2235940 * 10 - 7.3744856 * nf
x0L5cff = -2.9221399 * 10**2 + 1.8436214 * nf
x0L4cff = 7.3106077 * 10**3 - 3.7887135 * 10**2 * nf - 3.2438957 * 10 * nf2
# The resulting part of the function
P3GQ01 = (
+bfkl0 * (-(6 / (-1 + n) ** 4))
+ bfkl1 * 2 / (-1 + n) ** 3
+ x0L6cff * 720 / n**7
+ x0L5cff * -120 / n**6
+ x0L4cff * 24 / n**5
+ x1L4cff * lm14(n, S1, S2, S3, S4)
+ x1L5cff * lm15(n, S1, S2, S3, S4, S5)
+ y1L4cff * lm14m1(n, S1, S2, S3, S4)
+ y1L5cff * lm15m1(n, S1, S2, S3, S4, S5)
)
# The selected approximations for nf = 3, 4, 5
if nf == 3:
P3gqApp1 = (
P3GQ01
+ 6.0 * bfkl1 * (-(1 / (-1 + n) ** 2))
- 744384.0 * 1 / ((-1 + n) * n)
+ 2453640.0 * 1 / n
- 1540404.0 * (2 / (1 + n) + 1 / (2 + n))
+ 1933026.0 * -1 / n**2
+ 1142069.0 * 2 / n**3
+ 162196.0 * -6 / n**4
- 2172.1 * lm13(n, S1, S2, S3)
- 93264.1 * lm12(n, S1, S2)
- 786973.0 * lm11(n, S1)
+ 875383.0 * lm12m1(n, S1, S2)
)
P3gqApp2 = (
P3GQ01
+ 3.0 * bfkl1 * (-(1 / (-1 + n) ** 2))
+ 142414.0 * 1 / ((-1 + n) * n)
- 326525.0 * 1 / n
+ 2159787.0 * ((3 + n) / (2 + 3 * n + n**2))
- 289064.0 * -1 / n**2
- 176358.0 * 2 / n**3
+ 156541.0 * -6 / n**4
+ 9016.5 * lm13(n, S1, S2, S3)
+ 136063.0 * lm12(n, S1, S2)
+ 829482.0 * lm11(n, S1)
- 2359050.0 * (S1 - n * (zeta2 - S2)) / n**2
)
elif nf == 4:
P3gqApp1 = (
P3GQ01
+ 6.0 * bfkl1 * (-(1 / (-1 + n) ** 2))
- 743535.0 * 1 / ((-1 + n) * n)
+ 2125286.0 * 1 / n
- 1332472.0 * (2 / (1 + n) + 1 / (2 + n))
+ 1631173.0 * -1 / n**2
+ 1015255.0 * 2 / n**3
+ 142612.0 * -6 / n**4
- 1910.4 * lm13(n, S1, S2, S3)
- 80851.0 * lm12(n, S1, S2)
- 680219.0 * lm11(n, S1)
+ 752733.0 * lm12m1(n, S1, S2)
)
P3gqApp2 = (
P3GQ01
+ 3.0 * bfkl1 * (-(1 / (-1 + n) ** 2))
+ 160568.0 * 1 / ((-1 + n) * n)
- 361207.0 * 1 / n
+ 2048948.0 * ((3 + n) / (2 + 3 * n + n**2))
- 245963.0 * -1 / n**2
- 171312.0 * 2 / n**3
+ 163099.0 * -6 / n**4
+ 8132.2 * lm13(n, S1, S2, S3)
+ 124425.0 * lm12(n, S1, S2)
+ 762435.0 * lm11(n, S1)
- 2193335.0 * (S1 - n * (zeta2 - S2)) / n**2
)
elif nf == 5:
P3gqApp1 = (
P3GQ01
+ 6.0 * bfkl1 * (-(1 / (-1 + n) ** 2))
- 785864.0 * 1 / ((-1 + n) * n)
+ 285034.0 * 1 / n
- 131648.0 * (2 / (1 + n) + 1 / (2 + n))
- 162840.0 * -1 / n**2
+ 321220.0 * 2 / n**3
+ 12688.0 * -6 / n**4
+ 1423.4 * lm13(n, S1, S2, S3)
+ 1278.9 * lm12(n, S1, S2)
- 30919.9 * lm11(n, S1)
+ 47588.0 * lm12m1(n, S1, S2)
)
P3gqApp2 = (
P3GQ01
+ 3.0 * bfkl1 * (-(1 / (-1 + n) ** 2))
+ 177094.0 * 1 / ((-1 + n) * n)
- 470694.0 * 1 / n
+ 1348823.0 * ((3 + n) / (2 + 3 * n + n**2))
- 52985.0 * -1 / n**2
- 87354.0 * 2 / n**3
+ 176885.0 * -6 / n**4
+ 4748.8 * lm13(n, S1, S2, S3)
+ 65811.9 * lm12(n, S1, S2)
+ 396390.0 * lm11(n, S1)
- 1190212.0 * (S1 - n * (zeta2 - S2)) / n**2
)
else:
raise NotImplementedError("nf=6 is not available at N3LO")
# We return (for now) one of the two error-band representatives
# or the present best estimate, their average
if variation == 1:
P3GQA = P3gqApp1
elif variation == 2:
P3GQA = P3gqApp2
else:
P3GQA = 0.5 * (P3gqApp1 + P3gqApp2)
return -P3GQA