"""The :math:`O(a_{em}^2)` Altarelli-Parisi splitting kernels."""
import numba as nb
import numpy as np
from eko import constants
from ....harmonics import cache as c
from . import as1aem1
[docs]
@nb.njit(cache=True)
def gamma_phph(N, nf):
r"""Compute the :math:`O(a_{em}^2)` photon-photon singlet anomalous
dimension.
Implements Eq. (68) of :cite:`deFlorian:2016gvk`.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
Returns
-------
complex
:math:`O(a_{em}^2)` photon-photon singlet anomalous dimension
:math:`\\gamma_{\\gamma \\gamma}^{(0,2)}(N)`
"""
nu = constants.uplike_flavors(nf)
nd = nf - nu
return (
constants.NC
* (nu * constants.eu2**2 + nd * constants.ed2**2)
* (as1aem1.gamma_gph(N) / constants.CF / constants.CA + 4)
)
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@nb.njit(cache=True)
def gamma_uph(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` quark-photon anomalous dimension for up
quarks.
Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=u.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` quark-photon anomalous dimension :math:`\\gamma_{u \\gamma}^{(0,2)}(N)`
"""
return constants.eu2 * as1aem1.gamma_qph(N, nf, cache) / constants.CF
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@nb.njit(cache=True)
def gamma_dph(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` quark-photon anomalous dimension for
down quarks.
Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=d.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` quark-photon anomalous dimension :math:`\\gamma_{d \\gamma}^{(0,2)}(N)`
"""
return constants.ed2 * as1aem1.gamma_qph(N, nf, cache) / constants.CF
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@nb.njit(cache=True)
def gamma_phu(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` photon-quark anomalous dimension for up
quarks.
Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=u.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` photon-quark anomalous dimension :math:`\\gamma_{\\gamma u}^{(0,2)}(N)`
"""
nu = constants.uplike_flavors(nf)
nd = nf - nu
S1 = c.get(c.S1, cache, N)
tmp = (-16.0 * (-16.0 - 27.0 * N - 13.0 * N**2 - 8.0 * N**3)) / (
9.0 * (-1.0 + N) * N * (1.0 + N) ** 2
) - 16.0 * (2.0 + 3.0 * N + 2.0 * N**2 + N**3) / (
3.0 * (-1.0 + N) * N * (1.0 + N) ** 2
) * S1
eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2)
return constants.eu2 * as1aem1.gamma_phq(N, cache) / constants.CF + eSigma2 * tmp
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@nb.njit(cache=True)
def gamma_phd(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` photon-quark anomalous dimension for
down quarks.
Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=d.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` photon-quark anomalous dimension :math:`\\gamma_{\\gamma d}^{(0,2)}(N)`
"""
nu = constants.uplike_flavors(nf)
nd = nf - nu
S1 = c.get(c.S1, cache, N)
tmp = (-16.0 * (-16.0 - 27.0 * N - 13.0 * N**2 - 8.0 * N**3)) / (
9.0 * (-1.0 + N) * N * (1.0 + N) ** 2
) - 16.0 * (2.0 + 3.0 * N + 2.0 * N**2 + N**3) / (
3.0 * (-1.0 + N) * N * (1.0 + N) ** 2
) * S1
eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2)
return constants.ed2 * as1aem1.gamma_phq(N, cache) / constants.CF + eSigma2 * tmp
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@nb.njit(cache=True)
def gamma_nspu(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` singlet-like non-singlet anomalous
dimension for up quarks.
Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` singlet-like non-singlet anomalous dimension
:math:`\\gamma_{ns,+,u}^{(0,2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
nu = constants.uplike_flavors(nf)
nd = nf - nu
eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2)
tmp = (
2.0
* (-12.0 + 20.0 * N + 47.0 * N**2 + 6.0 * N**3 + 3.0 * N**4)
/ (9.0 * N**2 * (1.0 + N) ** 2)
- 80.0 / 9.0 * S1
+ 16.0 / 3.0 * S2
) * eSigma2
return constants.eu2 * as1aem1.gamma_nsp(N, cache) / constants.CF / 2.0 + tmp
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@nb.njit(cache=True)
def gamma_nspd(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` singlet-like non-singlet anomalous
dimension for down quarks.
Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` singlet-like non-singlet anomalous dimension
:math:`\\gamma_{ns,+,d}^{(0,2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
nu = constants.uplike_flavors(nf)
nd = nf - nu
eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2)
tmp = (
2.0
* (-12.0 + 20.0 * N + 47.0 * N**2 + 6.0 * N**3 + 3.0 * N**4)
/ (9.0 * N**2 * (1.0 + N) ** 2)
- 80.0 / 9.0 * S1
+ 16.0 / 3.0 * S2
) * eSigma2
return constants.ed2 * as1aem1.gamma_nsp(N, cache) / constants.CF / 2.0 + tmp
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@nb.njit(cache=True)
def gamma_nsmu(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` valence-like non-singlet anomalous
dimension for up quarks.
Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` valence-like non-singlet anomalous dimension
:math:`\\gamma_{ns,-,u}^{(0,2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
nu = constants.uplike_flavors(nf)
nd = nf - nu
eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2)
tmp = (
2.0
* (-12.0 + 20.0 * N + 47.0 * N**2 + 6.0 * N**3 + 3.0 * N**4)
/ (9.0 * N**2 * (1.0 + N) ** 2)
- 80.0 / 9.0 * S1
+ 16.0 / 3.0 * S2
) * eSigma2
return constants.eu2 * as1aem1.gamma_nsm(N, cache) / constants.CF / 2.0 + tmp
[docs]
@nb.njit(cache=True)
def gamma_nsmd(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` valence-like non-singlet anomalous
dimension for down quarks.
Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
complex
:math:`O(a_{em}^2)` valence-like non-singlet anomalous dimension
:math:`\\gamma_{ns,-,d}^{(0,2)}(N)`
"""
S1 = c.get(c.S1, cache, N)
S2 = c.get(c.S2, cache, N)
nu = constants.uplike_flavors(nf)
nd = nf - nu
eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2)
tmp = (
2.0
* (-12.0 + 20.0 * N + 47.0 * N**2 + 6.0 * N**3 + 3.0 * N**4)
/ (9.0 * N**2 * (1.0 + N) ** 2)
- 80.0 / 9.0 * S1
+ 16.0 / 3.0 * S2
) * eSigma2
return constants.ed2 * as1aem1.gamma_nsm(N, cache) / constants.CF / 2.0 + tmp
[docs]
@nb.njit(cache=True)
def gamma_ps(N, nf):
r"""Compute the :math:`O(a_{em}^2)` pure-singlet quark-quark anomalous
dimension.
Implements Eq. (59) of :cite:`deFlorian:2016gvk`.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
Returns
-------
complex
:math:`O(a_{em}^2)` pure-singlet quark-quark anomalous dimension
:math:`\\gamma_{ps}^{(0,2)}(N)`
"""
result = (
-4.0
* (2.0 + N * (5.0 + N))
* (4.0 + N * (4.0 + N * (7.0 + 5.0 * N)))
/ ((-1.0 + N) * N**3 * (1.0 + N) ** 3 * (2.0 + N) ** 2)
)
return 2 * nf * constants.CA * result
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@nb.njit(cache=True)
def gamma_singlet(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` singlet sector.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
:math:`O(a_{em}^2)` singlet anomalous dimension :math:`\\gamma_{S}^{(0,2)}`
"""
nu = constants.uplike_flavors(nf)
nd = nf - nu
vu = nu / nf
vd = nd / nf
e2m = constants.eu2 - constants.ed2
e2avg = (nu * constants.eu2 + nd * constants.ed2) / nf
e2m = constants.eu2 - constants.ed2
gamma_ph_u = gamma_phu(N, nf, cache)
gamma_ph_d = gamma_phd(N, nf, cache)
gamma_u_ph = gamma_uph(N, nf, cache)
gamma_d_ph = gamma_dph(N, nf, cache)
gamma_ns_p_u = gamma_nspu(N, nf, cache)
gamma_ns_p_d = gamma_nspd(N, nf, cache)
gamma_pure_singlet = gamma_ps(N, nf)
gamma_S_02 = np.array(
[
[0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j, 0.0 + 0.0j],
[
0.0 + 0.0j,
gamma_phph(N, nf),
vu * constants.eu2 * gamma_ph_u + vd * constants.ed2 * gamma_ph_d,
vu * (constants.eu2 * gamma_ph_u - constants.ed2 * gamma_ph_d),
],
[
0.0 + 0.0j,
vu * constants.eu2 * gamma_u_ph + vd * constants.ed2 * gamma_d_ph,
vu * constants.eu2 * gamma_ns_p_u
+ vd * constants.ed2 * gamma_ns_p_d
+ e2avg**2 * gamma_pure_singlet,
vu
* (
constants.eu2 * gamma_ns_p_u
- constants.ed2 * gamma_ns_p_d
+ e2m * e2avg * gamma_pure_singlet
),
],
[
0.0 + 0.0j,
vd * (constants.eu2 * gamma_u_ph - constants.ed2 * gamma_d_ph),
vd
* (
constants.eu2 * gamma_ns_p_u
- constants.ed2 * gamma_ns_p_d
+ e2m * e2avg * gamma_pure_singlet
),
vd * constants.eu2 * gamma_ns_p_u
+ vu * constants.ed2 * gamma_ns_p_d
+ vu * vd * e2m**2 * gamma_pure_singlet,
],
],
np.complex_,
)
return gamma_S_02
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@nb.njit(cache=True)
def gamma_valence(N, nf, cache):
r"""Compute the :math:`O(a_{em}^2)` valence sector.
Parameters
----------
N : complex
Mellin moment
nf : int
Number of active flavors
cache: numpy.ndarray
Harmonic sum cache
Returns
-------
numpy.ndarray
:math:`O(a_{em}^2)` valence anomalous dimension :math:`\\gamma_{V}^{(0,2)}`
"""
nu = constants.uplike_flavors(nf)
nd = nf - nu
vu = nu / nf
vd = nd / nf
gamma_ns_m_u = gamma_nsmu(N, nf, cache)
gamma_ns_m_d = gamma_nsmd(N, nf, cache)
gamma_V_02 = np.array(
[
[
vu * constants.eu2 * gamma_ns_m_u + vd * constants.ed2 * gamma_ns_m_d,
vu * (constants.eu2 * gamma_ns_m_u - constants.ed2 * gamma_ns_m_d),
],
[
vd * (constants.eu2 * gamma_ns_m_u - constants.ed2 * gamma_ns_m_d),
vd * constants.eu2 * gamma_ns_m_u + vu * constants.ed2 * gamma_ns_m_d,
],
],
np.complex_,
)
return gamma_V_02