def evaluate_kernel(self, zs, kTs, parent):
# Retrieve the collinear variables
z = zs[0]
kT = kTs[0]
# Instantiate the structure of the result
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [
None,
((parent, (kT, )), ),
],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
},
(1, 0): {
'finite': None
},
}
})
# The line below implements the g_{\mu\nu} part of the splitting kernel.
# Notice that the extra longitudinal terms included in the spin-correlation 'None'
# from the relation:
# \sum_\lambda \epsilon_\lambda^\mu \epsilon_\lambda^{\star\nu}
# = g^{\mu\nu} + longitudinal terms
# are irrelevant because Ward identities evaluate them to zero anyway.
# WARNING multiplied by two because of flavor factors
evaluation['values'][(0, 0)]['finite'] = 2. * self.TR
evaluation['values'][(
1, 0)]['finite'] = 2. * 4. * self.TR * z * (1. - z) / kT.square()
return evaluation
def evaluate_kernel(self, xs, kTs, parent):
# Retrieve the collinear variable x
x = xs[0]
# Instantiate the structure of the result
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [None],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
}
}
})
# The factor 'x' that should be part of the initial_state_crossing_factor cancels
# against the extra prefactor 1/x in the collinear factorisation formula
# (see Eq. (8) of NNLO compatible NLO scheme publication arXiv:0903.1218v2)
initial_state_crossing_factor = 1.
# Correct for the ratio of color-averaging factor between the real ME
# initial state flavor (gluon) and the one of the reduced Born ME (quark)
initial_state_crossing_factor *= (self.NC / float(self.NC**2 - 1))
z = 1. / x
norm = initial_state_crossing_factor * self.CF
# We re-use here the Altarelli-Parisi Kernel of the P_gq final state kernel without
# the soft-subtractio term 2./z since the gluon is here in the initial state
evaluation['values'][(0, 0)]['finite'] = norm * (1. + (1. - z)**2) / z
return evaluation
def evaluate_kernel(self, xs, kTs, parent):
# Retrieve the collinear variable x
x = xs[0]
kT = kTs[0]
# Instantiate the structure of the result
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [
None,
((parent, (kT, )), ),
],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
},
(1, 0): {
'finite': None
},
}
})
# The factor 'x' that should be part of the initial_state_crossing_factor cancels
# against the extra prefactor 1/x in the collinear factorisation formula
# (see Eq. (8) of NNLO compatible NLO scheme publication arXiv:0903.1218v2)
initial_state_crossing_factor = 1.
# Correct for the ratio of color-averaging factor between the real ME
# initial state flavor (gluon) and the one of the reduced Born ME (gluon)
initial_state_crossing_factor *= 1.
z = 1. / x
# The line below implements the g_{\mu\nu} part of the splitting kernel.
# Notice that the extra longitudinal terms included in the spin-correlation 'None'
# from the relation:
# \sum_\lambda \epsilon_\lambda^\mu \epsilon_\lambda^{\star\nu}
# = g^{\mu\nu} + longitudinal terms
# are irrelevant because Ward identities evaluate them to zero anyway.
# We re-use here the Altarelli-Parisi Kernel of the P_qg final state kernel, including
# its soft subtraction
# We must subtract the soft-collinear (CxS *not* SxC) from this contribution:
# P_gg = 2.*self.CA * ( (z/(1.-z)) + ((1.-z)/z) )
# CxS(P_gg) = 2.*self.CA * ( (z/(1.-z)) )
# P_gg-CxS(P_gg) = 2.*self.CA * ((1.-z)/z)
norm = initial_state_crossing_factor * 2. * self.CA
evaluation['values'][(0, 0)]['finite'] = norm * ((1. - z) / z)
evaluation['values'][(
1, 0)]['finite'] = -norm * 2. * z * (1. - z) / kT.square()
return evaluation
def evaluate_kernel(self, xs, kTs, parent):
# Retrieve the collinear variable x
x = xs[0]
kT = kTs[0]
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [
None,
((parent, (kT, )), ),
],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
},
(1, 0): {
'finite': None
},
}
})
# The factor 'x' that should be part of the initial_state_crossing_factor cancels
# against the extra prefactor 1/x in the collinear factorisation formula
# (see Eq. (8) of NNLO compatible NLO scheme publication arXiv:0903.1218v2)
initial_state_crossing_factor = -1.
# Correct for the ratio of color-averaging factor between the real ME
# initial state flavor (quark) and the one of the reduced Born ME (gluon)
initial_state_crossing_factor *= ((self.NC**2 - 1) / float(self.NC))
z = 1. / x
# We re-use here the Altarelli-Parisi Kernel of the P_q\bar{q} final state kernel
# The line below implements the g_{\mu\nu} part of the splitting kernel.
# Notice that the extra longitudinal terms included in the spin-correlation 'None'
# from the relation:
# \sum_\lambda \epsilon_\lambda^\mu \epsilon_\lambda^{\star\nu}
# = g^{\mu\nu} + longitudinal terms
# are irrelevant because Ward identities evaluate them to zero anyway.
norm = initial_state_crossing_factor * self.TR
evaluation['values'][(0, 0)]['finite'] = norm
evaluation['values'][(
1, 0)]['finite'] = norm * 4. * z * (1. - z) / kT.square()
return evaluation
def evaluate_kernel(self, zs, kTs, parent):
# Retrieve the collinear variables
z = zs[0]
kT = kTs[0]
# Instantiate the structure of the result
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [
None,
((parent, (kT, )), ),
],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
},
(1, 0): {
'finite': None
},
}
})
# The line below implements the g_{\mu\nu} part of the splitting kernel.
# Notice that the extra longitudinal terms included in the spin-correlation 'None'
# from the relation:
# \sum_\lambda \epsilon_\lambda^\mu \epsilon_\lambda^{\star\nu}
# = g^{\mu\nu} + longitudinal terms
# are irrelevant because Ward identities evaluate them to zero anyway.
# We must subtract the soft-collinear (CxS *not* SxC) from this contribution:
# P_gg = 2.*self.CA * ( (1.-z) / z + z / (1.- z) )
# CxS(P_gg) = 2.*self.CA * ( (1.-z) / z + z / (1.- z) )
# SxC(P_gg) = 2.*self.CA * ( 1 / z + 1 / (1.- z) )
# P_gg-CxS(P_gg) = 0
# P_gg-SxC(P_gg) = -4.*self.CA
evaluation['values'][(0, 0)]['finite'] = -4. * self.CA
evaluation['values'][(
1, 0)]['finite'] = -2. * self.CA * 2. * z * (1. - z) / kT.square()
return evaluation
def evaluate_kernel(self, zs, kTs, parent):
# Retrieve the collinear variables
z = zs[0]
# Instantiate the structure of the result
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [None],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
}
}
})
# We must subtract the soft-collinear (CxS *not* SxC) from this contribution:
# P_gq = self.CF * ((1.-z)**2 + 1.)/z
# CxS(P_gq) = self.CF * 2.*(1.-z) / z
# SxC(P_gq) = self.CF * 2. / z
# P_gq-CxS(P_gq) = self.CF * z
# P_gq-SxC(P_gq) = self.CF * ((1.-z)**2 - 1.)/z
evaluation['values'][(0, 0)]['finite'] = self.CF * z
return evaluation
def evaluate_kernel(self, xs, kTs, parent):
# Retrieve the collinear variable x
x = xs[0]
# Instantiate the structure of the result
evaluation = utils.SubtractionCurrentEvaluation({
'spin_correlations': [None],
'color_correlations': [None],
'values': {
(0, 0): {
'finite': None
}
}
})
# The factor 'x' that should be part of the initial_state_crossing_factor cancels
# against the extra prefactor 1/x in the collinear factorisation formula
# (see Eq. (8) of NNLO compatible NLO scheme publication arXiv:0903.1218v2)
initial_state_crossing_factor = 1.
# Correct for the ratio of color-averaging factor between the real ME
# initial state flavor (quark) and the one of the reduced Born ME (quark)
initial_state_crossing_factor *= 1.
z = 1. / x
# We re-use here the Altarelli-Parisi Kernel of the P_qg final state kernel, including
# its soft subtraction
# We must subtract the soft-collinear (CxS *not* SxC) from this contribution:
# P_qg = self.CF * ( (1.+z**2)/(1.-z) )
# CxS(P_qg) = self.CF * ( 2 / (x - 1) ) = self.CF * ( 2 z / (1 - z) )
# P_qg-CxS(P_qg) = self.CF * (1 + z**2 - 2*z) / (1 - z) = self.CF * ( 1 - z)
norm = initial_state_crossing_factor * self.CF
evaluation['values'][(0, 0)]['finite'] = norm * (1 - z)
return evaluation