226 Page 4 of 20 Eur. Phys. J. C (2018) 78:226
subcollision mass ˆs are increased in the process, the latter
by the same factor as x
r
. As with ISR, the increased x value
leads to an extra PDF weight
x
r
f
r
(x
r
, p
2
⊥
)
x
r
f
r
(x
r
, p
2
⊥
)
(9)
in the emission probability and Sudakov form factor. This
ensures a proper damping of radiation in the x
r
→ 1 limit.
So far Pythia has had no implementation of IF dipole
ends; all ISR is handled by the II approach. To first approxi-
mation this is no problem for the total emission rate, so long
as each incoming parton is allowed to radiate according to its
full colour charge. In more detail, however, one must beware
of a double- or undercounting of the full radiation pattern
when it is combined with the FI contribution. Note that this
pattern should depend on the scattering angle of the colour
flow in a hard process: if colour flows from an incoming par-
ton i to a final parton f then m
2
if
= E
i
E
f
(1 −cos θ
if
) sets
the phase space available for emission. In [32] an approxi-
mate prescription is introduced to dampen FI radiation that
otherwise could be doublecounted, but no corresponding pro-
cedure is implemented on the ISR side. What is done with
ISR, on the other hand, is to implement azimuthal asymme-
tries in the radiation pattern from colour coherence consid-
erations [39], that lines up radiation off the i parton with the
azimuthal angle of the f , in the same spirit as a dipole would,
but presumably not as accurately.
While it thus would seem that the dipole IF +FI approach
is superior to the global-recoil one, the issue is not always as
one-sided. The prime example is q +
q → γ
∗
/Z
0
produc-
tion. Once a gluon has been emitted from the original q +
q
II dipole, any further emission will be related to the resulting
q +g and g +
q dipoles represented in Fig. 4. Therefore the
γ
∗
/Z
0
only receives a recoil in the first step for the dipole
approach. With Feynman diagrams, on the other hand, the
γ
∗
/Z
0
takes a recoil that is modified as further gluon emis-
sions are considered. In this respect the global-recoil shower
strategy is analogous with how resummation techniques [40]
are used to sum up the effects of infinitely many gluon emis-
sions on the p
⊥
spectrum of the γ
∗
/Z
0
. This clear defect
of the dipole picture has been a main reason to maintain the
older global-recoil strategy, with modest improvements.
Nowadays showers are not used on their own when high
precision is required, however, but are matched/merged with
higher-order MEs [1]. With the kinematics of the hardest
four or so emissions based on MEs, and only subsequent
ones described by showers, it is reasonable to assume that
the γ
∗
/Z
0
p
⊥
spectrum is not impaired by the lack of further
recoils. On a philosophical level, it still reminds us that the
dipole picture also is an approximation, and that different
approaches should be developed as a means to assess uncer-
Z
0
FI/IF
FI/IF
q
g
q
Fig. 4 Colour flow for the process q + q → γ
∗
/Z
0
+ g. The dashed
lines represent the colour lines stretching between the dipole ends
tainties, as it has been done e.g. in [41] by using two different
recoil strategies.
Finally, it should be mentioned that Pythia also contains
a global-recoil option for FSR, not only for ISR. That is,
when one final parton radiates, all other final partons are
boosted, as a unit, so as to preserve total four-momentum.
This option is mainly intended to simplify matching/merging
with NLO results, the way they are calculated with the Mad-
Graph5_aMC@NLO program [42]. Typically global recoil
is therefore only used in the first one or two branchings,
whereafter one switches to the dipole picture. A similar strat-
egy could be envisioned for ISR, even if it has not been stud-
ied here.
3 The new approach
3.1 Kinematics for IF emissions
Let us consider a collision in the event frame between
two incoming partons b and d with four-momenta p
b,d
=
x
b,d
(
√
s/2)(1;0, 0, ±1), where
√
s is the total centre-of-
mass energy and x
b,d
are the four-momentum fractions. The
two partons are taken as massless. A sketch of the process is
given in Fig. 5a.
When evolving backwards in time, the parton b is seen
as coming from the branching a → b + c. The parton b
bd
z-axis
(a)
d
z-axis
c
a
(b)
Fig. 5 ISR kinematics. a Before branching: partons b and d incoming
in the ±z direction. b After the branching a → b + c, now with a and
d are along the z-axis
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