Gaining analytic control of parton showers Page: 4 of 12
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commonly found in parton shower algorithms.
Finally, we look at a double branch 1 -> 2 -> 4 where
each daughter further branches into two on-shell parti-
cles. In this case, the daughters' energy splittings zL,R,
or equivalently 0L,R, are additional free variables. The
complete phase space now is just an extension of Eq. (12),
tL+ R to , cos Bo < 1, Icos OL,RI < 1 . (14)
The limits on zL,R equivalent to cos 0L,RJ < 1 are anal-
ogous to Eq. (13) with the daughters' invariant masses
set to zero. Hence, the complete phase space in terms of
tL,R and zo,L,R reads
tL+ tR to,zo
1+ to
t -2oA(to, tL, tR),
0of - 21 /L
zL 2 C 2 '
zR - < R ,(15)
with
2 t2/E2 (16)
and ELR as in Eq. (4).
Eq. (15) explicitly shows the problem mentioned at
the end of the previous section. Initially, zo is generated
assuming tLR - 0, but since the limit on zo depends
on tL and tR, the generated value of zo has to be ad-
justed after (tL, zL) and (tR, zR) have been determined.
Changing zo, however, changes EL,R and L,R. This in
turn changes the limits on zL and zR, which can render
their values invalid. In addition, tL and tR are deter-
mined independently from one another, so the constraint
tL + tR < to can be violated as well.
III. A STANDARD PARTON SHOWER
To study a concrete example of a standard parton
shower, we consider the final-state parton shower of
Sherpa [8], which employs the same algorithm as the
Pythia virtuality-ordered parton shower [9, 10, 11].
Other algorithms which employ different ordering vari-
ables can be found in Refs. [15, 16, 17, 18].
A. The Algorithm
To be able to enforce four-momentum conservation,
the parton shower algorithm always branches two sisters
in pairs. That is, in each iteration it takes an existing
1 -> 2 branch, consisting of a mother and two unbranched
daughters, and converts it into a 1 -> 2 -> 4 double
branch by branching both daughters. To do so, the algo-
rithm proceeds in three steps as depicted in Fig. 2:1. Branch both daughters, each according to P(t, z).
2. Shuffle zo -> ze"(zo, tL, tR)-
3. Check kinematics in terms of new zoew:
(a) If successful, accept daughter branches.
(b) If failed, evolve daughter with larger t further
down and return to step 2.
In step 1, each daughter is branched separately, with val-
ues for (tL, zL) and (tR, zR) distributed according to the
single branch probability P(t, z). In step 2, zo is changed
to a new value zgOW(zo, tL, tR), which is derived from its
old value and takes into account the now nonzero values
of tL,R. In the mother's restframe this shuffling sim-
ply sets zo to the correct value zoM in Eq. (5). In a
general frame the form of zgOW(zo, tL, tR) is not dictated
by kinematics anymore, but is usually chosen to satisfy
Eq. (13). In step 3, the kinematics are checked, using
the new value zoeW. If they are satisfied, the daughter
branches are accepted. Otherwise, the algorithm takes
the daughter with the larger t, evolves it further down,
and goes back to step 2.
In the remainder of this section we discuss the details
of this algorithm. We first work out the precise form
of the single branch probability P(t, z) employed by the
algorithm. We then move to discuss in detail steps 2 and
3, which implement four-momentum conservation, but as
one can already see from Fig. 2 and the above discussion,
introduce a complicated correlation between tL and tR,
which clearly violates Eq. (3).
B. The Single Branch Probability
In this section we are only interested in a single 1 -> 2
branching of a mother into two daughters. This means
that each of the daughters in the algorithm described
above acts as the mother now, and similarly, the mother
and the other daughter in the algorithm now act as
grandmother and sister, respectively.
In the first step, the algorithm independently generates
two sets of values (tL, zL) and (tR, zR) according to the
single branch probability P(t, z), introduced in Eq. (1).
The precise form of P(t, z) that is actually used in the
algorithm can be written as
P(t, z) f(t, z) H(t, tmax) O(tmax t) O(t tcut)1-
X Ozcut(t) z -
with the Sudakov factorH(ti, t2) exp{- dt Jdz f(t, z)
Jt Ja-Zct(t)J(17)
(18)
In Eq. (17) we explicitly included all kinematic 0 func-
tions restricting the allowed ranges of t and z. All in-
formation on the precise form of P(t, z) is encoded in
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Tackmann, Frank; Bauer, Christian W. & Tackmann, Frank J. Gaining analytic control of parton showers, article, May 14, 2007; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc900691/m1/4/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.