#  A new standard for the logarithmic accuracy of parton showers

Melissa van Beekveld  Nikhef, Theory Group, Science Park 105, 1098 XG,
Amsterdam, The Netherlands  Mrinal Dasgupta  Department of Physics &
Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom  Basem
Kamal El-Menoufi  School of Physics and Astronomy, Monash University,
Wellington Rd, Clayton VIC-3800, Australia  Silvia Ferrario Ravasio  CERN,
Theoretical Physics Department, CH-1211 Geneva 23, Switzerland  Keith Hamilton
Department of Physics and Astronomy, University College London, London, WC1E
6BT, UK  Jack Helliwell  Rudolf Peierls Centre for Theoretical Physics,
Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK  Alexander Karlberg
CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland  Pier
Francesco Monni  CERN, Theoretical Physics Department, CH-1211 Geneva 23,
Switzerland  Gavin P. Salam  Rudolf Peierls Centre for Theoretical Physics,
Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK  All Souls College,
Oxford OX1 4AL, UK  Ludovic Scyboz  School of Physics and Astronomy, Monash
University, Wellington Rd, Clayton VIC-3800, Australia  Alba Soto-Ontoso
CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland  Gregory
Soyez  IPhT, Université Paris-Saclay, CNRS UMR 3681, CEA Saclay, F-91191 Gif-
sur-Yvette, France

###### Abstract

We report on a major milestone in the construction of logarithmically accurate
final-state parton showers, achieving next-to-next-to-leading-logarithmic
(NNLL) accuracy for the wide class of observables known as event shapes. The
key to this advance lies in the identification of the relation between
critical NNLL analytic resummation ingredients and their parton-shower
counterparts. Our analytic discussion is supplemented with numerical tests of
the logarithmic accuracy of three shower variants for more than a dozen
distinct event-shape observables in $Z\to q\bar{q}$ and Higgs $\to gg$ decays.
The NNLL terms are phenomenologically sizeable, as illustrated in comparisons
to data.

††preprint: CERN-TH-2024-057, OUTP-24-03P

Parton showers are essential tools for predicting QCD physics at colliders
across a wide range of momenta from the TeV down to the GeV regime [1, 2, 3,
4]. In the presence of such disparate momenta, the perturbative expansions of
quantum field theories have coefficients enhanced by large logarithms of the
ratios of momentum scales. One way of viewing parton showers is as automated
and immensely flexible tools for resumming those logarithms, thus correctly
reproducing the corresponding physics.

The accuracy of resummations is usually classified based on terms with the
greatest logarithmic power at each order in the strong coupling (leading
logarithms or LL), and then towers of terms with subleading powers of
logarithms at each order in the coupling (next-to-leading logarithms or NLL,
NNLL, etc.). Higher logarithmic accuracy for parton showers should make them
considerably more powerful tools for analysing and interpreting experimental
data at CERN’s Large Hadron Collider and potential future colliders. The past
years have seen major breakthroughs in advancing the logarithmic accuracy of
parton showers, with several groups taking colour-dipole showers from LL to
NLL [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. There has also been
extensive work on incorporating higher-order splitting kernels into showers
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and understanding the structure
of subleading-colour corrections, see e.g. Refs. [30, 31, 32, 33, 34, 35, 6,
36, 37, 38, 39, 40, 41].

Here, for the first time, we show how to construct parton showers with NNLL
accuracy for the broad class of event-shape observables at lepton colliders,
like the well-known Thrust [42, 43] (see e.g. Refs. [44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65] for
calculations at NNLL and beyond). This is achieved by developing a novel
framework that unifies several recent developments, on (a) the inclusive
structure of soft-collinear gluon emission [58, 66] up to third order in the
strong coupling $\alpha_{s}$; (b) the inclusive pattern of energetic (“hard”)
collinear radiation up to order $\alpha_{s}^{2}$ [67, 68]; and (c) the
incorporation of soft radiation fully differentially up to order
$\alpha_{s}^{2}$ in parton showers, ensuring correct generation of any number
of well-separated pairs of soft emissions [29].

We will focus the discussion on the $e^{+}e^{-}\to Z\to q\bar{q}$ process,
with the understanding that the same arguments apply also to $H\to gg$. Each
event has a set of emissions with momenta $\\{k_{i}\\}$ and we work in units
where the centre-of-mass energy $Q\equiv 1$. We will examine the probability
$\Sigma(v)$ that some global event shape, $V(\\{k_{i}\\})$, has a value
$V(\\{k_{i}\\})<v$. Event-shape observables have the property [69] that for a
single soft and collinear emission $k$, $V(k)\propto
k_{t}e^{-\beta_{\text{obs}}|y|}$, where $k_{t}$ ($y$) is the transverse
momentum (rapidity) of $k$ with respect to the Born event direction and
$\beta_{\text{obs}}$ depends on the specific observable, e.g.
$\beta_{\text{obs}}=1$ for Thrust. Whether considering analytic resummation or
a parton shower, for $v\ll 1$ we have

$\Sigma(v)\\!=\\!\mathcal{F}\exp\\!\left[\\!-4\\!\\!\int\\!\frac{dk_{t}}{k_{t}}\\!\\!\int_{k_{t}}^{1}\\!\\!\\!\\!dzP_{\\!gq}(z)M(k)\frac{\alpha_{\text{eff}}}{2\pi}\Theta(V(k)\\!>\\!v)\right]\\!\\!,$
(1)

with $P_{gq}(z)=C_{F}\frac{1+(1-z)^{2}}{z}$ and $M(k)$ a function that
accounts for next-to-leading order matching, with $M(k)\to 1$ for $k_{t}\to
0$. The exponential is a Sudakov form factor, encoding the suppression of
emissions with $V(k)>v$, cf. the grey region of Fig. 1. It brings the LL
contributions to $\ln\Sigma$, terms $\alpha_{s}^{n}L^{n+1}$ with $L=\ln v$, as
well as NLL ($\alpha_{s}^{n}L^{n}$), NNLL ($\alpha_{s}^{n}L^{n-1}$), etc.,
contributions. The function $\mathcal{F}$ accounts [69] for the difference
between the actual condition $V(\\{k_{i}\\})<v$ and the simplified single-
emission boundary $V(k)<v$ that is used in the Sudakov. It starts at NLL.

Figure 1: Schematic representation of the Lund plane [70]. A constraint on an
event shape that scales as $k_{t}e^{-\beta_{\text{obs}}|y|}$ implies that
shower emissions above the red line are mostly vetoed. At NNLL, one mechanism
that modifies this constraint is that subsequent branching may cause the
effective transverse momentum or rapidity to shift, as represented by the
arrows.

In Eq. (1), the effective coupling, $\alpha_{\text{eff}}$, can be understood
as the intensity of gluon emission, inclusive over possible subsequent
branchings of that emission and corresponding virtual corrections. We write it
as

$\alpha_{\text{eff}}=\alpha_{s}\\!\left[1\\!+\frac{\alpha_{s}}{2\pi}\left(K_{1}\\!+\\!\Delta
K_{1}(y)\\!+\\!B_{2}(z)\right)+\frac{\alpha_{s}^{2}}{4\pi^{2}}K_{2}\right]\\!\\!,$
(2)

with $\alpha_{s}\equiv\alpha_{s}^{\overline{\textsc{ms}}}(k_{t})$ and here the
rapidity $y=\ln z/k_{t}$.
$K_{1}=(\frac{67}{18}-\frac{\pi^{2}}{6})C_{A}-\frac{10}{9}n_{f}T_{R}$ (often
called $K_{\text{{cmw}}}$) [71] is required for NLL accuracy and the remaining
terms for NNLL. $\Delta K_{1}(y)$ is zero in the resummation literature, non-
zero at central rapidities for certain showers, and vanishes for $y\to\infty$
[29]; $B_{2}(z)$ affects the hard-collinear region and tends to zero in the
soft limit, $z\to 0$. In analytic resummation it is generally included as a
constant multiplying $\delta(1-z)$ [44]. It has been calculated in specific
resummation schemes in Refs. [67, 68], but is not yet known for the showers
that we consider, which also do not yet include the relevant triple-collinear
dynamics. At NNLL, $K_{2}$ is relevant in the whole soft-collinear region and
also so far calculated only for analytic resummation [58, 66]. Through shower
unitarity, $\alpha_{\text{eff}}$ is relevant also in the Sudakov-veto region
of Fig. 1, i.e. the region above the red line, even though that region
contains no emissions or subsequent branching.

It is straightforward to see from Eq. (1) that terms up to
$\alpha_{s}^{n}L^{n-1}$ in $\ln\Sigma(v)$ depend only on the integrals of
$\Delta K_{1}(y)$ and $B_{2}(z)$,

$\Delta
K_{1}^{\text{int}}\equiv\\!\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!dy\,\Delta
K_{1}(y)\,,\,\,B_{2}^{\text{int}}\equiv\\!\int_{0}^{1}dz\frac{P_{gq}(z)}{2C_{F}}B_{2}(z).$
(3)

One of the key observations of this Letter is that as long as a parton shower
correctly generates double-soft emissions in the soft-collinear region, it is
possible to identify relations between the $\Delta K_{1}^{\text{int}}$,
$B_{2}^{\text{int}}$ and $K_{2}$ as relevant for an NNLL event-shape
resummation and the corresponding constants needed for a parton shower. This
holds even if the shower does not reproduce the full relevant physics at
second order in the large-angle and hard-collinear regions and at third order
in the soft-collinear region. This is in analogy with the fact that including
the correct $K_{1}$ constant is sufficient to obtain NLL accuracy even without
the real double-soft contribution.

In the next few paragraphs we will identify the relations between individual
resummation and shower ingredients (neglecting terms beyond NNLL), and then
show how they combine to achieve overall NNLL shower accuracy. Let us start by
recalling how the $\mathcal{O}\left(\alpha_{s}^{2}\right)$ terms of Eq. (2)
come about. Consider a Born squared matrix element,
$\mathcal{B}_{\tilde{\imath}}$, for producing a gluon ${\tilde{\imath}}$
(multiplied by $\alpha_{s}/2\pi$). Schematically the
$\mathcal{O}\left(\alpha_{s}^{2}\right)$ terms involve the single-emission
virtual correction $\mathcal{V}_{{\tilde{\imath}}}$ and an integral over a
real ${\tilde{\imath}}\to ij$ branching phase space and matrix element,
$d\Phi_{ij|{\tilde{\imath}}}\mathcal{R}_{ij}$ (both multiply
$(\alpha_{s}/2\pi)^{2}$). The key difference between a resummation calculation
and a parton shower lies in the phase-space mapping that is encoded in
$d\Phi_{ij|{\tilde{\imath}}}$. For example, in many resummation calculations
$g_{\tilde{\imath}}\to q_{i}\bar{q}_{j}$ splitting implicitly conserves
transverse momentum ${k}_{t,i+j}={k}_{t{\tilde{\imath}}}$ and rapidity
$y_{i+j}=y_{{\tilde{\imath}}}$ with respect to the particle that emitted
${\tilde{\imath}}$ [58, 66]. A parton shower (ps) will organise the phase
space differently, and in a way that does not conserve these kinematic
quantities. The difference can be represented as an effective drift in one or
more kinematic variables $x$ (e.g. $x\equiv\ln k_{t}$, $x\equiv y$) of post-
versus pre-branching kinematics. The average drifts,
$\frac{\alpha_{s}}{2\pi}\langle\Delta_{x}\rangle$, are represented as arrows
in Fig. 1. For a soft-collinear (SC) gluon $k_{\tilde{\imath}}$, they are
independent of the kinematics of $k_{\tilde{\imath}}$. For the $C_{F}C_{A}$
and $C_{F}n_{f}$ colour channels they read

$\langle\Delta_{x}\rangle=\lim_{{\tilde{\imath}}\to\text{{sc}}}\frac{1}{\mathcal{B}_{\tilde{\imath}}}\int
d\Phi_{ij|{\tilde{\imath}}}^{\text{{ps}}}\,\mathcal{R}_{ij}\times(x_{i+j}-x_{\tilde{\imath}})\,.$
(4)

For the $C_{F}^{2}$ channel, one replaces $x_{i+j}$ with the $x$ value of that
of $i$ and $j$ that corresponds to the larger shower ordering variable
($v_{\text{{ps}}}=k_{t}e^{-\beta_{\text{{ps}}}|y|}$). Note that the sign of
$\langle\Delta_{y}\rangle$ depends on the sign of $y_{\tilde{\imath}}$ (below,
$y_{\tilde{\imath}}>0$).

To understand the relation of $\langle\Delta_{y}\rangle$ with $\Delta
K^{\text{int}}_{1}$, observe that a drift to large absolute rapidities
depletes radiation at central rapidities. However the shower must correctly
reproduce the total final amount of radiation integrated over any rapidity
window. That can only be achieved with a value for $\Delta K^{\text{int}}_{1}$
that generates just enough extra central radiation to compensate for the
drift-induced depletion. Quantitatively, the following relation can be proven
(Ref. [72], §.1)

$\Delta K_{1}^{\text{int,\text{{ps}}}}=2\langle\Delta_{y}\rangle\,.$ (5)

As a numerical check, Table 1 shows the result of $\Delta
K_{1}^{\text{int,\text{{ps}}}}$ as determined in Ref. [29], compared to
$\langle\Delta_{y}\rangle$ as determined for this paper. The results are given
for three variants [5, 29] of the PanGlobal shower. The $\text{PG}_{\beta=0}$
and $\text{PG}_{\beta=0}^{\text{sdf}}$ showers have $\beta_{\text{{ps}}}=0$
and differ in how the splitting probabilities are assigned between the two
dipole ends. For all three variants, one observes good agreement between
$\Delta K_{1}^{\text{int,\text{{ps}}}}$ and $2\langle\Delta_{y}\rangle$.

shower | colour | $\frac{1}{4\pi}\Delta K_{1}^{\text{int,\text{{ps}}}}$ | $\frac{1}{2\pi}\langle\Delta_{y}\rangle$ | $\frac{1}{2\pi}\langle\Delta_{\ln k_{t}}\rangle$
---|---|---|---|---
| $C_{F}$ | 0 | 0. | 000018(39) | -1. | 953481(1)
PG${}_{\beta=0}^{\text{sdf}}$ | $C_{A}$ | 0 | 0. | 000002(2) | 1. | 162602(2)
| $n_{f}T_{R}$ | 0 | -0. | 0000003(3) | -0. | 1048049(3)
| $C_{F}$ | 0. | 04967(3) | 0. | 049576(8) | -1. | 964624(6)
PGβ=0 | $C_{A}$ | 0. | 0323(5) | 0. | 032107(4) | 1. | 174900(4)
| $n_{f}T_{R}$ | 0. | 0040(1) | 0. | 003962(1) | -0. | 104655(1)
| $C_{F}$ | 1. | 6725(5) | 1. | 672942(9) | -1. | 749920(5)
PG${}_{\beta=\frac{1}{2}}$ | $C_{A}$ | 0. | 0172(11) | 0. | 015303(5) | 1. | 172042(5)
| $n_{f}T_{R}$ | 0. | 0535(2) | 0. | 053476(1) | -0. | 094205(1)
Table 1: The $\Delta K_{1}^{\text{int,\text{{ps}}}}$ and
$\langle\Delta_{y}\rangle$ and $\langle\Delta_{\ln k_{t}}\rangle$
coefficients, including the relevant leading-$N_{C}$ colour factors
($2C_{F}=C_{A}=3$ and $n_{f}=5$). The errors on $\Delta
K_{1}^{\text{int,\text{{ps}}}}$ are systematic dominated and estimated only to
within a factor of order $1$.

Turning to $B_{2}(z)$, the corresponding physics differentially in $z$ cannot
yet be included in our showers, insofar as they lack triple-collinear
splitting. However, we can use a constraint analogous to Eq. (5) to determine
the correct $B_{2}^{\text{int,\text{{ps}}}}$, starting from the NLO $1\to 2$
calculations of Refs. [67, 68], which conserve the light-cone momentum-
fraction $z=m_{t}e^{y}=\sqrt{k_{t}^{2}+m^{2}}e^{y}$. Specifically (Ref. [72],
§.2),

$\displaystyle
B_{2}^{\text{int,\text{{ps}}}}\\!\\!=B_{2}^{\text{int,{nlo}}}\\!\\!-\langle\Delta_{\ln
z}\rangle,\,$ (6)

with

$\langle\Delta_{\ln z}\rangle=\langle\Delta_{y}\rangle+\langle\Delta_{\ln
m_{t}}\rangle=\langle\Delta_{y}\rangle+\langle\Delta_{\ln
k_{t}}\rangle-\frac{\beta_{0}\pi^{2}}{12}.$ (7)

The $\beta_{0}=(11C_{A}-4n_{f}T_{R})/6$ term arises from the relation between
the drifts in $m_{t}$ and $k_{t}$, which is shower-independent [58, 66,
72].111In the $C_{F}$ channel one defines the drift from the single parton
with larger $k_{t}e^{-\beta_{\text{{ps}}}|y|}$, so $m_{t}=k_{t}$ and
$\langle\Delta_{\ln m_{t}}\rangle_{C_{F}}=\langle\Delta_{\ln
k_{t}}\rangle_{C_{F}}$.  Note that Eq. (6) does not constrain the functional
form of $B_{2}^{\text{{ps}}}(z)$. To do so meaningfully would require a shower
that incorporates triple-collinear splitting functions. Instead we take the
ansatz $B_{2}^{\text{{ps}}}(z)\propto z$, normalised so as to satisfy Eq. (6).
Note that in an analytical resummation, Eq. (1) would use
$B_{2}^{\text{int,\text{resum}}}=B_{2}^{\text{int,{nlo}}}+\frac{\beta_{0}\pi^{2}}{12}$
(the $\frac{\beta_{0}\pi^{2}}{12}$ term has the same origin as in Eq. (7)).

The next ingredient that we need is $K_{2}$, which, for resummations, has been
calculated in two schemes [58, 66]. We adopt the scheme in which transverse
momentum is conserved and consider the amount of radiation in a (fixed-
rapidity) transverse-momentum window $k_{tb}<k_{t}<k_{ta}$, where $k_{t}$ is
the post-branching pair transverse momentum. The total amount of radiation in
the window should be the same in the resummation and the shower. In the shower
specifically, one should account for the $\ln k_{t}$ drifts through the lower
and upper edges of the window, which involve $\alpha_{s}$ at scales $k_{tb}$
and $k_{ta}$ respectively. Defining
$T_{n}(k_{tb},k_{ta})=\int_{k_{tb}}^{k_{ta}}\frac{dk_{t}}{k_{t}}\frac{\alpha_{s}^{n}(k_{t})}{(2\pi)^{n}}$,
that yields the constraint

$K_{2}^{\text{resum}}\,T_{3}(k_{tb},k_{ta})=K_{2}^{\text{{ps}}}\,T_{3}(k_{tb},k_{ta})\,+\\\
+\left(\frac{\alpha_{s}^{2}(k_{tb})}{4\pi^{2}}-\frac{\alpha_{s}^{2}(k_{ta})}{4\pi^{2}}\right)\langle\Delta_{\ln
k_{t}}\rangle\,,$ (8)

where the second line accounts for the drift contributions at the edges.
Setting

$K_{2}^{\text{{ps}}}=K_{2}^{\text{resum}}-4\beta_{0}\langle\Delta_{\ln
k_{t}}\rangle,$ (9)

ensures Eq. (8) is satisfied for all NNLL terms
$\alpha_{s}^{2+n}\ln^{n}k_{t1}/k_{t2}$, noting that for 1-loop running,

$2n\beta_{0}T_{n+1}(k_{tb},k_{ta})=[\alpha_{s}^{n}(k_{tb})-\alpha_{s}^{n}(k_{ta})]/(2\pi)^{n}.$
(10)

The final element in the connection with analytic resummation is
$\mathcal{F}$, which encodes the effect of emissions near the boundary
$V(k)\sim v$. The shower generates this factor through the interplay between
real and virtual emission. However $\mathcal{F}^{\text{{ps}}}$ differs from
$\mathcal{F}^{\text{resum}}$ because of relative drifts across the boundary
(Ref. [72], §.3)

$\frac{\mathcal{F}^{\text{{ps}}}}{\mathcal{F}^{\text{resum}}}=1+8C_{F}T_{2}(v,v_{\text{hc}})\left[\langle\Delta_{y}\rangle-\frac{1}{\beta_{\text{obs}}}\langle\Delta_{\ln
k_{t}}\rangle\right],$ (11)

with $v_{\text{hc}}\equiv v^{\frac{1}{1+\beta_{\text{obs}}}}$. Concentrating
on the right-hand half of the Lund plane in Fig. 1, it encodes the fact that a
positive $y$ drift increases the number of events that pass the constraint
$V(\\{k\\})<v$, because emissions to the left of the boundary move to the
right of the boundary, and vice-versa for a positive $\ln k_{t}$ drift.

We are now in a position to write the ratio of $\Sigma(v)$ in the shower as
compared to a resummation. Assembling the contributions discussed above into
Eq. (1) yields

$\frac{\Sigma^{\text{{ps}}}(v)}{\Sigma^{\text{resum}}(v)}-1=8C_{F}\bigg{\\{}-\langle\Delta_{y}\rangle
T_{2}(v,1)\\\ +\left[\langle\Delta_{y}\rangle+\langle\Delta_{\ln
k_{t}}\rangle\right]T_{2}\left(v_{\text{hc}},1\right)\\\ +\langle\Delta_{\ln
k_{t}}\rangle\left[\frac{1}{\beta_{\text{obs}}}T_{2}\left(v,v_{\text{hc}}\right)-T_{2}\left(v_{\text{hc}},1\right)\right]\\\
+\left[\langle\Delta_{y}\rangle-\frac{1}{\beta_{\text{obs}}}\langle\Delta_{\ln
k_{t}}\rangle\right]T_{2}\left(v,v_{\text{hc}}\right)\bigg{\\}}=0\,,$ (12)

up to NNLL. The lines account, respectively, for the shower contributions to
$\Delta K_{1}$, $B_{2}$, $K_{2}$ (using Eq. (10) and then trading rapidity and
$k_{t}$ integrations) and $\mathcal{F}$. The fact that they add up to zero
ensures shower NNLL accuracy for arbitrary global event shapes. The connection
with the ARES NNLL formalism [51, 52, 58] is discussed in Ref. [72], §.4.

Besides the analytic proof, we also carry out a series of numerical
verifications of the NNLL accuracy of several parton showers with the above
elements, using a leading-colour limit $2C_{F}=C_{A}=3$. These tests help
provide confidence both in the overall picture and in our specific
implementation for final-state showers. Fig. 2 shows a suitably normalised
logarithm of the ratio of the cumulative shower and resummed cross sections,
for a specific observable, the two-to-three jet resolution parameter,
$y_{23}$, for the Cambridge jet algorithm [73] in $Z\to q\bar{q}$ (left) and
$H\to gg$ (right) processes. Focusing on the
PG${}_{\beta_{\text{{ps}}}=0}^{\text{sdf}}$ shower, the plots show results
with various subsets of ingredients. A zero result indicates NNLL accuracy.
Only with 2-jet NLO matching [74], double-soft corrections [29], $B_{2}$ [67,
68] terms, 3-loop running of $\alpha_{s}$ [75, 76], $K_{2}$ contributions [58,
66], and the drift correction of this Letter does one obtain agreement with
the known NNLL predictions [52, 77].

Figure 2:  Test of NNLL accuracy of the PanGlobal
(PG${}_{\beta=0}^{\text{sdf}}$) shower for the cumulative distribution of the
Cambridge $y_{23}$ resolution variable, compared to known results for $Z\to
q\bar{q}$ [52] (left) and $H\to gg$ [77] (right). The curves show the
difference relative to NNLL for various subsets of ingredients. Starting from
the red curve, DS additionally includes double soft contributions and $2$-jet
NLO matching; $3\ell$ includes 3-loop running of $\alpha_{s}$ and the
$K_{2}^{\text{resum}}$ term. Including all effects (blue line) gives a result
that is consistent with zero, i.e. in agreement with NNLL.  Figure 3: Summary
of NNLL tests across observables and shower variants. Results consistent with
zero (shown in green) are in agreement with NNLL. The observables correspond
to the event shapes used in Ref. [5] and they are grouped according to the
power ($\beta_{\text{obs}}$) of their dependence on the emission angle. All
showers that include the corrections of this Letter agree with NNLL.

Tests across a wider range of observables and shower variants are shown in
Fig. 3 for a fixed value of $\lambda=\alpha_{s}\ln v=-0.4$. With the drifts
and all other contributions included, there is good agreement with the NNLL
predictions [45, 46, 47, 48, 49, 50, 51, 52, 58, 61, 77].

Earlier work on NLL accuracy had found that the coefficients of NLL violations
in common showers tended to be moderate for relatively inclusive observables
like event shapes [5]. In contrast, here we see that non-NNLL showers differ
from NNLL accuracy with coefficients of order one. That suggests a potential
non-negligible phenomenological effect.

Figure 4: Results for the Thrust and Durham $y_{23}$ [78] observables with the
PanGlobal showers compared to ALEPH data [79], using
$\alpha_{s}(M_{Z})=0.118$. The lower (middle) panel shows the ratios of the
NNLL (NLL) shower variants to data.

Fig. 4 compares three PanGlobal showers with ALEPH data [79] using Rivet v3
[80], illustrating the showers in their NLL and NNLL variants, with
$\alpha_{s}^{\overline{\textsc{ms}}}(M_{Z})=0.118$ for both. We use $2$-jet
NLO matching [74], and the NODS colour scheme [6], which guarantees full-
colour accuracy in terms up to NLL for global event shapes. Our showers are
implemented in a pre-release of PanScales [81] v0.2.0, interfaced to Pythia
v8.311 [3] for hadronisation, with non-perturbative parameters tuned to ALEPH
[82, 79] and L3 [83] data (starting from the Monash 13 tune [84], cf. Ref.
[72] §.5; the tune has only a modest impact on the observables of Fig. 4). The
impact of the NNLL terms is significant and brings the showers into good
agreement with ALEPH data [79], both in terms of normalisation and shape. Some
caution is required in interpreting the results: given that the logarithms are
not particularly large at LEP energies, NLO 3-jet corrections (not included)
may also play a significant role and should be studied in future work.
Furthermore, the PanGlobal showers do not include finite quark-mass effects.
Still, Fig. 4 suggests that NNLL terms have the potential to resolve a long-
standing issue in which a number of dipole showers (including notably the
Pythia 8 shower, but also the PanGlobal NLL shower) required an anomalously
large value of $\alpha_{s}(m_{Z})\gtrsim 0.130$ [84] to achieve agreement with
the data.

The parton showers developed here are expected to achieve NNLL (leading-
colour) accuracy also for non-global event shapes such as hemisphere or jet
observables, and $\alpha_{s}^{n}L^{n-1}$ (NSL) accuracy [54, 85, 62, 86, 63,
64, 68] for the soft-drop [87, 88] family of observables, in the limit where
either their $z_{\text{cut}}$ parameter is taken small or
$\beta_{\text{{sd}}}>0$. (We have not carried out corresponding logarithmic-
accuracy tests, because the small $z_{\text{cut}}$ limit renders them somewhat
more complicated than those of Figs. 2–3. In the case of non-global event
shapes, there exist no reference calculations.) This is in addition to the NSL
accuracy for energy-flow in a slice [89, 90, 91] and $\alpha_{s}^{n}L^{2n-2}$
(NNDL) accuracy for subjet multiplicities [92] that was already achieved with
the inclusion of double-soft corrections [29].

Next objectives in the programme of bringing higher logarithmic accuracy to
parton showers should include incorporation of full triple-collinear splitting
functions (as relevant for experimentally important observables such as
fragmentation functions), the extension to initial-state radiation, and
logarithmically consistent higher-order matching for a variety of hadron-
collider processes. The results presented here, a significant advance in their
own right, also serve to give confidence in the feasibility and value of this
broad endeavour.

###### Acknowledgements.

We are grateful to Peter Skands for discussions and helpful suggestions on
non-perturbative tunes in Pythia and to Silvia Zanoli for comments on the
manuscript. This work has been funded by the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 788223, MD, KH, JH, GPS, GS) and under its Horizon Europe
programme (grant agreement No. 101044599, PM), by a Royal Society Research
Professorship (RP$\backslash$R1$\backslash$231001, GPS) and by the Science and
Technology Facilities Council (STFC) under grants ST/T000864/1 (GPS),
ST/X000761/1 (GPS), ST/T000856/1 (KH) and ST/X000516/1 (KH), ST/T001038/1 (MD)
and ST/00077X/1 (MD). LS is supported by the Australian Research Council
through a Discovery Early Career Researcher Award (project number
DE230100867). BKE is supported by the Australian Research Council via
Discovery Project DP220103512. We also thank each others’ institutes for
hospitality during the course of this work. Views and opinions expressed are
those of the authors only and do not necessarily reflect those of the European
Union or the European Research Council Executive Agency. Neither the European
Union nor the granting authority can be held responsible for them.

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oneΔ

## Supplemental material

###  .1 Demonstration of relation between $\Delta K$ and drift

To help understand why Eq. (5) holds, we choose here to focus on the “non-
Abelian” or “correlated-emission” channels, which for a quark emitter
correspond to the terms involving $C_{F}n_{f}$ and $C_{F}C_{A}$ colour
factors. In general, in the soft limit, we define the correlated contribution
as being the part of the matrix element that remains after subtraction of the
double independent-emission contribution, which in the case of a quark emitter
corresponds to the $C_{F}^{2}$ component of the matrix element.

The starting point is a definition for $\Delta K_{1}(y_{\tilde{\imath}})$, to
be understood with a renormalisation scale of $\mu=k_{t{\tilde{\imath}}}$,

$K_{1}+\Delta
K_{1}(y_{\tilde{\imath}})=\frac{1}{\mathcal{B}_{\tilde{\imath}}}\left(\mathcal{V}_{{\tilde{\imath}}}+\int
d\Phi_{ij|{\tilde{\imath}}}^{\text{{ps}}}\,\mathcal{R}_{ij}\right).$ (13)

It is convenient to define

$R(y_{{\tilde{\imath}}},y_{d})=\frac{1}{\mathcal{B}_{\tilde{\imath}}}\left(\mathcal{V}_{{\tilde{\imath}}}\,\delta(y_{\tilde{\imath}}-y_{d})+\int
d\Phi_{ij|{\tilde{\imath}}}^{\text{{ps}}}\,\mathcal{R}_{ij}\,\delta(y_{d}(i,j)-y_{d})\right),$
(14)

where $y_{d}(i,j)$ denotes the effective rapidity of the $i,j$ descendants, as
produced by the shower. In the correlated-emission channels, it is to be taken
as the rapidity of $i+j$. The discussion can be adapted to the independent-
emission channel by instead taking $y_{d}$ to be the rapidity of either $i$ or
$j$, choosing the one with the larger shower ordering variable
($v_{\text{{ps}}}=k_{t}e^{-\beta_{\text{{ps}}}|y|}$).

It is useful to introduce a few properties of $R(y_{{\tilde{\imath}}},y_{d})$.
Firstly, a trivial rewriting of Eq. (13) is that

$K_{1}+\Delta
K_{1}(y_{\tilde{\imath}})=\int_{-\infty}^{+\infty}dy_{d}\,R(y_{{\tilde{\imath}}},y_{d})\,.$
(15)

For the integral to converge, $R(y_{{\tilde{\imath}}},y_{d})$ must vanish
sufficiently fast when $|y_{d}-y_{\tilde{\imath}}|$ is large,

$R(y_{{\tilde{\imath}}},y_{d})\to 0\,,\qquad\text{for
}|y_{d}-y_{\tilde{\imath}}|\gg 1\,,$ (16)

which can be seen as a consequence of the fact that for large rapidity
separations between $i$ and $j$, the shower reduces to independent emission
(cf. the PanScales conditions [5]). A second property of
$R(y_{{\tilde{\imath}}},y_{d})$ is that

$K_{1}=\int_{-\infty}^{+\infty}dy_{{\tilde{\imath}}}\,R(y_{{\tilde{\imath}}},y_{d})\,,$
(17)

independently of $y_{d}$. This corresponds to the statement that if one
integrates over $y_{{\tilde{\imath}}}$, the shower will generate the correct
rate of soft-parton pairs for any $y_{d}$ — it is an essential property of a
shower that has the correct double-soft contributions as in Ref. [29]. At
large rapidities (but where the particles are still soft),
$R(y_{{\tilde{\imath}}},y_{d})$ becomes a function of just
$y_{d}-y_{{\tilde{\imath}}}$. Denoting that function as $\bar{R}$, we have

$\displaystyle R(y_{{\tilde{\imath}}},y_{d})$
$\displaystyle\to\bar{R}(y_{d}-y_{\tilde{\imath}}),\qquad\text{for
}y_{{\tilde{\imath}}},y_{d}\gg 1\,,$ (18a) $\displaystyle
R(y_{{\tilde{\imath}}},y_{d})$
$\displaystyle\to\bar{R}(y_{\tilde{\imath}}-y_{d}),\qquad\text{for
}-y_{{\tilde{\imath}}},-y_{d}\gg 1\,.$ (18b)

This then implies that $\Delta K_{1}(y_{\tilde{\imath}})$ tends to zero at
large $|y_{\tilde{\imath}}|$, because Eqs. (15) and (17) become equivalent.

Now let us verify Eq. (5), which can be written as the large-$Y$ limit of

$\displaystyle\Delta
K_{1}^{\text{int}}(Y)=\int_{-Y}^{+Y}dy_{\tilde{\imath}}\,\Delta
K_{1}(y_{\tilde{\imath}})$
$\displaystyle=\int_{-Y}^{+Y}dy_{\tilde{\imath}}\int_{-\infty}^{+\infty}dy_{d}\,R(y_{{\tilde{\imath}}},y_{d})-\int_{-Y}^{+Y}dy_{d}\int_{-\infty}^{+\infty}dy_{\tilde{\imath}}\,R(y_{{\tilde{\imath}}},y_{d})\,,$
(19)

where, on the right-hand side, the last term is obtained by writing $2YK_{1}$
in terms of Eq. (17) for the factor of $K_{1}$ and an integral over $y_{d}$
for the factor $2Y$. Noting that the integration region where both rapidities
are in the range $-Y$ to $Y$ cancels between the two terms, we get

$\displaystyle\frac{1}{2}\Delta
K_{1}^{\text{int}}(Y)=\int_{-Y}^{+Y}dy_{\tilde{\imath}}\int_{Y}^{+\infty}dy_{d}\,R(y_{{\tilde{\imath}}},y_{d})-\int_{-Y}^{+Y}dy_{d}\int_{Y}^{+\infty}dy_{\tilde{\imath}}\,R(y_{{\tilde{\imath}}},y_{d})\,,$
(20)

where the factor of $1/2$ accounts for the fact that we consider just the
positive infinite rapidity range (the negative infinite rapidity range gives
identical results, notably since
$R(y_{\tilde{\imath}},y_{d})=R(-y_{\tilde{\imath}},-y_{d})$). Next, we make
use of the property that the integrand is dominated by a region where
$y_{\tilde{\imath}}$ and $y_{d}$ are not too different (cf. Eq. (16)), change
variables to $\Delta_{y}\equiv y_{d}-y_{\tilde{\imath}}$ and
$\bar{y}=y_{\tilde{\imath}}-Y$, take the limit $Y\to\infty$ and use Eq. (18a)
so as to write

$\frac{1}{2}\Delta K_{1}^{\text{int}}\equiv\lim_{Y\to\infty}\frac{1}{2}\Delta
K_{1}^{\text{int}}(Y)=\int_{0}^{+\infty}d\Delta_{y}\int_{-\Delta_{y}}^{0}d\bar{y}\,\bar{R}(\Delta_{y})-\int_{-\infty}^{0}d\Delta_{y}\int_{0}^{-\Delta_{y}}d\bar{y}\,\bar{R}(\Delta_{y})\,.$
(21)

The $d\bar{y}$ integrations can be performed trivially and assembling both
terms, we obtain

$\frac{1}{2}\Delta
K_{1}^{\text{int}}=\int_{-\infty}^{+\infty}d\Delta_{y}\Delta_{y}\,\bar{R}(\Delta_{y})=\lim_{{\tilde{\imath}}\to\text{{sc}}}\frac{1}{\mathcal{B}_{\tilde{\imath}}}\int
d\Phi_{ij|{\tilde{\imath}}}^{\text{{ps}}}\,\mathcal{R}_{ij}\times(y_{d}-y_{\tilde{\imath}})\,,$
(22)

which coincides with Eq. (5).

###  .2 Relation of $B_{2}$ coefficients with Refs. [67, 68]

####  .2.1 Overview of quark case

Refs. [67, 68] consider a collinear $q\to qg$ splitting probability, with
original quark energy $E$, splitting opening angle $\theta$ and outgoing quark
energy $\zeta E$. Eq. (2.10) of Ref. [68] defines the splitting probability as

$\frac{\alpha_{s}(\mu)}{2\pi}\mathcal{P}_{q}(\zeta,\theta)\equiv\frac{\alpha_{s}(\mu)}{2\pi}\left\\{\frac{2C_{F}}{1-\zeta}\\!\left[1+\frac{\alpha_{s}(\mu)}{2\pi}\\!\left(\\!K_{1}+\beta_{0}\ln\frac{\mu^{2}}{(1-\zeta)^{2}E^{2}\theta^{2}}\right)\right]+\mathcal{B}_{1}^{q}(\zeta)+\frac{\alpha_{s}(\mu)}{2\pi}\\!\left(\\!\mathcal{B}_{2}^{q}(\zeta)+\mathcal{B}_{1}^{q}(\zeta)\beta_{0}\ln\frac{\mu^{2}}{E^{2}\theta^{2}}\right)\\!\right\\}\\!,$
(23)

where $\mathcal{B}_{1}^{q}(\zeta)=-C_{F}(1+\zeta)$ is the finite part of the
$q\to qg$ splitting function and $\mathcal{B}_{2}^{q}(\zeta)$ is to be found
in Refs. [67, 68]. Specifically, we will need the integral of
$\mathcal{B}_{2}^{q}(\zeta)$, Eqs. (3.6), (3.4) and (3.7) of Ref. [68],

$\int_{0}^{1}d\zeta\,\mathcal{B}_{2}^{q}(\zeta)=-\gamma_{q}^{(2)}+\beta_{0}X_{\theta^{2}}^{q}\,,$
(24)

with

$\gamma_{q}^{(2)}=C_{F}^{2}\left(\frac{3}{8}-\frac{\pi^{2}}{2}+6\zeta_{3}\right)+C_{F}C_{A}\left(\frac{17}{24}+\frac{11\pi^{2}}{18}-3\zeta_{3}\right)-C_{F}n_{f}T_{R}\left(\frac{1}{6}+\frac{2\pi^{2}}{9}\right),\qquad
X_{\theta^{2}}^{q}=C_{F}\left(\frac{2\pi^{2}}{3}-\frac{13}{2}\right).$ (25)

In that same collinear limit, up to order $\alpha_{s}^{2}$, let us write the
parton shower’s splitting probability as

$\frac{\alpha_{\text{eff}}}{2\pi}P_{gq}(z)=\frac{\alpha_{s}(k_{t})}{2\pi}P_{gq}(z)\left[1+\frac{\alpha_{s}(k_{t})}{2\pi}\left(K_{1}+B_{2}^{\text{{nlo}}}(z)+B_{2}^{\text{drift}}(z)\right)\right],$
(26)

where we have separated $B_{2}(z)$ of Eq. (2) into two pieces. The piece
$P_{gq}(z)B_{2}^{\text{{nlo}}}(z)$ will integrate to
$2C_{F}B_{2}^{\text{int,\text{{nlo}}}}$, while
$P_{gq}(z)B_{2}^{\text{drift}}(z)$ will integrate to
$-2C_{F}\langle\Delta_{\ln z}\rangle$ in Eq. (6). The fundamental constraint
that we will impose is that the integrals of Eqs. (23) and (26) should yield
identical results provided that they cover equivalent phase space regions and
include the full hard collinear domain.

Let us first examine the equivalence requirement if we consider limits just on
the $1\to 2$ phase space, with $z\equiv 1-\zeta$ (the breaking of this
equivalence at the level of the full $1\to 3$ phase space will be examined
below and included through $B_{2}^{\text{drift}}(z)$). In doing so, it is
important to identify the relation between $k_{t}$ in the shower and the
kinematics of a hard collinear splitting. For the PanGlobal family of showers,
it is straightforward to show that

$k_{t}^{\text{PG}}\equiv zE\theta\,.$ (27)

Setting $\mu=(1-\zeta)E\theta$ in Eq. (23), we can then write

$\lim_{\epsilon\to
0}\int_{\epsilon}^{1-\epsilon}d\zeta\left[\frac{2C_{F}}{1-\zeta}K_{1}+\mathcal{B}_{2}^{q}(\zeta)+2\mathcal{B}_{1}^{q}(\zeta)\beta_{0}\ln(1-\zeta)\right]=\lim_{\epsilon\to
0}\int_{\epsilon}^{1-\epsilon}dzP_{gq}(z)(K_{1}+B_{2}^{\text{{nlo}}}(z))\,,$
(28)

from which we deduce

$2C_{F}B_{2}^{\text{int,NLO}}=\int_{0}^{1}dz\left(\mathcal{B}^{q}_{2}(1-z)+\mathcal{B}^{q}_{1}(1-z)\left[2\beta_{0}\ln
z-K_{1}\right]\right)=-\gamma_{q}^{(2)}+C_{F}\beta_{0}\left(\frac{2\pi^{2}}{3}-3\right)+\frac{3}{2}C_{F}K_{1}\,.$
(29)

Recall that the $z$ dependence that we assign to $B_{2}^{\text{{nlo}}}(z)$ is
arbitrary for event-shape NNLL accuracy, as long as the integral in Eq. (3)
converges, i.e. $B_{2}^{\text{{nlo}}}(z)$ has to vanish sufficiently fast as
$z\to 0$. Our choice is

$B_{2}^{\text{{nlo}}}(z)=\frac{3z}{2}B_{2}^{\text{int,NLO}}\,.$ (30)

####  .2.2 Overview of gluon case

The case of $B_{2}^{\text{{nlo}}}(z)$ for gluon jets essentially follows the
same steps as that in the quark case modulo a few subtleties that require
extra care. For the sake of conciseness, we only discuss the main steps in the
paragraphs below.

In the gluon case, the splitting probability is given by Eq. (7.2) of Ref.
[68]. The $z\leftrightarrow 1-z$ symmetry of gluon splittings, also equivalent
to the fact that parton showers generate gluon splitting by summing over two
dipoles, can be used to recast this equation as

$\displaystyle\frac{\alpha_{s}(k_{t})}{2\pi}\mathcal{P}_{g}(\zeta,\theta)\equiv\frac{\alpha_{s}(k_{t})}{2\pi}\left\\{\frac{2C_{A}}{1-\zeta}\\!\left(1+\frac{\alpha_{s}(k_{t})}{2\pi}K_{1}\right)+\mathcal{B}_{1}^{g}(\zeta)+\frac{\alpha_{s}(k_{t})}{2\pi}\left[\mathcal{B}_{2}^{g}(\zeta)+2\mathcal{B}_{1}^{g}(\zeta)\beta_{0}\ln(1-\zeta)\right]\right\\},$
(31)

with $\mu=k_{t}\equiv(1-\zeta)\theta E$. Here, we have explicitly summed the
$g\to gg$ and $g\to q\bar{q}$ channels by introducing
$\mathcal{B}_{1}^{g}=\mathcal{B}_{1}^{gg}+\mathcal{B}_{1}^{qg}=C_{A}(\zeta(1-\zeta)-2)+n_{f}T_{R}(\zeta^{2}+(1-\zeta)^{2})$
and $\mathcal{B}_{2}^{g}=\mathcal{B}_{2}^{gg}+\mathcal{B}_{2}^{qg}$ so that

$\int_{0}^{1}d\zeta\,\mathcal{B}_{2}^{g}(\zeta)=-\gamma_{g}^{(2)}+\beta_{0}X_{\theta^{2}}^{g}\,,$
(32)

with

$\gamma_{g}^{(2)}=\left(\frac{8}{3}+3\zeta_{3}\right)C_{A}^{2}-\frac{4}{3}C_{A}n_{f}T_{R}-C_{F}n_{f}T_{R}\,,\qquad
X_{\theta^{2}}^{g}=-\left(\frac{67}{9}-\frac{2\pi^{2}}{3}\right)C_{A}+\frac{23}{9}n_{f}T_{R}\,.$
(33)

In writing this set of equations, we have performed a few simplifications.
First, we have taken into account that, as in the quark case, the coupling for
PanGlobal is evaluated at the scale $\mu=k_{t}\equiv
k_{t}^{\text{PG}}=zE\theta$. Then, Eq. (32) differs from the corresponding
equation (4.2) in Ref. [68] in that it does not include the
$\mathcal{F}_{\text{clust.}}^{C_{A}^{2}}$ contribution. This contribution was
an artefact of the use of an mMDT [87] procedure to define $z$ and $\theta$ in
the $C_{A}^{2}$ channel, which does not apply in our case. We also refer to
the discussion in Appendix E of Ref. [68] for more details.

With this in hand, the procedure follows exactly what we did above in the
quark case, yielding

$2C_{A}B_{2}^{\text{int,NLO}}=\int_{0}^{1}dz\left(\mathcal{B}_{2}^{g}(1-z)+\mathcal{B}_{1}^{g}(1-z)\left[2\beta_{0}\ln
z-K_{1}\right]\right)=-\gamma_{g}^{(2)}+\left[\left(\frac{2\pi^{2}}{3}-\frac{67}{18}\right)C_{A}+\frac{10}{9}n_{f}T_{R}\right]\beta_{0}+\beta_{0}K_{1}\,.$
(34)

This contribution has to be distributed over the two dipoles that contribute
to the emission rate from a gluon. When implementing this in a shower, we also
have the freedom to distribute this contribution over the $g\to gg$ and $g\to
q\bar{q}$ channels as well as the freedom, already present in the quark case,
to choose any explicit $z$ dependence as long as the gluon-splitting analogue
of Eq. (3) is satisfied. In practice, we choose to have a unique
$B_{2,g}^{\text{{nlo}}}(z)$ common to both splitting channels and take

$B_{2,g}^{\text{{nlo}}}(z)=\frac{24C_{A}}{15C_{A}+2n_{f}T_{R}}z\,B_{2}^{\text{int,NLO}},$
(35)

where the prefactor accompanies the shower-specific partitioning prescription
for the splitting functions $P_{gg}(z)=C_{A}\frac{1+(1-z)^{3}}{2z}$ and
$P_{qg}(z)=n_{f}T_{R}(1-z)^{2}$, for each of the two dipoles contributing to a
given gluon. Note that, for event shapes, we are allowed to take a unique
$B_{2,g}^{\text{{nlo}}}(z)$ across $g\to gg$ and $g\to q\bar{q}$ splittings
since its contribution only appears in the Sudakov exponent where it is summed
over flavour channels.

####  .2.3 Drift contributions

Concentrating once again on the correlated-emission channels, let us
schematically write $B_{2}^{\text{{nlo}}}(z)$ as

$K_{1}+B_{2}^{\text{{nlo}}}(z)\equiv\frac{1}{\mathcal{B}_{{\tilde{\imath}}{\tilde{k}}}}\left(\mathcal{V}_{{\tilde{\imath}}{\tilde{k}}}+\int
d\Phi_{ijk|{\tilde{\imath}}{\tilde{k}}}^{\text{{nlo}}}\,\mathcal{R}_{ijk}\right),$
(36)

where in the quark branching channel ${\tilde{\imath}}$ is a gluon,
${\tilde{k}}$ is a quark,
$z=E_{\tilde{\imath}}/(E_{\tilde{\imath}}+E_{\tilde{k}})$ and we assume the
same scale choice $\mu=zE\theta$ as above. This is the extension of Eq. (13)
to the collinear region noting that (a) in the soft-collinear region, $\Delta
K_{1}(y)$ and $B_{2}(z)$ both go to zero and (b) the “Born” starting point is
a collinear ${\tilde{\imath}}{\tilde{k}}$ splitting, rather than just a soft
${\tilde{\imath}}$ emission. In the correlated-emission channels, the
$d\Phi_{ijk|{\tilde{\imath}}{\tilde{k}}}^{\text{{nlo}}}$ phase space map is
organised such that the energy fraction carried by the $ij$ pair is equal to
that carried by ${\tilde{\imath}}$, $E_{i}+E_{j}=E_{\tilde{\imath}}$, where
${\tilde{\imath}}$ is the emitted gluon (the calculation also preserves
$\theta_{ij,k}=\theta_{{\tilde{\imath}}{\tilde{k}}}$).222In the independent-
emission channel, we would instead have $E_{i}=E_{\tilde{\imath}}$ and
$\theta_{i,jk}=\theta_{{\tilde{\imath}}{\tilde{k}}}$.

Let us now suppose that we had a parton shower with tree-level $1\to 3$
splittings and 1-loop $1\to 2$ splittings. That shower would implicitly
involve a $B_{2}^{\text{{ps}}}(z)$. In what follows, we will work out the
relation between the integral of $B_{2}^{\text{{ps}}}(z)$ and that of
$B_{2}^{\text{{nlo}}}(z)$ and observe that this difference depends only on the
behaviour of the shower in the soft collinear region, where the triple-
collinear structure reduces to the double-soft structure. This will be useful,
because event-shape NNLL accuracy will be sensitive just to the integral of
$B_{2}(z)$, through its impact on the Sudakov form factor. Thus, even if our
actual shower does not have the full triple-collinear structure, the integral
of $B_{2}^{\text{{ps}}}(z)$ that we determine for an imaginary shower that
does have that structure will be sufficient to achieve NNLL event-shape
accuracy in the actual shower.333An analogous statement holds for the large-
angle region, where global NNLL event-shape accuracy could have been obtained
without the full large-angle double-soft structure, as long as the double-soft
structure is still correct in the soft-collinear limit and the shower has the
correct integral of $\Delta K_{1}(y)$. We explicitly tested this by modifying
$\Delta K_{1}(y)$ to be some function proportional to $e^{-|y|}$ with the
constraint that its integral should be correct. This gave event shape results
that were consistent with NNLL accuracy.

The $B_{2}^{\text{{ps}}}(z)$ function can be expressed as

$K_{1}+B_{2}^{\text{{ps}}}(z)\equiv\frac{1}{\mathcal{B}_{{\tilde{\imath}}{\tilde{k}}}}\left(\mathcal{V}_{{\tilde{\imath}}{\tilde{k}}}+\int
d\Phi_{ijk|{\tilde{\imath}}{\tilde{k}}}^{\text{{ps}}}\,\mathcal{R}_{ijk}\right),$
(37)

where $d\Phi_{ijk|{\tilde{\imath}}{\tilde{k}}}^{\text{{ps}}}$ reflects the
parton-shower kinematic map and associated partitioning of phase space. In
particular the shower map will not in general have
$E_{i}+E_{j}=E_{\tilde{\imath}}$ or
$\theta_{ij,k}=\theta_{{\tilde{\imath}}{\tilde{k}}}$. Next, defining
$B_{2}^{\text{drift}}=B_{2}^{\text{{ps}}}(z)-B_{2}^{\text{{nlo}}}(z)$ so as to
obtain Eq. (26), we have

$2C_{F}B_{2}^{\text{int,drift}}\equiv\int_{0}^{1}dzP_{gq}(z)B_{2}^{\text{drift}}(z)=\lim_{\epsilon\to
0}\int_{\epsilon}^{1}dz\,P_{gq}(z)\left[B_{2}^{\text{{ps}}}(z)-B_{2}^{\text{{nlo}}}(z)\right].$
(38)

It is now convenient to introduce a representation of the integrand of Eq.
(36) that is differential in the energy fraction $z_{d}$ of the descendants
(e.g. $z_{d}=z_{i}+z_{j}$ in the non-Abelian channel)

$K_{1}+B_{2}^{\text{{nlo}}}(z)=\int dz_{d}R^{\text{{nlo}}}(z,z_{d})\,,\qquad
R^{\text{{nlo}}}(z_{\tilde{\imath}},z_{d})=\frac{1}{\mathcal{B}_{{\tilde{\imath}}{\tilde{k}}}}\left(\mathcal{V}_{{\tilde{\imath}}{\tilde{k}}}\delta(z_{d}-z_{\tilde{\imath}})+\int
d\Phi_{ijk|{\tilde{\imath}}{\tilde{k}}}^{\text{{nlo}}}\,\mathcal{R}_{ijk}\delta(z_{d}-z_{d}(i,j))\right)\,.$
(39)

For the NLO phase-space map discussed above $R^{\text{{nlo}}}(z,z_{d})$ is
equal to $(K_{1}+B_{2}^{\text{{nlo}}}(z))\delta(z-z_{d})$. One can also define
an analogue for the parton shower, $R^{\text{{ps}}}(z,z_{d})$, which will have
a more complex structure. We can rewrite Eq. (38) as

$2C_{F}B_{2}^{\text{int,drift}}=\lim_{\epsilon\to
0}\int_{0}^{1}dz_{d}\int_{0}^{1}dzP_{gq}(z)\left[R^{\text{{ps}}}(z,z_{d})-R^{\text{{nlo}}}(z,z_{d})\right]\Theta(z-\epsilon)\,.$
(40)

Within our assumption that the shower has the correct triple collinear (and
double-collinear virtual) content, the total amount of radiation at a given
$z_{d}$ will be reproduced by the parton shower, i.e.

$\int_{0}^{1}dzP_{gq}(z)R^{\text{{ps}}}(z,z_{d})=\int_{0}^{1}dzP_{gq}(z)R^{\text{{nlo}}}(z,z_{d})\,.$
(41)

Furthermore, as in the case of section .1, any sensible shower that satisfies
the PanScales conditions will have the property that the integral over
$R^{\text{{ps}}}(z,z_{d})$ is dominated by the region of $z\sim z_{d}$. For
finite $z_{d}$, the $\Theta(z-\epsilon)$ term in Eq. (40) is irrelevant.
Therefore if we place an upper limit on the $z_{d}$ integral,
$z_{d}\ll\epsilon_{d}$, with $\epsilon\ll\epsilon_{d}\ll 1$, the result will
be unchanged,

$2C_{F}B_{2}^{\text{int,drift}}=\lim_{\begin{subarray}{c}\epsilon_{d}\to 0\\\
\epsilon/\epsilon_{d}\to
0\end{subarray}}\int_{0}^{\epsilon_{d}}dz_{d}\int_{0}^{1}dzP_{gq}(z)\left[R^{\text{{ps}}}(z,z_{d})-R^{\text{{nlo}}}(z,z_{d})\right]\Theta(z-\epsilon)\,.$
(42)

Given that the integral now just involves a region of small $z$ and $z_{d}$,
i.e. we are in the soft-collinear limit, we can use the property that aside
from an overall $1/z_{d}$ factor, $R^{\text{{ps}}}(z,z_{d})$ is a function
just of the ratio of $z_{d}/z$

$\displaystyle R^{\text{{ps}}}(z,z_{d})$ $\displaystyle\overset{z,z_{d}\ll
1}{\longrightarrow}\frac{1}{z_{d}}\bar{R}^{\text{{ps}}}(z_{d}/z)\,,$ (43a)
$\displaystyle R^{\text{{nlo}}}(z,z_{d})$ $\displaystyle\overset{z,z_{d}\ll
1}{\longrightarrow}\frac{K_{1}}{z_{d}}\delta(z_{d}/z-1)\,.$ (43b)

In this limit, we can also use the soft limit of $P_{gq}(z)$ as $z\sim
z_{d}\ll 1$, allowing us to write

$\displaystyle 2C_{F}B_{2}^{\text{int,drift}}$
$\displaystyle=2C_{F}\lim_{\begin{subarray}{c}\epsilon_{d}\to 0\\\
\epsilon/\epsilon_{d}\to
0\end{subarray}}\int_{0}^{\epsilon_{d}}\frac{dz_{d}}{z_{d}}\int_{0}^{1}\frac{dz}{z}\left[\bar{R}^{\text{{ps}}}(z_{d}/z)-K_{1}\delta(z_{d}/z-1)\right]\Theta(z-\epsilon)$
(44a) $\displaystyle=2C_{F}\lim_{\begin{subarray}{c}\epsilon_{d}\to 0\\\
\epsilon/\epsilon_{d}\to
0\end{subarray}}\int_{0}^{\infty}\frac{d\xi}{\xi}\ln\frac{\epsilon_{d}}{\xi\epsilon}\left[\bar{R}^{\text{{ps}}}(\xi)-K_{1}\delta(\xi-1)\right]\,.$
(44b)

To get the second line, we have replaced $z\to z_{d}/\xi$, used the fact that
$R^{\text{{ps}}}(\xi)$ vanishes for $\xi\to 0$ to replace the lower limit of
the $\xi$ integrand with $0$, and exchanged the order of $z_{d}$ and $\xi$
integrations.

Next, exploiting the fact that
$\int_{0}^{\infty}d\xi/\xi\bar{R}^{\text{{ps}}}(\xi)=K_{1}$, we can see that
the $\ln\epsilon_{d}/\epsilon$ contribution vanishes, leaving

$2C_{F}B_{2}^{\text{int,drift}}=2C_{F}\int_{0}^{\infty}\frac{d\xi}{\xi}\ln\frac{1}{\xi}\,\bar{R}^{\text{{ps}}}(\xi)\equiv-2C_{F}\langle\Delta_{\ln
z}\rangle\,,$ (45)

which is the basis for Eq. (6).

Eq. (7) expresses the $\langle\Delta_{\ln z}\rangle$ in terms of
$\langle\Delta_{y}\rangle$ and $\langle\Delta_{\ln m_{t}}\rangle$. The
relation between $\langle\Delta_{\ln m_{t}}\rangle$ and $\langle\Delta_{\ln
k_{t}}\rangle$ is given by

$\langle\Delta_{\ln m_{t}}\rangle-\langle\Delta_{\ln k_{t}}\rangle=\int
d\Phi^{\text{{ps}}}_{ij|{\tilde{\imath}}}\,\mathcal{R}(i,j)\,\frac{1}{2}\ln\frac{m_{t,ij}^{2}}{k_{t,ij}^{2}}\,.$
(46)

Note that aside from the phase space map, the integrand does not depend on
${\tilde{\imath}}$. Accordingly, as long as the phase space map covers the
full double-soft region, which it does for any soft collinear
${\tilde{\imath}}$, we are free to replace the parton-shower phase space map
with an analytic one that preserves $k_{t}$ and rapidity. One can then deduce
the result for Eq. (46) from the corresponding expressions in the literature,
e.g. Eqs. (3.7)–(3.12) of Ref. [58],

$\langle\Delta_{\ln m_{t}}\rangle-\langle\Delta_{\ln
k_{t}}\rangle=-\beta_{0}\frac{\pi^{2}}{12}\,.$ (47)

We have also verified this numerically for each of our shower maps.

###  .3 Multiple emission contribution

In our definition, the function $\mathcal{F}$ in Eq. (1) accounts for the
difference between applying a separate condition
$\Theta(V(k_{\tilde{\imath}})<v)$ on each primary emission ${\tilde{\imath}}$,
versus applying a single condition $\Theta(V(\\{k_{i}\\})<v)$ on the full set
of emissions after branching. It starts at order $\alpha_{s}^{2}$ and its NNLL
component (starting from $\alpha_{s}^{2}L$) differs from the corresponding
$\delta\mathcal{F}$ contribution which appears at NNLL accuracy in ARES [58],
in that the latter also includes certain order $\alpha_{s}$ terms (cf. Eq.
(2.45) of Ref. [58]). Let us start by defining a primary inclusive emission
density

$d\rho_{k}=2\frac{dk_{t}}{k_{t}}\,dzP_{gq}(z)\,\frac{\alpha_{\text{eff}}}{2\pi}\Theta(z>k_{t})\,,$
(48)

which allows us to write the Sudakov factor from Eq. (1) as

$S(v)=\exp\left[-2\int d\rho_{k}\,M(k)\Theta(V(k)>v)\right].$ (49)

The factor of two in the Sudakov reflects the presence of two hemispheres. The
matching factor $M(k)$ will be irrelevant in the remainder of this section
because it tends to $1$ for $k_{t}\to 0$, but we will need it later in section
.4. Up to NLL, one way of writing $\mathcal{F}$ is as

$\mathcal{F}_{\text{{nll}}}=\frac{S(\epsilon
v)}{S(v)}\sum_{n=0}^{\infty}\frac{1}{n!}\prod_{{\tilde{\imath}}=1}^{n}\left[2\int
d\rho_{k_{\tilde{\imath}}}\,\Theta(V(k_{\tilde{\imath}})>\epsilon
v)\right]\Theta(V(k_{\tilde{1}},\ldots,k_{\tilde{n}})<v),$ (50)

where $S(\epsilon v)$ exponentiates the virtual corrections for no emission
down to scale $\epsilon v$, the denominator $S(v)$ simply divides out the
Sudakov factor already included in Eq. (1), while the sum and product account
for the real emission of any number $n$ of primary particles
${\tilde{\imath}}$ at scales above $\epsilon v$. The final $\Theta$-function
represents the constraint on the emissions from the requirement for the event-
shape observable to have a value less than $v$. The parameter $\epsilon$ is to
be taken small, such that $\ln\epsilon$ is kept finite or, equivalently,
$v\ll\epsilon\ll 1$, so that one does not need to resum $\ln\epsilon$-enhanced
terms, for example in the ratio $S(\epsilon v)/S(v)$.

Now let us extend Eq. (50) to NNLL. In the various manipulations along the
way, we will discard contributions that are beyond NNLL (i.e. that are
$\alpha_{s}^{n}L^{n-2}$ or higher). We start by writing

$\mathcal{F}_{\text{{nnll}}}=\frac{S(\epsilon
v)}{S(v)}\sum_{n=0}^{\infty}\bigg{\\{}\frac{1}{n!}\prod_{{\tilde{\imath}}=1}^{n}\left[2\int
d\rho_{k_{\tilde{\imath}}}\left(u_{\tilde{\imath}}+\frac{\alpha_{s}}{2\pi}\int
d\Phi_{i_{a}i_{b}|{\tilde{\imath}}}\,\frac{\mathcal{R}_{i_{a}i_{b}}}{\mathcal{B}_{{\tilde{\imath}}}}\left[u_{i_{a}}u_{i_{b}}-u_{\tilde{\imath}}\right]\right)\Theta(V(k_{\tilde{\imath}})>\epsilon
v)\right]\times\\\
\times\Theta(V(\\{k_{1}\\},\\{k_{2}\\}\ldots,\\{k_{n}\\})<v)\bigg{\\}},$ (51)

where the $u_{\tilde{\imath}}$ notation, inspired by generating-functionals,
means that in the second line, $\\{k_{i}\\}$ is just $k_{\tilde{\imath}}$,
while $u_{i_{a}}u_{i_{b}}$ means that $\\{k_{i}\\}$ is to be understood as
$k_{i_{a}},k_{i_{b}}$, i.e.

$\displaystyle
u_{{\tilde{\imath}}}\Theta(V(\left\\{k_{1}\right\\},\dots,\left\\{k_{i}\right\\},\dots,\left\\{k_{n}\right\\})<v)$
$\displaystyle\equiv\Theta(V(\left\\{k_{1}\right\\},\dots,k_{\tilde{\imath}},\dots,\left\\{k_{n}\right\\})<v)\,,$
(52a) $\displaystyle\quad
u_{i_{a}}u_{i_{b}}\Theta(V(\left\\{k_{1}\right\\},\dots,\left\\{k_{i}\right\\},\dots,\left\\{k_{n}\right\\})<v)$
$\displaystyle\equiv\Theta(V(\left\\{k_{1}\right\\},\dots,k_{i_{a}},k_{i_{b}},\dots,\left\\{k_{n}\right\\})<v)\,.$
(52b)

Note that the ${\tilde{\imath}}\to i_{a}i_{b}$ branching is unitary in the
soft-collinear region that dominates $\mathcal{F}$. The individual emission
constraint $\Theta(V(k_{\tilde{\imath}})>\epsilon v)$ preserves this
unitarity, and so one should similarly constrain just $V(k)>\epsilon v$ in the
definition of $S(\epsilon v)$ of Eq. (51).

Next, we take advantage of freedom in how to define the unresolved emissions
(e.g. $V(k_{\tilde{\imath}})<\epsilon v$), as long as we use consistent
definitions of unresolved in the Sudakov and real emission contributions. In
particular, we choose to place the resolution condition on final
($i_{a},i_{b}$) particles rather than the (potentially) intermediate
${\tilde{\imath}}$ particle, which gives

$\mathcal{F}_{\text{{nnll}}}=\frac{\bar{S}(\epsilon
v)}{S(v)}\sum_{n=0}^{\infty}\bigg{\\{}\frac{1}{n!}\prod_{{\tilde{\imath}}=1}^{n}\left[2\int
d\rho_{k_{\tilde{\imath}}}\left(u_{\tilde{\imath}}+\frac{\alpha_{s}}{2\pi}\int
d\Phi_{i_{a}i_{b}|{\tilde{\imath}}}\,\frac{\mathcal{R}_{i_{a}i_{b}}}{\mathcal{B}_{{\tilde{\imath}}}}\left[u_{i_{a}}u_{i_{b}}-u_{\tilde{\imath}}\right]\right)\Theta(V(k_{i_{ab}})>\epsilon
v)\right]\times\\\
\times\Theta(V(\\{k_{1}\\},\\{k_{2}\\}\ldots,\\{k_{n}\\})<v)\bigg{\\}}.$ (53)

Relative to Eq. (51) there are two changes. Firstly, at the end of the square
bracket on the first line, $\Theta(V(k_{{\tilde{\imath}}})>\epsilon v)$ has
been replaced by $\Theta(V(k_{i_{ab}})>\epsilon v)$, where $k_{i_{ab}}$ is
defined to be $k_{\tilde{\imath}}$ for the $u_{\tilde{\imath}}$ contribution,
while for the $u_{i_{a}}u_{i_{b}}$ contribution it is equal to a massless
momentum with the same transverse components and rapidity as
$k_{i_{a}}+k_{i_{b}}$. The shower and resummation have different maps from
${\tilde{\imath}}$ to $i_{a}i_{b}$. However, both maps have the property that
if one integrates over all possible ${\tilde{\imath}}$, one obtains identical
final distributions of $i_{a}i_{b}$. Since the $\Theta(V(k_{i_{ab}})>\epsilon
v)$ condition is a constraint on those final momenta, not on the (sometimes)
intermediate ${\tilde{\imath}}$ momenta, the sum and product in Eq. (53) will
be the same in the shower and in the resummation (at least up to NNLL). The
second change to note in Eq. (53) relative to Eq. (51) is that $S(\epsilon v)$
has been substituted with $\bar{S}(\epsilon v)$, defined as

$\bar{S}(\epsilon v)=\exp\left[-2\int d\rho_{k}\left(\Theta(V(k)>\epsilon
v)+\frac{\alpha_{s}}{2\pi}\int
d\Phi_{ab|k}\,\frac{\mathcal{R}_{ab}}{\mathcal{B}_{k}}\left[\Theta(V(k_{ab})>\epsilon
v)-\Theta(V(k)>\epsilon v)\right]\right)\right].$ (54)

As compared to the $S(\epsilon v)$ of Eq. (49), the $\Theta(V(k)>\epsilon v)$
factor has been replaced by conditions that are specified in terms of final
momenta, either $k$ when there was no secondary branching or $k_{ab}$ when
there was a branching (with $k_{ab}$ defined in analogy to $k_{i_{ab}}$
above). This modification is necessary in order for the exponentiated virtual
corrections in $\bar{S}(\epsilon v)$ to exactly match the phase space in the
real sum and product of Eq. (53), as required by unitarity.

Since the sum-product contribution in Eq. (53) is the same between the parton
shower and the resummation, it will cancel in the ratio
$\mathcal{F}^{\text{{ps}}}/\mathcal{F}^{\text{resum}}$, leaving us with

$\displaystyle\frac{\mathcal{F}^{\text{{ps}}}}{\mathcal{F}^{\text{resum}}}=\frac{\bar{S}^{\text{{ps}}}(\epsilon
v)}{\bar{S}^{\text{resum}}(\epsilon v)}$ $\displaystyle=\exp\left[-2\int
d\rho_{k}\left(\frac{\alpha_{s}}{2\pi}\int
d\Phi_{ab|k}^{\text{{ps}}}\,\frac{\mathcal{R}_{ab}}{\mathcal{B}_{k}}[\Theta(V(k_{ab})>\epsilon
v)-\Theta(V(k)>\epsilon v)]\right)\right].$ (55)

To reach that result, it is useful to note that in $\bar{S}^{\text{resum}}$,
the map $d\Phi_{ab|k}^{\text{resum}}$ has the property that $V(k_{ab})$ and
$V(k)$ are identical, giving $\bar{S}^{\text{resum}}(\epsilon v)=S(\epsilon
v)$. Our next step is to write $d\rho_{k}$ explicitly in the soft-collinear
limit as $4C_{F}dk_{t}/k_{t}dy\alpha_{s}(k_{t})/2\pi$ and to use $V(k)\propto
k_{t}\exp[-\beta_{\text{obs}}y]$. We then perform the $k_{t}$ integration
within the limits set by the difference of $\Theta$-functions in Eq. (55). In
doing so, it is convenient to approximate the scale of the coupling as
$k_{t}\sim ve^{\beta_{\text{obs}}y}$, which is legitimate up to and including
NNLL when focusing on the ratio of
$\mathcal{F}^{\text{{ps}}}/\mathcal{F}^{\text{resum}}$. We then have

$\displaystyle\frac{\mathcal{F}^{\text{{ps}}}}{\mathcal{F}^{\text{resum}}}$
$\displaystyle=\exp\left[-8C_{F}\int_{0}^{\ln
1/v_{\text{hc}}}\\!\\!\\!\\!dy\int\frac{dk_{t}}{k_{t}}\delta\\!\left(\ln\frac{k_{t}}{ve^{\beta_{\text{obs}}y}}\right)\frac{\alpha_{s}^{2}(k_{t})}{(2\pi)^{2}}\int
d\Phi_{ab|k}^{\text{{ps}}}\,\frac{\mathcal{R}_{ab}}{\mathcal{B}_{k}}\left(\ln\frac{k_{t,ab}}{k_{t}}-\beta_{\text{obs}}(y_{ab}-y)\right)\right]$
(56a)
$\displaystyle=1-\frac{8C_{F}}{\beta_{\text{obs}}}T_{2}(v,v_{\text{hc}})\left[\langle\Delta_{\ln
k_{t}}\rangle-\beta_{\text{obs}}\langle\Delta_{y}\rangle\right],$ (56b)

where we have ignored any $\epsilon$ dependence (since we ignore $\ln\epsilon$
contributions) and in the second line we have transformed the $dy$ integral
into a $1/\beta_{\text{obs}}dk_{t}/k_{t}$ integral so as to obtain the
$T_{2}(v,v_{\text{hc}})$ function. The final result corresponds to Eq. (11).
Note that for $\beta_{\text{obs}}=0$, the result is to be understood as the
$\beta_{\text{obs}}\to 0$ limit of Eq. (56b), with
$1/\beta_{\text{obs}}T_{2}(v,v_{\text{hc}})\to(\frac{\alpha_{s}(v)}{2\pi})^{2}\ln
1/v$.

###  .4 Equivalence with NNLL resummation

####  .4.1 Relation to the ARES approach

In the previous sections, we have given analytic arguments showing (i) how we
can incorporate into the shower the hard-collinear $\mathcal{B}_{2}(z)$
computed analytically in Refs. [67, 68], (ii) how the various drift
contributions emerge in the parton shower Sudakov and ultimately compensate
for the effects of the shower’s double-soft map close to the observable
boundary, cf. Eq. (12). In this section, we show that these ingredients,
together with 3-loop running coupling, the CMW scheme ($K_{1}$ and $K_{2}$)
and full double-soft matrix-element corrections are sufficient for the shower
to achieve NNLL accuracy, for which we take ARES [58] as our reference.444The
discussion below is presented at full colour accuracy. Our shower double-soft
corrections are currently implemented only at leading-colour accuracy,
consequently the NNLL terms in the shower are also only leading-colour
accurate. Were they to be upgraded to full colour, we would expect to achieve
full-colour NNLL accuracy, at least for processes with two coloured Born legs.

Our starting point is the shower result, Eq. (1). Since the
$\langle\Delta_{\ln k_{t}}\rangle$ and $\langle\Delta_{y}\rangle$
contributions all cancel, cf. Eq. (12) and sections .1, .2.3 and .3, we will
leave them out in the rest of this section, or equivalently work as if they
were zero. We then write Eq. (1) using the shorthand Eq. (49)

$\displaystyle\Sigma^{\textsc{ps}}_{\textsc{nnll}}(v)=\mathcal{F}^{\textsc{ps}}_{\textsc{nnll}}(v)\,S^{\textsc{ps}}(v)\,.$
(57)

It is useful to write the Sudakov form factor $S^{\text{{ps}}}$ separated into
terms of different logarithmic order

$\ln S^{\text{{ps}}}=(\ln S)_{\text{{ll}}}+(\ln S)_{\text{{nll}}}+(\ln
S)_{\text{{nnll}}}+\ldots$ (58)

We can write the individual orders as integrals either over $k_{t}$ and $z$
(as in Eq. (1) and in typical resummations), or as an integral over the shower
ordering variable $v_{\text{{ps}}}$ and $z$. The latter gives slightly more
complicated expressions, but helps connect with the actual shower algorithm.
On a first pass, readers may wish to set $\beta_{\text{{ps}}}=0$, in which
case $v_{\text{{ps}}}\equiv k_{t}$. One can also check explicitly that any
dependence on $\beta_{\text{{ps}}}$ cancels separately at each logarithmic
order. We take an observable that in the soft or collinear limit behaves as
$V(k)=f(z)g(y)k_{t}e^{-\beta_{\text{obs}}y}$ with $f(z)\to 1$ for $z\to 0$ and
$g(y)\to 1$ for $y\to\infty$. Using $z=k_{t}e^{y}$, we can write

$k_{t}=z^{\frac{\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}}v_{\textsc{ps}}^{\frac{1}{1+\beta_{\text{{ps}}}}},\qquad
V(k)=f(z)\,g(y)\,V_{\text{sc}}(k)\,,\quad\text{ with }\quad
V_{\text{sc}}(k)=z^{\frac{\beta_{\text{{ps}}}-\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}\,.$
(59)

At LL, $P_{gq}(z)$ can be approximated as $2C_{F}/z$, the 3-jet matrix-element
correction factor $M(k)$ can be ignored, and the observable can be
approximated by its soft-collinear limit, yielding

$(\ln
S)_{\text{{ll}}}\equiv-\frac{4}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dz\frac{\alpha_{s}(z^{\frac{\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{2\pi}\frac{2C_{F}}{z}\,\Theta(z^{\frac{\beta_{\text{{ps}}}-\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v)\,.$
(60)

Note that we choose to include $3$-loop running for the coupling even in the
LL contribution. Strictly speaking, this introduces subleading contributions
in $(\ln S)_{\text{{ll}}}$, but it simplifies the expressions for the higher-
order contributions, without compromising any of the arguments of this
section. At NLL, we have

$\displaystyle(\ln S)_{\text{{nll}}}\equiv$
$\displaystyle-\frac{4K_{1}}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dz\frac{\alpha_{s}^{2}(z^{\frac{\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{(2\pi)^{2}}\frac{2C_{F}}{z}\,\Theta(z^{\frac{\beta_{\text{{ps}}}-\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v)$
$\displaystyle-\frac{4B_{1}}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\textsc{ps}}}{v_{\textsc{ps}}}\frac{\alpha_{s}(v_{\textsc{ps}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{2\pi}\,\Theta(v_{\textsc{ps}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v),$
(61)

where we have included the $K_{1}$ NLL contribution and introduced
$B_{1}=\int_{0}^{1}dz\mathcal{B}_{1}(1-z)$. In the second line, we have
substituted $z=1$ in the argument of $\alpha_{s}$ and the observable so as to
have purely $\alpha_{s}^{n}L^{n}$ terms (aside from the higher-loop running
coupling contributions). The independence of $(\ln S)_{\text{{ll}}}$ and $(\ln
S)_{\text{{nll}}}$ of $\beta_{\text{{ps}}}$ can be verified, e.g. by changing
variables from $v_{\text{{ps}}}$ to $v_{\text{obs}}\equiv V_{\text{sc}}(k)$.

The NNLL contributions can be written as

$\displaystyle(\ln S)_{\text{{nnll}}}\equiv$
$\displaystyle-\frac{4K_{2}^{\text{resum}}}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dz\frac{\alpha_{s}^{3}(z^{\frac{\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{(2\pi)^{3}}\frac{2C_{F}}{z}\,\Theta(z^{\frac{\beta_{\text{{ps}}}-\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v)$
$\displaystyle-\frac{4\tilde{B}_{2}^{\text{{ps}}}}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\textsc{ps}}}{v_{\textsc{ps}}}\frac{\alpha_{s}^{2}(v_{\textsc{ps}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{(2\pi)^{2}}\,\Theta(v_{\textsc{ps}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v)+H_{1}^{\text{{ps}}}\frac{\alpha_{s}(Q)}{2\pi}+2C_{1,\text{hc}}^{\text{{ps}}}\frac{\alpha_{s}(v^{\frac{1}{1+\beta_{\text{obs}}}})}{2\pi}+C_{1,\text{wa}}^{\text{{ps}}}\frac{\alpha_{s}(v)}{2\pi}\,.$
(62)

The $K_{2}^{\text{resum}}$ coefficient has already been introduced, and we
elaborate on the others below. Note that each of the integrals is independent
of $\beta_{\text{{ps}}}$, but some of the coefficients multiplying them depend
on $\beta_{\text{{ps}}}$. Below, we will verify that this
$\beta_{\text{{ps}}}$ dependence disappears when summing over all
contributions.

Let us start with $H_{1}^{\text{{ps}}}$, which involves the $3$-jet matrix
element correction factor $M(k)$,

$\displaystyle H_{1}^{\text{{ps}}}$ $\displaystyle=\lim_{\epsilon\to
0}\left[\frac{4}{1+\beta_{\text{{ps}}}}\int_{\epsilon}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dzP_{gq}(z)M(k)-\frac{4}{1+\beta_{\text{{ps}}}}\int_{\epsilon}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dzP_{gq}(z)\right]$
(63a)
$\displaystyle=\frac{4}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dzP_{gq}(z)\left(M(k)-1\right)\,.$
(63b)

Note that $M(k)$ also accounts for the Jacobian associated with the
integration variables and exact phase space limits. In the second line, the
lower limit of the integration on $v_{\text{{ps}}}$ could be taken to zero,
since $M(k)\to 1$ when $k_{t}\to 0$. Through unitarity, the shower is such
that $\Sigma(v)$ is normalised to one, i.e. $\Sigma(v=1)=1$, which ensures
that $H_{1}^{\text{{ps}}}$ can be determined fully from the real $3$-jet
matrix element. The result is independent of the choice of observable,

$H_{1}^{\text{{ps}}}=\frac{-C_{F}}{1+\beta_{\text{{ps}}}}\left(\frac{\pi^{2}}{3}(1+\beta_{\text{{ps}}})+\frac{7}{2}(1-\beta_{\text{{ps}}})\right).$
(64)

Next we examine $C_{1,\text{hc}}^{\text{{ps}}}$,

$\displaystyle C_{1,\text{hc}}^{\text{{ps}}}$
$\displaystyle=\frac{-2}{1+\beta_{\text{obs}}}\int_{0}^{1}dzP_{gq}(z)\log
f(z)-\frac{2}{1+\beta_{\text{obs}}}\int_{0}^{1}dz\mathcal{B}_{1}(1-z)\left(\frac{\beta_{\text{{ps}}}-\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}\log
z\right)$ (65a)
$\displaystyle=\frac{-2}{1+\beta_{\text{obs}}}\int_{0}^{1}dzP_{gq}(z)\log
f(z)+\frac{\beta_{\text{obs}}-\beta_{\text{{ps}}}}{(1+\beta_{\text{obs}})(1+\beta_{\text{{ps}}})}\frac{7C_{F}}{2}.$
(65b)

This has a contribution that accounts for the exact boundary of the observable
in the hard-collinear limit (the terms involving $f(z)$) and another that
accounts for the fact that in the second line of Eq. (61) the boundary
condition on $v_{\text{{ps}}}$ was imposed for $z=1$ rather than integrated
over the actual $z$ dependence of Eq. (59).

One can proceed the same way for the wide-angle coefficient,
$C_{1,\text{wa}}^{\text{{ps}}}$. In this limit, we can approximate
$P_{gq}(z)=2C_{F}/z$, and after changing variable from $z$ to $y$ one gets

$C_{1,\text{wa}}^{\text{{ps}}}=-8C_{F}\int_{0}^{\infty}dy\log g(y).$ (66)

We now focus on the term of $(\ln S)_{\text{{nnll}}}$ proportional to
$B_{\text{{ps}}}^{(2)}$. This hard-collinear $\mathcal{O}(\alpha_{s}^{2}L)$
term receives contributions from three sources: (i) $B_{2}(z)$ in Eq. (2),
(ii) a contribution involving $K_{1}\mathcal{B}_{1}(1-z)$, with
$\mathcal{B}_{1}$ coming from the product of the finite part of the splitting
function in Eq. (48) with $K_{1}$ from Eq. (2), and (iii) a leftover running-
coupling correction from the NLL term. The latter stems from the fact that the
scale of the running coupling,
$k_{t}=z^{\beta_{\text{{ps}}}/(1+\beta_{\text{{ps}}})}v^{1/(1+\beta_{\text{{ps}}})}$,
is evaluated at the scale $v^{1/(1+\beta_{\text{{ps}}})}$ in the NLL
contribution $(\ln S)_{\text{{nll}}}$ given above. Summing these
contributions, we get

$\displaystyle\tilde{B}_{2}^{\text{{ps}}}$
$\displaystyle=\int_{0}^{1}dz\left(P_{gq}(z)B_{2}^{\text{{ps}}}(z)+K_{1}\mathcal{B}_{1}(1-z)-\mathcal{B}_{1}(1-z)\frac{2\beta_{0}\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}\log
z\right)$ (67a)
$\displaystyle=-\gamma_{q}^{(2)}-\beta_{0}X^{\text{{ps}}}+C_{F}\beta_{0}\frac{\pi^{2}}{6}\,,\qquad\text{
with }\quad
X^{\text{{ps}}}=C_{F}\left(\frac{7}{2}\frac{\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}+3-\frac{2\pi^{2}}{3}\right).$
(67b)

In the second line, we have explicitly separated out a $\pi^{2}/6$ which can
be traced back to the $-\beta_{0}\pi^{2}/12$ term in Eq. (7)

In order to ease the comparison to the ARES formalism, it is helpful to use
Eq. (10) in order to rewrite the $\beta_{0}X^{\text{{ps}}}$ contribution to
$\tilde{B}_{2}^{\text{{ps}}}$ in terms of an extra contribution to
$H_{1}^{\text{{ps}}}$ and to $C_{1,\text{hc}}^{\text{{ps}}}$. This gives

$\displaystyle(\ln S)_{\text{{nnll}}}\equiv$
$\displaystyle-\frac{4K_{2}^{\text{resum}}}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\text{{ps}}}}{v_{\text{{ps}}}}\int_{v_{\text{{ps}}}}^{1}dz\frac{\alpha_{s}^{3}(z^{\frac{\beta_{\text{{ps}}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{(2\pi)^{3}}\frac{2C_{F}}{z}\,\Theta(z^{\frac{\beta_{\text{{ps}}}-\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}v_{\text{{ps}}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v)$
$\displaystyle-\frac{4\bar{B}^{\text{{ps}}}_{2}}{1+\beta_{\text{{ps}}}}\int_{0}^{1}\frac{dv_{\textsc{ps}}}{v_{\textsc{ps}}}\frac{\alpha_{s}^{2}(v_{\textsc{ps}}^{\frac{1}{1+\beta_{\text{{ps}}}}})}{(2\pi)^{2}}\,\Theta(v_{\textsc{ps}}^{\frac{1+\beta_{\text{obs}}}{1+\beta_{\text{{ps}}}}}>v)+\bar{H}_{1}^{\text{{ps}}}\frac{\alpha_{s}(Q)}{2\pi}+2\bar{C}_{1,\text{hc}}^{\text{{ps}}}\frac{\alpha_{s}(v^{\frac{1}{1+\beta_{\text{obs}}}})}{2\pi}+C_{1,\text{wa}}^{\text{{ps}}}\frac{\alpha_{s}(v)}{2\pi}\,,$
(68)

with

$\displaystyle\bar{B}_{2}^{\text{{ps}}}$
$\displaystyle=\tilde{B}_{2}^{\text{{ps}}}+\beta_{0}X^{\text{{ps}}}=-\gamma_{q}^{(2)}+C_{F}\beta_{0}\frac{\pi^{2}}{6},$
(69a) $\displaystyle\bar{H}_{1}^{\text{{ps}}}$
$\displaystyle=H_{1}^{\text{{ps}}}-2X^{\text{{ps}}}=C_{F}\left(\pi^{2}-\frac{19}{2}\right),$
(69b) $\displaystyle\bar{C}_{1,\text{hc}}^{\text{{ps}}}$
$\displaystyle=C_{1,\text{hc}}^{\text{{ps}}}+X^{\text{{ps}}}=\frac{-2}{1+\beta_{\text{obs}}}\int_{0}^{1}dz\,P_{gq}(z)\log
f(z)+C_{F}\left(3+\frac{7}{2}\frac{\beta_{\text{obs}}}{1+\beta_{\text{obs}}}-\frac{2\pi^{2}}{3}\right).$
(69c)

One sees that this reorganisation explicitly removes the residual dependence
on $\beta_{\text{{ps}}}$.

We are now in a position to connect the above ingredients to the corresponding
ones in the ARES NNLL formalism where the cumulative distribution can be
recast as follows (neglecting N3LL corrections, as elsewhere in this section),

$\displaystyle\Sigma_{\textsc{nnll}}(v)=e^{-R(v)}\left[1+\frac{\alpha_{s}(Q)}{2\pi}H^{(1)}+\frac{\alpha_{s}(v^{\frac{1}{1+\beta_{\text{obs}}}})}{2\pi}2C^{(1)}_{\text{hc}}\right]\mathcal{F}_{\textsc{NNLL}}\,,$
(70)

cf. Eq. (2.45) of Ref. [58], which can be viewed as a short-hand notation for
Eq. (2.39) of that same reference. At LL and NLL, changing variables from
$v_{\textsc{ps}}$ to $k_{t}$ is sufficient to see that Eqs. (60) and (61)
correspond to the ARES LL and NLL contributions to the Sudakov radiator, and
also capture the $3$-loop running-coupling contributions to the NNLL Sudakov,
as already discussed above.

The NNLL contributions in Eq. (.4.1) also map directly onto a series of
ingredients in ARES: the $K_{2}^{\text{resum}}$ term in Eq. (.4.1) reproduces
the corresponding CMW in the ARES Sudakov; the $\bar{H}_{1}^{\text{{ps}}}$ is
identical to the $H^{(1)}$ coefficient in ARES (cf. Eq. (2.44) of Ref. [58]);
the $\gamma_{q}^{(2)}$ part of $\bar{B}_{2}^{\text{{ps}}}$ corresponds to the
one-loop $\gamma_{\ell}^{(1)}$ contribution to the hard-collinear radiator,
cf. section 3.2 of Ref. [58], noting the different sign convention between
$\gamma_{q}^{(2)}$ and $\gamma_{\ell}^{(1)}$; and the
$C_{F}\beta_{0}\pi^{2}/6$ part of $\bar{B}_{2}^{\text{{ps}}}$ reproduces the
$\delta g_{3}^{(\ell)}$ contribution to the ARES Sudakov given in Eq. (3.28).
Eq. (3.7) therein further showcases the similar origin of this contribution in
the shower and ARES approaches.

Having dealt with the above shower contributions, we are only left with those
in $\mathcal{F}_{\textsc{nnll}}$, $\bar{C}_{1,\text{hc}}^{\text{{ps}}}$ and
$C_{1,\text{wa}}^{\text{{ps}}}$. For these, it is helpful to consider also Eq.
(2.39) of Ref. [58] and Eq. (53) above.

Most of the structure is common thanks to the fact that the shower reproduces
the relevant real kinematic configurations, namely, a single hard-collinear
emission, or a single soft-wide-angle emission, or a double-soft pair of
emissions, each accompanied by an arbitrary number of well-separated soft-
collinear emissions.

The $\mathcal{F}_{\textsc{nnll}}$ in ARES also receives a contribution from
the 1-loop correction to the soft-collinear gluon emission, see the second
line of Eq. (2.39) of Ref. [58], ultimately contributing to $\delta
F_{\text{sc}}$. This is implicitly present in the shower formalism which
produces this term, through unitarity, by including a factor
$1+\frac{\alpha_{s}(k_{t})}{2\pi}K_{1}$ in $\alpha_{\text{eff}}$. In practice,
this gives rise to the term proportional to $K^{(1)}\equiv K_{1}$ in
$R^{\prime}_{\text{NNLL},\ell}$, cf. Eq. (3.32) of Ref. [58].

With these considerations in place, we are now only left with the soft-wide-
angle and hard-collinear contributions to $\mathcal{F}_{\textsc{nnll}}$, and
with the $\bar{C}_{1,\text{hc}}^{\text{{ps}}}$ and
$C_{1,\text{wa}}^{\text{{ps}}}$ coefficients. To address these, it is helpful
to note that the Sudakov in Eq. (49) differs from the convention used in ARES
as the latter defines it using the observable computed in the soft-collinear
limit. As already discussed in section .3, as long as this is done
consistently in $S(\epsilon v)$ and in the phase-space condition for resolved
real emissions, cf. e.g. Eq. (53), there is a degree of flexibility in
defining the Sudakov form factor, which also appears as a $1/S(v)$ prefactor
in $\mathcal{F}_{\textsc{nnll}}$. This flexibility amounts to reshuffling
contributions between $S$ and $\mathcal{F}_{\textsc{nnll}}$.

Due to the way the soft-collinear approximation is used for the observable in
the ARES approach, the $C_{1,\text{wa}}^{\text{{ps}}}$ term of our Eq. (66)
does not appear in the ARES Sudakov factor. Instead, a corresponding
contribution appears in the $\delta\mathcal{F}_{\text{wa}}$ term of Eq. (2.46)
of Ref. [58], as defined in Eqs. (3.33)–(3.35) of Ref. [51]. The
$f_{\text{wa}}(\eta,\phi)$ that appears there corresponds to our $g(y)$. It is
easy to verify the exact equivalence for the specific example of an additive
observable, cf. Eq. (C.23) of Ref. [51], which gives

$\frac{\alpha_{s}(Q)}{\pi}\delta\mathcal{F}_{\text{wa}}=8C_{F}\frac{\alpha_{s}(v)}{2\pi}\mathcal{F}_{\text{{nll}}}\int_{0}^{\infty}dy\,g(y)\,,$
(71)

where on the right-hand side, we have translated to our notation. Since our
$C_{1,\text{wa}}^{\text{{ps}}}$, as part of $S$, multiplies $\mathcal{F}$ in
Eq. (57), the ARES and shower wide-angle contributions are identical. For more
general observables, the additional contributions that start at
$\alpha_{s}^{2}$ are contained in $\mathcal{F}$ both in ARES and in our
analytic analysis of the shower’s prediction.

A similar but slightly more involved argument holds in the hard-collinear
region. There, the $\bar{C}_{1,\text{hc}}^{\text{{ps}}}$ term corresponds to
two contributions in ARES: one from the hard-collinear constant
$C^{(1)}_{\text{hc},\ell}$ in Eq. (2.41) of Ref. [58], and a second one from
the $\mathcal{O}(\alpha_{s})$ contribution,
$\delta\mathcal{F}^{(1)}_{\text{rec}}$, to $\delta\mathcal{F}_{\text{rec}}$ in
Eq. (2.46). The $C^{(1)}_{\text{hc},\ell}$ coefficient is obtained when
imposing the condition

$z^{-\beta_{\text{obs}}}k_{t,\textsc{ares}}^{1+\beta_{\text{obs}}}>v\,,$ (72)

where, for a collinear $\tilde{q}\to qg$ splitting,
$k_{t,\textsc{ares}}=z(1-z)\theta_{qg}E_{\tilde{q}}=(1-z)k_{t}$. This is to be
supplemented [51, 58] with a recoil term $\delta\mathcal{F}_{\text{rec}}$,
which accounts for the ratio between Eq. (72) and the actual observable
condition $f(z)k_{t}e^{-\beta_{\text{obs}}y}>v$. Making use of $e^{y}=z/k_{t}$
and the relation between $k_{t}^{\text{{ares}}}$ and $k_{t}$, one can rewrite
the actual observable condition as

$f(z)z^{-\beta_{\text{obs}}}\left(\frac{k_{t,\textsc{ares}}}{1-z}\right)^{1+\beta_{\text{obs}}}>v\,,$
(73)

which allows us to determine

$\delta\mathcal{F}_{\text{rec}}^{(1)}=\frac{2}{1+\beta_{\text{obs}}}\int_{0}^{1}dz\,P_{gq}(z)\log\left(\frac{(1-z)^{1+\beta_{\text{obs}}}}{f(z)}\right).$
(74)

The sum of these two contributions gives

$C^{(1)}_{\text{hc},\ell}+\delta\mathcal{F}_{\text{rec}}^{(1)}=\frac{-2}{1+\beta_{\text{obs}}}\int_{0}^{1}dz\,P_{gq}(z)\log
f(z)+C_{F}\left(3+\frac{7}{2}\frac{\beta_{\text{obs}}}{1+\beta_{\text{obs}}}-\frac{2\pi^{2}}{3}\right)=\bar{C}_{1,\text{hc}}^{\text{{ps}}},$
(75)

reproducing Eq. (69c). Once this is satisfied, the ARES and shower hard-
collinear contributions only differ by a reshuffling similar to the one done
at large angle in Eq. (71), hence not affecting NNLL accuracy. Note that the
critical connection between an integrated $B_{2}(z)$ ($\beta_{0}X$ type
terms), $C^{(1)}_{\text{hc},\ell}$ and $\delta\mathcal{F}_{\text{rec}}$ has
been commented on before in Ref. [63] in the context of NNLL calculations for
groomed jet observables.

With this, all the terms have now been mapped between our shower and the ARES
formalisms, guaranteeing that, with the new ingredients introduced in this
paper, our shower algorithms achieve NNLL accuracy.

####  .4.2 Expressions for 3-loop CMW running coupling

For practical implementation in our showers, we have implemented Eq. (2)
factorising the genuine CMW running coupling from additional NNLL
contributions, namely

$\alpha_{\text{eff}}(k_{t})=\alpha_{s}^{\textsc{cmw}}(k_{t})\times\left\\{1+\tanh\left[\frac{\alpha_{s}^{\overline{\textsc{ms}}}(k_{t})}{2\pi}\left(\Delta
K_{1}(y)+B_{2}(z)\right)+\left(\frac{\alpha_{s}^{\overline{\textsc{ms}}}(k_{t})}{2\pi}\right)^{2}\Delta
K_{2}\right]\right\\}\,,$ (76)

with $\Delta K_{2}=K_{2}^{\text{{ps}}}-K_{2}^{\text{resum}}$. For the first
factor, we use an expansion valid at NNLL accuracy:

$\displaystyle\alpha_{s}^{\textsc{cmw}}(k_{t})=$
$\displaystyle\frac{\alpha_{s}}{1+t}+\frac{\alpha_{s}^{2}}{(1+t)^{2}}\left(\frac{-b_{1}}{b_{0}}\ln(1+t)+\frac{K_{1}}{2\pi}\right)$
(77)
$\displaystyle+\frac{\alpha_{s}^{3}}{(1+t)^{3}}\left[\frac{-b_{2}}{b_{0}}t+\frac{b_{1}^{2}}{b_{0}^{2}}\left(t-\ln(1+t)+\ln^{2}(1+t)\right)+\frac{K_{2}^{\text{resum}}}{4\pi^{2}}-\frac{b_{1}K_{1}}{b_{0}\pi}\ln(1+t)\right],$

with $\alpha_{s}\equiv\alpha_{s}^{\overline{\textsc{ms}}}(Q)$,
$t=2\alpha_{s}b_{0}\ln(k_{t}/Q)$, and the following coefficients for the QCD
$\beta$-function and CMW $K_{2}$ coefficient

$\displaystyle b_{0}$ $\displaystyle=\frac{11C_{A}-4n_{f}T_{R}}{12\pi},$ (78a)
$\displaystyle b_{1}$
$\displaystyle=\frac{17C_{A}^{2}-10C_{A}n_{f}T_{R}-6C_{F}n_{f}T_{R}}{24\pi^{2}},$
(78b) $\displaystyle b_{2}$
$\displaystyle=\frac{2857C_{A}^{3}+(54C_{F}^{2}-615C_{F}C_{A}-1415C_{A}^{2})2n_{f}T_{R}+(66C_{F}+79C_{A})4n_{f}^{2}T_{R}^{2}}{3456\pi^{3}},$
(78c) $\displaystyle K_{2}^{\text{resum}}$
$\displaystyle=\left(\frac{245}{24}-\frac{67\pi^{2}}{54}+\frac{11}{6}\zeta_{3}+\frac{11\pi^{4}}{180}\right)C_{A}^{2}+\left(4\zeta_{3}-\frac{55}{12}\right)C_{F}n_{f}T_{R}+\left(\frac{10\pi^{2}}{27}-\frac{209}{54}-\frac{14\zeta_{3}}{3}\right)C_{A}n_{f}T_{R}$
$\displaystyle\phantom{=}\;-\frac{4}{27}n_{f}^{2}T_{R}^{2}+\frac{1}{2}\pi
b_{0}\left[\left(\frac{808}{27}-28\zeta_{3}\right)C_{A}-\frac{224}{27}n_{f}T_{R}\right].$
(78d)

The factor in the curly brackets of Eq. (76) incorporates all the shower
contributions beyond the CMW running. The specific form we have used only
introduces subleading contributions and guarantees both that the correction is
never negative and that it can easily be treated as an acceptance factor in
the Sudakov veto algorithm of the shower.

###  .5 Non-perturbative tuning

parameter | PG${}_{0}^{\text{sdf}}$-24A | PG0-24A | PG1/2-24A | PG${}_{0}^{\text{sdf}}$-M13 | Monash13
---|---|---|---|---|---
$\alpha_{s}(M_{Z})$ | 0. | 118 | 0. | 118 | 0. | 118 | 0. | 118 | 0. | 1365
use CMW for $\alpha_{s}$ | true | true | true | true | false
$n$ loops for $\alpha_{s}$ | 3 | 3 | 3 | 3 | 1
$k_{t,\min}$ shower cutoff | $0.5\,\mathrm{GeV}$ | $0.5\,\mathrm{GeV}$ | $0.5\,\mathrm{GeV}$ | $0.5\,\mathrm{GeV}$ | $0.5\,\mathrm{GeV}$
StringPT:sigma | 0. | 3026 | 0. | 294 | 0. | 29 | 0. | 335 | 0. | 335
StringPT:enhancedFraction | 0. | 0084 | 0. | 0107 | 0. | 0196 | 0. | 01 | 0. | 01
StringPT:enhancedWidth | 1. | 6317 | 1. | 5583 | 2. | 0 | 2. | 0 | 2. | 0
StringZ:aLund | 0. | 6553 | 0. | 7586 | 0. | 6331 | 0. | 68 | 0. | 68
StringZ:bLund | 0. | 7324 | 0. | 7421 | 0. | 5611 | 0. | 98 | 0. | 98
StringZ:aExtraDiquark | 0. | 9713 | 0. | 7267 | 0. | 8707 | 0. | 97 | 0. | 97
Table 2:  Top rows: parameters for the QCD running coupling used in each of
our showers, and the corresponding values in the Monash 13 tune for the Pythia
8 shower. Bottom rows: parameters used in the Pythia 8.311 hadronisation model
when interfaced to each of our showers. Other non-perturbative parameters
coincide with the Monash13 tune.

Figure 5: Results from the PG${}^{\text{sdf}}_{\beta_{\text{{ps}}}=0}$ NNLL
parton shower, supplemented with Pythia 8.311 hadronisation, compared to a
range of data, using both the PG${}_{0}^{\text{sdf}}$-24A and the
PG${}_{0}^{\text{sdf}}$-M13 tunes from Table 2. The two tunes have identical
perturbative parameters, but different non-perturbative parameters and the
plots illustrate that predictions for infrared safe observables (top three
rows) are largely unaffected by the change in non-perturbative parameters,
except at the edge of the perturbative region.

Table 2 shows the non-perturbative parameters of Pythia 8.311 that have been
tuned relative to the default Monash 2013 tune [84]. The tunes are based on
ALEPH [82, 79] and L3 [83] data and have been produced with our own proof-of-
concept tuning framework.

In addition to modifying non-perturbative parameters, we also take
$\alpha_{s}(M_{Z})=0.1180$ with a 3-loop running coupling. In contrast, the
Monash 2013 tune uses $\alpha_{s}(M_{Z})=0.1365$ with a 1-loop running
coupling. Note that the latter does not include a $K_{1}$ contribution. If one
interprets $\alpha_{s}(M_{Z})=0.1365$ as a CMW-scheme coupling, the
corresponding $\overline{\textsc{ms}}$ value would be
$\alpha_{s}^{\overline{\textsc{ms}}}(M_{Z})=0.1276$.

To appreciate the impact of the non-perturbative parameter choices, Fig. 5
shows results with the PG${}^{\text{sdf}}_{\beta_{\text{{ps}}}=0}$ shower and
two tunes from Table 2: the Monash13 tune (PG${}_{0}^{\text{sdf}}$-M13, with
$\alpha_{s}(M_{Z})=0.1180$) and the dedicated tune presented here
(PG${}_{0}^{\text{sdf}}$-24A). (results for the other showers are broadly
similar). For infrared safe observables (top three rows) the impact of the
change in parameters is negligible except (a) where experimental uncertainties
grow large and (b) deep in the 2-jet region where the Sudakov suppression
involves non-perturbative physics. This gives confidence that the broad
agreement that we see for these observables has not simply been artificially
engineered by the tuning of the non-perturbative parameters. For all
observables in the top two rows and the Thrust Major on the third row, our
showers bring NNLL accuracy. The remaining observables on the third row, i.e.
the Thrust minor, $y_{34}$ and $y_{45}$ have the property that they are non-
zero starting only from four or more particles and for these we do not claim
NNLL accuracy.555For the Thrust minor and $y_{34}$ ($y_{45}$) the usual
$\ln\Sigma$ accuracy classification applies only if one requires three (four)
hard jets. NNLL parton shower accuracy would then additionally require the
shower to have 3-jet (4-jet) NLO accuracy.  Agreement remains generally good,
though notably in the $4$ and $5$-jet regions this is at least in part
accidental, given the lack of corresponding fixed-order matrix elements in the
shower. For infrared unsafe observables (bottom row), such as the distribution
of the number of charged tracks ($N_{\text{ch}}$), the particle momentum
distribution ($\xi_{p}=-\ln(2|\vec{p}|/Q)$) or the rapidity of particles with
respect to the Thrust axis ($y_{T}$), the tunes have a significant impact.

We stress that this tuning exercise should be considered as exploratory. For
example, we have not made any effort to address the question of theory
uncertainties. From a perturbative viewpoint, we do not include any heavy-
quark effects (all quarks, including charm and bottom, are treated as
massless). Also, we have made no efforts to tune the non-perturbative
parameters affecting the rates of various identified particles ($\pi^{0}$,
$\pi^{\pm}$, $K$, etc…). Nevertheless, this tuning exercise does show that the
good agreement with infrared safe LEP $Z$-pole observables is not
significantly affected by variations of the non-perturbative parameters and
that distributions sensitive to non-perturbative physics improve after tuning.