To understand the likelihood of this occurring, we first introduce the
shorthand

$\ell_{ij}=\ln\frac{p_{ti}}{p_{tj}}\,.$ (33)

Let us first consider $p_{t4}<p_{t2}$ ($\ell_{24}>0$). The $d_{24}$ will be
the smallest one when

$\displaystyle d_{24}<d_{23}$ $\displaystyle\to\ell_{14}+{\cal O}\\!\left(\ln
z_{41}\right)<\frac{1}{z_{23}},$ (34a) $\displaystyle d_{24}<d_{4B}$
$\displaystyle\to\ell_{14}+{\cal O}\\!\left(\ln
z_{41}\right)<\frac{e^{-\ell_{24}}}{z_{41}}.$ (34b)

Ignoring azimuthal integrals (and the rapidity of $2$ and $3$), we now have to
integrate over four phase-space variables, which we take to be $\ell_{14}$,
$\ell_{24}$, $z_{23}$ and $z_{41}$. We have found that we can ignore the
${\cal O}\\!\left(\ln z_{41}\right)$ terms in Eq. (34) and rewrite the limits
as

$z_{23}<\frac{1}{\ell_{14}}\,,\quad
z_{41}<\frac{e^{-\ell_{24}}}{\ell_{14}}\,,\quad[\ell_{24}>0]\,.$ (35)

Since both $z$ fractions will be small, we will perform the integrations over
$z_{23}$ and $z_{41}$ using the constant small-$z$ limit of $P_{g\to
q\bar{q}}(z)dz$. The overall rate of $24$ clustering (with $l_{24}>0$) is then
given by

$N_{24,\text{flav-}k_{t}}^{(\ell_{24}>0)}\sim\alpha_{s}^{3}\int_{0}^{\infty}\\!\\!d\ell_{14}\int_{0}^{\ell_{14}}\\!\\!d\ell_{24}\int_{0}^{\frac{1}{\ell_{14}}}\\!\\!dz_{23}\int_{0}^{\frac{e^{-\ell_{24}}}{\ell_{14}}}\\!\\!dz_{41}\\\
\sim\alpha_{s}^{3}\int^{\infty}\frac{d\ell_{14}}{\ell_{14}^{2}}\,,$ (36)

where in setting the lower limits of the $\ell_{14}$ and $\ell_{24}$ integrals
to zero, we are ignoring any constraints from the interplay, e.g., with the
$z_{23}$ integral. A similar analysis can be carried out for $\ell_{24}<0$ (or
equivalently $\ell_{42}>0$), giving

$z_{41}<\frac{1}{\ell_{14}}\,,\quad
z_{23}<\frac{e^{-\ell_{42}}}{\ell_{14}}\,,\quad[\ell_{42}>0]\,,$ (37)

and yielding

$N_{24,\text{flav-}k_{t}}^{(\ell_{42}>0)}\sim\alpha_{s}^{3}\int^{\infty}\frac{d\ell_{14}}{\ell_{14}^{2}}\,.$
(38)

This and Eq. (36) both converge in the infrared, i.e. for
$\ell_{14}\to\infty$, however this convergence is extremely slow. In
particular, if one places an upper limit $p_{t4}>\epsilon$, the result
converges as $(\ln 1/\epsilon)^{-1}$, which is consistent also with what we
find in our numerical tests, cf. the magenta ($\alpha=2$) points in Fig. 16.
While this is strictly IRC safe at this order, one should worry that at the
next order there may be logarithmic enhancements proportional to $\ell_{14}$
(for example from running-coupling effects), which would be sufficient to make
the integral diverge. Accordingly, it would seem wise for future uses of the
flavour-$k_{t}$ algorithm to adopt the same kind of $\Delta
R_{ij}^{2}\to\Omega_{ij}^{2}$ replacement as used in our IFN algorithm, and
similarly for any other algorithms that make use of similarly defined
distances, e.g. the GHS flavour-dressing algorithm.

Figure 16: Failure rate of the flavour-$k_{t}$ algorithm for the configuration
of Fig. 15, in particular illustrating results for $\alpha=2$ (magenta points)
that are qualitatively consistent with the expected $1/\ln p_{t,\max}$
behaviour (dashed line). Also shown are results for $\alpha=1$, as well as the
results for flavour-$k_{t,\Omega}$, i.e. the adaptation with an $\Omega_{ij}$
angular distance, illustrating the much faster drop of the failure rate.

A final comment concerns the $\alpha=1$ case. The analysis is somewhat more
involved than for $\alpha=2$, and it is also clear from Fig. 16 that the issue
is reduced with $\alpha=1$. Our investigations are consistent with a
$1/\ln^{p}\epsilon$ scaling, with a larger value of $p$ than for the
$\alpha=2$ case. One might wish to investigate this point further, however it
would anyway seem wise to use the $\Delta R_{ij}^{2}\to\Omega_{ij}^{2}$
replacement also for $\alpha=1$.

###  C.2 IHC2 issue at $\alpha_{s}^{2}$ for CMP

Figure 17:  Example $\mathcal{O}(\alpha_{s}^{2})$ configuration that yields an
issue for the CMP algorithm. There are two oppositely-flavoured initial-state
hard-collinear splittings ($b$ and $\bar{b}$, labelled 1 and 2), and a hard
particle $3$ at central rapidity.

An issue arises in the CMP algorithm at order $\alpha_{s}^{2}$ for a
configuration like the one shown in Fig. 17. We consider two initial-state
hard-collinear emissions, a $q$ and a $\bar{q}$ (labelled 1 and 2), from the
forward and backward beams respectively, and additionally one hard large-angle
particle (numbered 3 in the figure). The initial-state hard-collinear
emissions have a very small transverse momentum ($p_{t1},p_{t2}\ll p_{t3}$)
but large energies ($E_{1},E_{2}\sim E_{3}$). Let us assume $p_{t3}=E_{3}=1$,
and particle $3$ is simply aligned along the $x$-axis. Then we have that
$y_{1}\sim-\ln p_{t1}$ and $y_{2}\sim\ln p_{t2}$ in the hard-collinear limit.
For simplicity, we will work with $R=1$.

The CMP algorithm will strongly favour clustering $1$ and $2$ together first.
The global scale ($p_{t,\text{global-max}}$ in Eq. (5)) is set by the $p_{t}$
of the hardest pseudojet currently available, $p_{t3}$, so the value of
$\kappa_{12}$ is small,

$\kappa_{12}=\frac{1}{2a}\frac{p_{t1}^{2}+p_{t2}^{2}}{p_{t3}^{2}}\ll 1\,,$
(39)

and the distance between the oppositely-flavoured particles $1$ and $2$ is
thus given by

$\displaystyle d_{12}$
$\displaystyle=\frac{1}{\max(p_{t1}^{2},p_{t2}^{2})}\Delta
R_{12}^{2}\left(1-\cos\left(\frac{\pi}{2}\kappa_{12}\right)\right)$ (40a)
$\displaystyle\simeq\frac{1}{\max(p_{t1}^{2},p_{t2}^{2})}\Delta
R_{12}^{2}\frac{1}{2}\left(\frac{\pi}{2}\kappa_{12}\right)^{2}$ (40b)
$\displaystyle\simeq\frac{\pi^{2}}{32a^{2}}\Delta
R_{12}^{2}\frac{\max(p_{t1}^{2},p_{t2}^{2})}{p_{t3}^{4}},$ (40c)
$\displaystyle\simeq\frac{\pi^{2}}{32a^{2}}\left(\ln\frac{p_{t3}}{p_{t1}}+\ln\frac{p_{t3}}{p_{t2}}\right)^{2}\frac{\max(p_{t1}^{2},p_{t2}^{2})}{p_{t3}^{4}},$
(40d)

where $\Delta R_{12}$ is dominated by the large rapidity difference. The other
distances are

$\displaystyle d_{iB}$ $\displaystyle=\frac{1}{p_{ti}^{2}}\,,\qquad
i=\\{1,2,3\\}\,,$ (41a) $\displaystyle d_{i3}$
$\displaystyle\sim\frac{y_{i}^{2}}{p_{t3}^{2}}\,,\qquad i=\\{1,2\\}\,.$ (41b)

When $p_{t1},p_{t2}\ll p_{t3}$, it is straightforward to see that
$d_{12}<d_{1B},d_{2B}$ and $d_{12}<d_{13},d_{23}$ (the logarithms in Eq. (40d)
have no impact on this). Therefore the first step of the algorithm will be to
cluster particles $1$ and $2$, giving a flavourless pseudojet with transverse
momentum, rapidity and squared invariant mass of

$\displaystyle p_{t,(1+2)}$ $\displaystyle\sim\max(p_{t1},p_{t2})\,,$ (42a)
$\displaystyle y_{(1+2)}$
$\displaystyle\simeq\frac{1}{2}\ln\frac{E_{1}}{E_{2}}\,,$ (42b) $\displaystyle
m_{(1+2)}^{2}$ $\displaystyle\simeq 4E_{1}E_{2}\,.$ (42c)

From the point of view of standard jet clustering, the $(1+2)$ pseudojet is
unusual, because its transverse momentum is much smaller than its invariant
mass.

The $(1+2)$ pseudojet will cluster with particle $3$ if it is within a
distance $\Delta R_{(1+2),3}<R=1$. For any $p_{t1},p_{t2}\ll p_{t3}$, there is
always a finite azimuthal and $E_{1},E_{2}$ phase space region such that that
condition is satisfied, and the resulting $1+2+3$ cluster will have
significantly different kinematics than particle $3$, because of the extra
energy brought by particles $1$ and $2$. Thus the jets can differ between the
$1,2,3$ event and the event with just particle $3$. The rate for this to
happen is given by constants from the azimuthal and energy integrations
multiplying divergent integrals over $p_{t1}$ and $p_{t2}$,

$N\sim\alpha_{s}^{2}\int^{p_{t3}}_{\epsilon}\frac{dp_{t1}}{p_{t1}}\int^{p_{t3}}_{\epsilon}\frac{dp_{t2}}{p_{t2}}=\alpha_{s}^{2}\ln^{2}\epsilon,$
(43)

where we have explicitly included a cutoff scale $\epsilon$ in order to make
the nature of the divergence manifest.

In Sec. IV.4, we proposed a modification of the CMP algorithm, Eq. (11). For a
generic $\omega$, Eq. (40d) in particular is replaced by

$\displaystyle d_{12}^{(\Omega)}$
$\displaystyle\sim\left(\frac{p_{t3}^{2}}{p_{t1}p_{t2}}\right)^{\omega}\frac{\max(p_{t1}^{2},p_{t2}^{2})}{p_{t3}^{4}},$
(44a)
$\displaystyle\sim\frac{\max(p_{t1}^{2},p_{t2}^{2})}{p_{t1}p_{t2}p_{t3}^{2}}\cdot\left(\frac{p_{t3}^{2}}{p_{t1}p_{t2}}\right)^{\omega-1}\,.$
(44b)

When $\omega=1$, $d_{12}^{(\Omega)}$ will be of same order as $d_{3B}$ if
$p_{t1}\sim p_{t2}$. Schematically that suggests that there can still be a
divergence of the form

$N\sim\alpha_{s}^{2}\int^{p_{t3}}_{\epsilon}\frac{dp_{t1}}{p_{t1}}\int^{p_{t3}}_{\epsilon}\frac{dp_{t2}}{p_{t2}}\delta\left(\ln
p_{t2}-\ln p_{t1}\right)=\alpha_{s}^{2}\ln\epsilon,$ (45)

i.e. with one power of the logarithm. With a more complete calculation one can
verify that that divergence is indeed present for $\omega=1$. For $\omega>1$,
the second factor in Eq. (44b) instead ensures that $d_{12}^{(\Omega)}\gg
d_{3B}$, thus ensuring particle $3$ becomes a jet before any $1+2$ clustering,
and resolving the IRC safety issue with the kinematics of jet $3$.171717Once
particle $3$ has been declared a jet and removed from the clustering, the
$1+2$ cluster, if formed, would become a jet in its own right. Standard jet
analyses place a cut on the jet $p_{t}$, which ensures that a residual lone
massive, but low-$p_{t}$ $1+2$ cluster does not count as a hard jet. However,
one could imagine a scenario where one cuts not on $p_{t}$, but on
$p_{t}^{2}+m^{2}$, and in this case the $1+2$ cluster would count as an
additional hard jet. In such a case, one could envisage that this would cause
IRC unsafety even for $\omega>1$. We have not explored this question further,
insofar as analyses do not normally cut on $p_{t}^{2}+m^{2}$.

Figure 18:  Results of the numerical tests for CMP for a configuration as in
Fig. 17, with the $\Delta R^{2}$ angular factor in the distance measure, as in
the original algorithm (in blue), and with a corrected angular factor as in
Eq. (11) where $\omega=1$ (in orange), and $\omega=2$ (in red). Miniature
plots on the right depict the integrated failure rate on a linear scale to
help read the functional form of the divergence. The original algorithm
suffers from a quadratic divergence, the corrected measure with $\omega=1$
from a linear divergence, and the fix with $\omega=2$ is IRC safe.

Results of the numerical IRC safety tests are presented in Fig. 18 for the
original algorithm (with an angular factor $\Delta R^{2}$ in the distance
measure, in blue) and for the algorithm with an IRC-safe angular distance (for
values of the parameter $\omega=1$, in orange, and $\omega=2$, in red). The
results confirm the above analyses, showing a quadratic divergence in the
original algorithm, while the divergence is linear for $\omega=1$ and fully
lifted for $\omega=2$.181818Note that Fig. 1 (left) of Ref. [10] has studied
$pp\to\ell^{+}\ell^{-}bb\bar{b}$ (with massless $b$’s), which should include
the configuration of our Fig. 17. That figure does not appear to show a
divergence as the technical cutoff is reduced. At first sight that may seem
surprising, however two considerations should be kept in mind. Firstly, the
configuration of Fig. 17 requires an initial-state gluon to originate from a
$b\to gb$ splitting, which is responsible for only a small fraction of
incoming gluons. Secondly, the impact of the IRC unsafety is to smear the
rapidity distribution of the $b$-jet and a smearing of broad distribution
tends to have a limited impact on the integral of the broad distribution
within some window (Ref. [10] used $|y_{b}|<2.4$). As a result, it is
conceivable that the expected squared logarithmic divergence in Fig. 1 (left)
of Ref. [10] might be too small to clearly see in that figure.

Finally, note that the flavour-$k_{t}$ algorithm does not suffer from the
issue presented here thanks to the form of its beam distance: the initial-
state emissions $1$ and $2$ would be declared as beam jets and so be removed
from further consideration early in the clustering sequence.

###  C.3 IHC$\times$IDS issue at $\alpha_{s}^{3}$ for CMP

The subtlety from Sec. C.1 for the flavour-$k_{t}$ and GHS algorithms has an
interesting manifestation in the CMP algorithm. We consider again the scenario
of Fig. 15, with the same set of variables and in a configuration where
$p_{t4}\ll p_{t2}$. We will concentrate on the two distances that are
smallest, which, neglecting $\mathcal{O}(1)$ factors, read

$d_{23}\sim\frac{p_{t2}^{2}}{z_{23}^{2}p_{t1}^{4}},\quad
d_{24}\sim\frac{p_{t2}^{2}}{p_{t1}^{4}}y_{4}^{2}\,.$ (46)

As before, the probability for a 2+4 recombination must be finite for IRC
safety. The 2+4 recombination will occur if

$z_{23}\lesssim\frac{1}{y_{4}}=\frac{1}{\ell_{14}+\ln 2z_{41}}.$ (47)

We neglect the $\ln 2z_{41}$ term in the denominator (and take the $z_{41}$
integral to give a constant), integrate over all $p_{t4}$ values, over
$p_{t2}>p_{t4}$ and over the allowed $z_{23}$ range (with the same constant
splitting function approximation as in App. C.1). We then obtain the
probability for a $2+4$ clustering

$N_{24}\sim\alpha_{s}^{3}\int_{0}^{\infty}d\ell_{14}\int_{0}^{\ell_{14}}d\ell_{24}\int_{0}^{1/\ell_{14}}dz_{23},$
(48)

which is divergent. If we regulate the upper integration region of the
$\ell_{14}$ integral with $\infty\to\ln\frac{1}{\epsilon}$, the probability
scales as

$N_{24}\sim\alpha_{s}^{3}\ln\frac{1}{\epsilon}.$ (49)

As with the other occurrences of this kind of issue, the replacement Eq. (11)
solves the problem, as can be seen from the numerical IRC safety tests for
this configuration shown in Fig. 19.

Figure 19: Failure rate of the CMP algorithm for the configuration of Fig. 15,
both for the original formulation and our modification, Eq. (11), showing a
divergence for the former and none for the latter.

###  C.4 FHC2 issue at $\alpha_{s}^{2}$ for GHS

For the discussion here and in App. C.5, we assume a version of the GHS
algorithm with an $\Omega_{ij}^{2}$ style angular distance in the dressing
(flavour-$k_{t}$-like) phase, since we know from Sec. C.1 that this is
required for the flavour-$k_{t}$ algorithm, which is the basis of the
algorithm’s dressing step. To make this clear, in plots where this
modification is used, we refer to the algorithm as GHSΩ.

Let us consider a hard event as that in Fig. 20. The event has four particles,
which we will call $g_{1}$, $q$, $\bar{q}$ and $g_{2}$, starting from the
left. We will work through the algorithm to see what happens if a hard, but
collinear gluon $g$, is emitted from the $q$, and splits collinearly to
$q^{\prime}\bar{q}^{\prime}$ (with $\Delta R_{q^{\prime}\bar{q}^{\prime}}\ll
1$ much smaller than any other scale in the problem).

Figure 20:  Example $\mathcal{O}(\alpha_{s}^{2})$ configuration that yields an
issue for the GHS algorithm. There are four hard particles (that one can
imagine recoiling against a hard gluon or electroweak system on the other side
of the event), a collinear emission of a hard gluon $g$ from one of the
flavoured particles (the $q$), which then splits collinearly to a flavoured
pair $q^{\prime}\bar{q}^{\prime}$.

Focusing on the hard event first (i.e. without the emission of the hard
collinear gluon $g$), we assume that the anti-$k_{t}$ algorithm clusters the
four particles into two jets ($j_{1}$ and $j_{2}$), as indicated in the
figure. We can further assume that $\Delta R_{g_{1}q},\Delta
R_{g_{2}\bar{q}}>R_{\text{cut}}\sim 0.1$, so that the hard gluons $g_{1}$ and
$g_{2}$ are not accumulated into $q$ and $\bar{q}$ in that phase of the
algorithm.

First we will consider the case $\alpha<2$. For any angular structure of the
event satisfying the above limits, we take the momenta of $g_{1},g_{2},q$ and
$\bar{q}$ such that the event without the $g\to q^{\prime}\bar{q}^{\prime}$
emission has the following properties

$\displaystyle d_{{q}j_{1}}>d_{{q}{\bar{q}}}>d_{{\bar{q}}j_{2}}\,,$ (50a)
$\displaystyle p_{tq}<p_{t\bar{q}}\,.$ (50b)

As a result, the first dressing step is for the $\bar{q}$ flavour to be
assigned to jet $j_{2}$, followed by the $q$ flavour being assigned to jet
$j_{1}$. Thus both $j_{1}$ and $j_{2}$ are flavoured. Note that for a full
analysis, one should also take into account $d_{iB}$ beam distances for all
flavoured particles $i$. To help understand why we can ignore it, suppose that
all the hard particles have rapidities close to zero, which results in
$p_{tB}(0)$ in Eq. (29) being approximately the scalar sum of all the
particles’ transverse momenta. That scale will tend to be a few times larger
than the transverse momenta of any of the individual particles, which ensures
that the distance of any cluster to its jet will be smaller than the $d_{iB}$,
as will the $d_{q\bar{q}}$ if the two jets are not too far away in angle.

Next we consider the impact of the emission of the collinear hard gluon from
$q$ with $p_{tg}=zp_{tq}$, followed by its splitting into a collinear
$q^{\prime}\bar{q}^{\prime}$ pair. Recall that we work with $\Delta
R_{q^{\prime}\bar{q}^{\prime}}\ll R_{\text{cut}}$ so that it is the smallest
angular distance in the event. The algorithm goes through the accumulation
step, and will identify four flavour clusters: $\hat{q}^{\prime}$,
$\hat{\bar{q}}^{\prime}$, and the original $\hat{q}$ and $\hat{\bar{q}}$. The
angular structure is otherwise unchanged, so we get no further flavour
accumulation. To lighten the notation, below we will leave out the explicit
“hats” for the flavour clusters, especially as the flavour clusters coincide
with the original particles.

The final step is the flavour dressing: the ${q}^{\prime},{\bar{q}}^{\prime}$
pair will annihilate first, as they should because they came from a common
parent gluon. These flavour clusters (including their kinematics) are
discarded from further consideration, and any distance involving them is
removed from the list. The remaining distances ($d^{\prime}$) for the event
with the $g\to q^{\prime}\bar{q}^{\prime}$ splitting are then given in terms
of the hard-event’s distances ($d$) as

$\displaystyle d^{\prime}_{{q}j_{1}}=(1-z)^{2-\alpha}d_{{q}j_{1}}\,,$ (51a)
$\displaystyle d^{\prime}_{{q}{\bar{q}}}=(1-z)^{2-\alpha}d_{{q}{\bar{q}}}\,,$
(51b) $\displaystyle d^{\prime}_{{\bar{q}}j_{2}}=d_{{\bar{q}}j_{2}}\,,$ (51c)

where the $(1-z)^{2-\alpha}$ factor arises because of the reduction in
transverse momentum of the $q$ after emission of the $g\to
q^{\prime}\bar{q}^{\prime}$ (which carries a fraction $z$ of its original $q$
momentum). The potentially dangerous scenario is that where the ordering of
distances, Eq. (50a), is modified,

$d^{\prime}_{{q}{\bar{q}}}<\min\left(d^{\prime}_{{q}j_{1}},d^{\prime}_{{\bar{q}}j_{2}}\right)\,,$
(52)

because then $q$ and $\bar{q}$’s flavours will annihilate, leaving flavourless
hard jets, associated with a squared logarithmic divergence from the two
nested hard collinear divergences. There is a finite range of $z$ in which
this occurs,

$1-z<\left(\frac{d_{{\bar{q}}j_{2}}}{d_{{q}{\bar{q}}}}\right)^{\frac{1}{2-\alpha}}\,,$
(53)

thus confirming the presence of IRC unsafety from the configuration of Fig. 20
for $\alpha<2$.

When $\alpha=2$, we instead consider a hard event satisfying
$p_{tq}>p_{t\bar{q}}$ rather than the inequality in Eq. (50b), in which case
we have

$\displaystyle d^{\prime}_{{q}j_{1}}=d_{{q}j_{1}}\,,$ (54a) $\displaystyle
d^{\prime}_{{q}{\bar{q}}}=\max\left((1-z)^{2},\frac{p_{t\bar{q}}^{2}}{p_{tq}^{2}}\right)d_{{q}{\bar{q}}}\,,$
(54b) $\displaystyle d^{\prime}_{{\bar{q}}j_{2}}=d_{{\bar{q}}j_{2}}\,.$ (54c)

Again, there is the possibility of $d^{\prime}_{{q}{\bar{q}}}$ becoming the
smallest of the three distances, with the outcome that the $q$ and $\bar{q}$
flavours would annihilate, leaving flavourless hard jets, with a squared
logarithmic divergence associated with the collinear splittings.

Figure 21: Failure rate of the GHS algorithm for the FHC$\times$FHC
configuration of Fig. 20, illustrating the quadratic divergence, specifically
for $\alpha=1$, $\beta=1$ and $\alpha=2$, $\beta=2$. Other parameters are
$R_{\text{cut}}=0.1$, $z_{\text{cut}}=0.1$ and
$p_{t,\text{cut}}=100\,\text{GeV}$. The jet radius $R$ has been sampled in the
range $0.3{-}1.57$. The version of the GHS algorithm used is one where $\Delta
R_{ij}^{2}$ in the dressing stage has been replaced with $\Omega_{ij}^{2}$
using $\omega=3-\alpha$ (the original $\Delta R_{ij}^{2}$ similarly gives a
squared logarithmic divergence).

The set of distances in the argument above is perhaps somewhat complicated,
with angular factors to consider, the beam distances and the extra subtleties
of the $\alpha=2$ case. Therefore, in Fig. 21 we show the outcome of our IRC
safety tests, illustrating that the divergence is indeed present for the two
combinations $\alpha=1$, $\beta=1$ and $\alpha=2$, $\beta=2$. We leave to
future work the possibility of identifying a concrete modification of the
algorithm that solves this problem, nevertheless we anticipate that one line
of investigation could be to allow accumulation of kinematics within a jet
during the dressing stage.

A final comment is that this configuration can appear at NNLO for a process
such as fully hadronic $t\bar{t}$ production, however only if one asks for two
massive $b$-tagged jets. It also appears at N4LO for a process such as
$Zb\bar{b}$ production.

###  C.5 IDS$\times$FDS issue at $\alpha_{s}^{4}$ for GHS

Figure 22:  An FDS$\times$IDS kinematic configuration that causes problems for
GHS algorithms for $\alpha\beta\geq 2$.

The GHS algorithm exhibits an interesting interplay between initial-state and
final-state double-soft emissions at order $\alpha_{s}^{4}$ if
$\alpha\beta\geq 2$. The configuration that we consider here is that
represented in Fig. 22, involving a hard Born event with one or more
unflavoured jets. That event is then supplemented with a double-soft pair
($1$,$2$) that is collinear to an (originally) unflavoured jet and an
additional large-angle double-soft pair ($3$,$4$) outside the jet.

We are specifically interested in the situation where

$\displaystyle\theta_{1g}$
$\displaystyle\lesssim\\{\theta_{2g},\theta_{12}\\}\ll 1\,,$ (55a)
$\displaystyle p_{t1}$ $\displaystyle\sim p_{t2}\ll p_{tg}\,,$ (55b)
$\displaystyle p_{t3}$ $\displaystyle\sim p_{t4}\ll p_{tg}\,,$ (55c)
$\displaystyle\theta_{23}$ $\displaystyle\sim\theta_{34}\sim 1\,.$ (55d)

During the accumulation step, there is a possibility that $1$ clusters with
$g$, giving a hard $\widehat{g1}$ flavour cluster, leaving an unclustered,
much softer $\hat{2}$ flavour cluster. If during the subsequent dressing
phase, $\hat{2}$ goes on to annihilate with $\hat{3}$ rather than with
$\widehat{g1}$, then the resulting hard jet will be flavoured.

The SoftDrop condition for $1$ to cluster with $g$ is given by

$\frac{p_{t1}}{p_{tg}}>\theta_{1g}^{\beta}\,,$ (56)

where throughout our discussion here we neglect factors of order $1$ (e.g.
$R_{\text{cut}}$ and $z_{\text{cut}}$). There is no further accumulation since
all particles are now flavoured and the flavoured clusters will be
$\widehat{g1}$, $\hat{2}$, $\hat{3}$ and $\hat{4}$. The angle between the
$\widehat{g1}$ and the jet direction will be given by

$\theta_{j\widehat{g1}}\sim\frac{p_{t2}}{p_{tg}}\theta_{2g}\,.$ (57)

Without loss of generality we can consider the case where
$d_{\hat{2}\hat{3}}<d_{\hat{2}\hat{4}}$, and then the distances to take into
account during the dressing phase are

$\displaystyle d_{j\widehat{g1}}$ $\displaystyle\sim
p_{t2}^{2}\theta_{2g}^{2}\,,$ (58a) $\displaystyle d_{\widehat{g1}\hat{2}}$
$\displaystyle\sim d_{j\hat{2}}\sim
p_{tg}^{2}\left(\frac{p_{t2}}{p_{tg}}\right)^{2-\alpha}\theta_{2g}^{2}\,,$
(58b) $\displaystyle d_{\hat{2}\hat{3}}$
$\displaystyle\sim\max(p_{t2},p_{t3})^{\alpha}\min(p_{t2},p_{t3})^{2-\alpha},$
(58c) $\displaystyle d_{\hat{3}\hat{4}}$ $\displaystyle\sim p_{t3}^{2},$ (58d)

where, again we have ignored factors of order $1$, e.g. from angular
distances. The hard jet will acquire a flavour if the SoftDrop condition of
Eq. (56) is satisfied and if additionally $\hat{2}$ fails to annihilate the
flavour of the $\widehat{g1}$ cluster. This will occur if $d_{23}<d_{34}$ and
$d_{23}<d_{j\hat{2}}\simeq d_{\widehat{g1}2}$.191919 Note that the
$d_{23}<d_{34}$ condition implies that $p_{t2}$ cannot be substantially larger
than $p_{t3}$, which leads to $d_{j\widehat{g1}}$ being the smallest of all
the distances. Consequently, the first step of the dressing is that the
flavour of $\widehat{g1}$ is assigned to the jet and the $\widehat{g1}$
cluster is removed from consideration.  In determining whether these
conditions are satisfied, it is helpful to introduce shorthands

$\displaystyle\ell_{i}$ $\displaystyle=\ln\frac{p_{tg}}{p_{ti}},$ (59a)
$\displaystyle\ell_{\theta}$ $\displaystyle=\ln\frac{1}{\theta_{2g}},$ (59b)

and to observe that in the SoftDrop condition Eq. (56), we can replace $1\to
2$, since this only affects ${\cal O}\\!\left(1\right)$ terms.

Figure 23: Failure rate of the GHS algorithm for the $\alpha_{s}^{4}$
FDS$\times$IDS configuration of Fig. 22, illustrating the cubic divergence for
parameter choices involving $\alpha\beta>2$, the linear divergence for
$\alpha\beta=2$ and convergence for $\alpha\beta<2$. In its dressing stage,
the GHS implementation for these runs uses an $\Omega_{ij}^{2}$ angular
distance instead of $\Delta R_{ij}^{2}$, where $\omega=3-\alpha$. Other
parameters are $R_{\text{cut}}=0.1$, $z_{\text{cut}}=0.1$ and
$p_{t,\text{cut}}=100\,\text{GeV}$.

Our conditions then become (to within ${\cal O}\\!\left(1\right)$ offsets)

SD $\displaystyle:\quad\ell_{2}<\beta\ell_{\theta}\,,$ (60a) $\displaystyle
d_{34}>d_{23}$ $\displaystyle:\quad\ell_{2}>\ell_{3}\,,$ (60b) $\displaystyle
d_{g2}>d_{23}$
$\displaystyle:\quad(2-\alpha)\ell_{2}+2\ell_{\theta}<\alpha\ell_{3}+(2-\alpha)\ell_{2}\,,$
(60c)

where the last line already underwent some simplification (using the second
line), and can then be further simplified to read

$2\ell_{\theta}<\alpha\ell_{3}\,.$ (61)

Assembling all inequalities, we obtain

$2\ell_{\theta}<\alpha\ell_{3}<\alpha\ell_{2}<\alpha\beta\ell_{\theta}\,.$
(62)

We immediately see that if $\alpha\beta<2$, there is no available logarithmic
integration region, and so no IRC divergence from this configuration.
Conversely, if $\alpha\beta>2$, we expect to see a cubic logarithmic
divergence from integrals over $\ell_{\theta}$, $\ell_{3}$ and $\ell_{2}$. For
$\alpha\beta=2$, the ${\cal O}\\!\left(1\right)$ factors become critical and
it is easiest to carry out a numerical study, but it is reasonable to expect a
divergence with a single logarithm.

The results of the numerical study are shown in Fig. 23 for four combinations
of $\alpha$ and $\beta$. They confirm our expectations and suggest that if one
wishes to employ a GHS-style algorithm, one should use it with
$\alpha\beta<2$. Nevertheless, one would still need to find a solution to the
separate issue identified in App. C.4 (which cannot be resolved just through
parameter choices), and then verify that the resulting algorithm passes a full
set of IRC safety tests.

##  Appendix D Summary plots for IRC-safe algorithms

In this appendix, we present summary plots from our IRC safety tests for the
three approaches that have passed all those tests: IFN, CMPΩ and
flavour-$k_{t,\Omega}$. In Figs. 24 and 25, we show the results from the IRC-
safety tests for the anti-$k_{t}$ and C/A algorithms with IFN, i.e. each of
the algorithms labelled as safe in Table 1. Figs. 26 and 27 shows
corresponding results for our adaptations of the flavour-$k_{t}$ and CMP
algorithms, supporting the conclusion that they too are IRC safe.

Figure 24: Summary of IRC safety test results at orders $\alpha_{s}$ to
$\alpha_{s}^{6}$ for the anti-$k_{t}$ algorithm with IFN.

Figure 25: Same as Fig. 24, for the C/A algorithm with IFN.

Figure 26: Same as Fig. 24, for the flavour-$k_{t,\Omega}$ algorithm.

Figure 27: Same as Fig. 24, for the CMPΩ algorithm.

## References

  * [1] A. Banfi, G. P. Salam, and G. Zanderighi, Infrared safe definition of jet flavor, Eur. Phys. J. C 47 (2006) 113–124, [hep-ph/0601139].
  * [2] A. Banfi, G. P. Salam, and G. Zanderighi, Accurate QCD predictions for heavy-quark jets at the Tevatron and LHC, JHEP 07 (2007) 026, [arXiv:0704.2999].
  * [3] S. Catani, Y. L. Dokshitzer, M. Olsson, G. Turnock, and B. R. Webber, New clustering algorithm for multi-jet cross-sections in $e^{+}e^{-}$ annihilation, Phys. Lett. B 269 (1991) 432–438.
  * [4] S. Catani, Y. L. Dokshitzer, M. H. Seymour, and B. R. Webber, Longitudinally invariant $K_{t}$ clustering algorithms for hadron hadron collisions, Nucl. Phys. B 406 (1993) 187–224.
  * [5] S. D. Ellis and D. E. Soper, Successive combination jet algorithm for hadron collisions, Phys. Rev. D 48 (1993) 3160–3166, [hep-ph/9305266].
  * [6] M. Cacciari, G. P. Salam, and G. Soyez, The anti-$k_{t}$ jet clustering algorithm, JHEP 04 (2008) 063, [arXiv:0802.1189].
  * [7] R. Gauld, A. Gehrmann-De Ridder, E. W. N. Glover, A. Huss, and I. Majer, Predictions for $Z$-Boson Production in Association with a $b$-Jet at $\mathcal{O}(\alpha_{s}^{3})$, Phys. Rev. Lett. 125 (2020), no. 22 222002, [arXiv:2005.03016].
  * [8] S. Caletti, A. J. Larkoski, S. Marzani, and D. Reichelt, Practical jet flavour through NNLO, Eur. Phys. J. C 82 (2022), no. 7 632, [arXiv:2205.01109].
  * [9] S. Caletti, A. J. Larkoski, S. Marzani, and D. Reichelt, A fragmentation approach to jet flavor, JHEP 10 (2022) 158, [arXiv:2205.01117].
  * [10] M. Czakon, A. Mitov, and R. Poncelet, Infrared-safe flavoured anti-kT jets, JHEP 04 (2023) 138, [arXiv:2205.11879].
  * [11] R. Gauld, A. Huss, and G. Stagnitto, Flavor Identification of Reconstructed Hadronic Jets, Phys. Rev. Lett. 130 (2023), no. 16 161901, [arXiv:2208.11138].
  * [12] S. Weinzierl, The Forward-backward asymmetry at NNLO revisited, Phys. Lett. B 644 (2007) 331–335, [hep-ph/0609021].
  * [13] Z. Trócsányi, G. Somogyi, and F. Tramontano, Fully Differential Decay Rate of a Standard Model Higgs Boson into Two $b$-jets at NNLO, Acta Phys. Polon. B 46 (2015), no. 11 2097.
  * [14] G. Ferrera, G. Somogyi, and F. Tramontano, Associated production of a Higgs boson decaying into bottom quarks at the LHC in full NNLO QCD, Phys. Lett. B 780 (2018) 346–351, [arXiv:1705.10304].
  * [15] F. Caola, G. Luisoni, K. Melnikov, and R. Röntsch, NNLO QCD corrections to associated $WH$ production and $H\to b\bar{b}$ decay, Phys. Rev. D 97 (2018), no. 7 074022, [arXiv:1712.06954].
  * [16] R. Gauld, A. Gehrmann-De Ridder, E. W. N. Glover, A. Huss, and I. Majer, Associated production of a Higgs boson decaying into bottom quarks and a weak vector boson decaying leptonically at NNLO in QCD, JHEP 10 (2019) 002, [arXiv:1907.05836].
  * [17] M. Czakon, A. Mitov, M. Pellen, and R. Poncelet, NNLO QCD predictions for W+c-jet production at the LHC, JHEP 06 (2021) 100, [arXiv:2011.01011].
  * [18] H. B. Hartanto, R. Poncelet, A. Popescu, and S. Zoia, Next-to-next-to-leading order QCD corrections to $Wb\bar{b}$ production at the LHC, Phys. Rev. D 106 (2022), no. 7 074016, [arXiv:2205.01687].
  * [19] H. B. Hartanto, R. Poncelet, A. Popescu, and S. Zoia, Flavour anti-$k_{\text{T}}$ algorithm applied to $Wb\bar{b}$ production at the LHC, arXiv:2209.03280.
  * [20] M. Czakon, A. Mitov, M. Pellen, and R. Poncelet, A detailed investigation of W+c-jet at the LHC, JHEP 02 (2023) 241, [arXiv:2212.00467].
  * [21] R. Gauld, A. Gehrmann-De Ridder, E. W. N. Glover, A. Huss, A. R. Garcia, and G. Stagnitto, NNLO QCD predictions for Z-boson production in association with a charm jet within the LHCb fiducial region, Eur. Phys. J. C 83 (2023), no. 4 336, [arXiv:2302.12844].
  * [22] Y. L. Dokshitzer, G. D. Leder, S. Moretti, and B. R. Webber, Better jet clustering algorithms, JHEP 08 (1997) 001, [hep-ph/9707323].
  * [23] M. Wobisch and T. Wengler, Hadronization corrections to jet cross-sections in deep inelastic scattering, in Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting Meeting), pp. 270–279, 4, 1998.  hep-ph/9907280.
  * [24] S. Hoeche, F. Krauss, S. Schumann, and F. Siegert, QCD matrix elements and truncated showers, JHEP 05 (2009) 053, [arXiv:0903.1219].
  * [25] A. Karlberg, G. P. Salam, L. Scyboz, and R. Verheyen, Spin correlations in final-state parton showers and jet observables, Eur. Phys. J. C 81 (2021), no. 8 681, [arXiv:2103.16526].
  * [26] M. van Beekveld, S. Ferrario Ravasio, K. Hamilton, G. P. Salam, A. Soto-Ontoso, G. Soyez, and R. Verheyen, PanScales showers for hadron collisions: all-order validation, JHEP 11 (2022) 020, [arXiv:2207.09467].
  * [27] G. P. Salam and G. Soyez, A Practical Seedless Infrared-Safe Cone jet algorithm, JHEP 05 (2007) 086, [arXiv:0704.0292].
  * [28] A. Buckley and C. Pollard, QCD-aware partonic jet clustering for truth-jet flavour labelling, Eur. Phys. J. C 76 (2016), no. 2 71, [arXiv:1507.00508].
  * [29] JADE Collaboration, W. Bartel et al., Experimental Studies on Multi-Jet Production in e+ e- Annihilation at PETRA Energies, Z. Phys. C 33 (1986) 23.
  * [30] JADE Collaboration, S. Bethke et al., Experimental Investigation of the Energy Dependence of the Strong Coupling Strength, Phys. Lett. B 213 (1988) 235–241.
  * [31] A. Behring, W. Bizoń, F. Caola, K. Melnikov, and R. Röntsch, Bottom quark mass effects in associated $WH$ production with the $H\to b\bar{b}$ decay through NNLO QCD, Phys. Rev. D 101 (2020), no. 11 114012, [arXiv:2003.08321].
  * [32] A. J. Larkoski, S. Marzani, G. Soyez, and J. Thaler, Soft Drop, JHEP 05 (2014) 146, [arXiv:1402.2657].
  * [33] B. Andersson, G. Gustafson, L. Lonnblad, and U. Pettersson, Coherence Effects in Deep Inelastic Scattering, Z. Phys. C 43 (1989) 625\.
  * [34] M. Cacciari, G. P. Salam, and G. Soyez, FastJet User Manual, Eur. Phys. J. C 72 (2012) 1896, [arXiv:1111.6097].
  * [35] Y. Hida, X. S. Li, and D. H. Bailey, Quad-double arithmetic: Algorithms, implementation, and application, in 15th IEEE Symposium on Computer Arithmetic, pp. 155–162, 2000.
  * [36] M. Dasgupta, F. A. Dreyer, K. Hamilton, P. F. Monni, G. P. Salam, and G. Soyez, Parton showers beyond leading logarithmic accuracy, Phys. Rev. Lett. 125 (2020), no. 5 052002, [arXiv:2002.11114].
  * [37] K. Hamilton, R. Medves, G. P. Salam, L. Scyboz, and G. Soyez, Colour and logarithmic accuracy in final-state parton showers, JHEP 03 (2021), no. 041 041, [arXiv:2011.10054].
  * [38] CMS Collaboration, A. M. Sirunyan et al., Evidence for the Higgs boson decay to a bottom quark–antiquark pair, Phys. Lett. B 780 (2018) 501–532, [arXiv:1709.07497].
  * [39] ATLAS Collaboration, G. Aad et al., Measurements of $WH$ and $ZH$ production in the $H\rightarrow b\bar{b}$ decay channel in $pp$ collisions at 13 TeV with the ATLAS detector, Eur. Phys. J. C 81 (2021), no. 2 178, [arXiv:2007.02873].
  * [40] ATLAS Collaboration, G. Aad et al., Measurement of the associated production of a Higgs boson decaying into $b$-quarks with a vector boson at high transverse momentum in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, Phys. Lett. B 816 (2021) 136204, [arXiv:2008.02508].
  * [41] CMS Collaboration, A. M. Sirunyan et al., Inclusive search for highly boosted Higgs bosons decaying to bottom quark-antiquark pairs in proton-proton collisions at $\sqrt{s}=$ 13 TeV, JHEP 12 (2020) 085, [arXiv:2006.13251].
  * [42] J. M. Butterworth, A. R. Davison, M. Rubin, and G. P. Salam, Jet substructure as a new Higgs search channel at the LHC, Phys. Rev. Lett. 100 (2008) 242001, [arXiv:0802.2470].
  * [43] S. Marzani, G. Soyez, and M. Spannowsky, Looking inside jets: an introduction to jet substructure and boosted-object phenomenology, vol. 958.  Springer, 2019.
  * [44] T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159–177, [arXiv:1410.3012].
  * [45] C. Bierlich et al., A comprehensive guide to the physics and usage of PYTHIA 8.3, arXiv:2203.11601.
  * [46] R. Corke and T. Sjostrand, Interleaved Parton Showers and Tuning Prospects, JHEP 03 (2011) 032, [arXiv:1011.1759].
  * [47] P. Skands, S. Carrazza, and J. Rojo, Tuning PYTHIA 8.1: the Monash 2013 Tune, Eur. Phys. J. C 74 (2014), no. 8 3024, [arXiv:1404.5630].
  * [48] P. Gras, S. Höche, D. Kar, A. Larkoski, L. Lönnblad, S. Plätzer, A. Siódmok, P. Skands, G. Soyez, and J. Thaler, Systematics of quark/gluon tagging, JHEP 07 (2017) 091, [arXiv:1704.03878].
  * [49] M. Dasgupta, F. Dreyer, G. P. Salam, and G. Soyez, Small-radius jets to all orders in QCD, JHEP 04 (2015) 039, [arXiv:1411.5182].
  * [50] S. Catani and M. H. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291–419, [hep-ph/9605323]. [Erratum: Nucl.Phys.B 510, 503–504 (1998)].
  * [51] M. van Beekveld, W. Beenakker, E. Laenen, and C. D. White, Next-to-leading power threshold effects for inclusive and exclusive processes with final state jets, JHEP 03 (2020) 106, [arXiv:1905.08741].