$\bigg{(}1+\frac{\kappa+\sqrt{8\mathtt{z}^{2}+\kappa^{2}}}{4\mathtt{z}}\bigg{)}(n_{+}-n_{-})=2p_{1}\,,$
(6.16)

which is a non-trivial Diophantine equation. One example of solution is given
by the set of values $\kappa=-1$, $\mathtt{z}=6$, $n_{+}-n_{-}=6$, $p_{1}=5$
and we have checked that other combinations exist. Moreover, when $\kappa=-1$
it is possible to smoothly take the limit $\mathtt{z}\to 0$, in which case the
equation reduces to $n_{+}-n_{-}=2p_{1}$.

We can now uplift the solution (6.5) to massive type IIA using the recipe
described in section 2.2. For simplicity, we will only present the relevant
ingredients for the computation of the entropy, namely the metric and the
dilaton, that read

$\displaystyle\begin{split}\mathrm{d}s_{\text{s.f.}}^{2}&=\lambda^{2}\mu_{0}^{-1/3}\biggl{\\{}\tilde{\Delta}^{1/2}\biggl{[}\mathrm{e}^{-2C}L_{\mathrm{AdS}_{4}}^{2}\biggl{(}\frac{y^{2}}{4}\,\mathrm{d}s_{\mathrm{AdS}_{2}}^{2}+\frac{y^{2}}{q(y)}\,\mathrm{d}y^{2}+\frac{q(y)}{4y^{2}}\,\mathrm{d}z^{2}\biggr{)}+\mathrm{e}^{2C}\mathrm{d}s_{\Sigma_{\mathrm{g}}}^{2}\biggr{]}\\\
&+g^{-2}\tilde{\Delta}^{-1/2}k_{8}^{-1/4}\bigl{[}k_{8}^{1/2}\mathrm{d}\mu_{0}^{2}+k_{2}^{-1/2}\bigl{(}\mathrm{d}\mu_{1}^{2}+\mu_{1}^{2}\sigma_{1}^{2}\bigr{)}+k_{2}^{1/2}\bigl{(}\mathrm{d}\mu_{2}^{2}+\mu_{2}^{2}\sigma_{2}^{2}\bigr{)}\bigr{]}\biggr{\\}}\,,\end{split}$
(6.17) $\displaystyle\mathrm{e}^{\Phi}$
$\displaystyle=\lambda^{2}\mu_{0}^{-5/6}\tilde{\Delta}^{1/4}k_{8}^{-1/8}\,,$
(6.18)

with $\sigma_{i}=\mathrm{d}\phi_{i}-gA_{i}$ and $\tilde{\Delta}$ given in
(2.38). Expressing the metric in string frame as

$\mathrm{d}s_{\text{s.f.}}^{2}=\mathrm{e}^{2A}\bigl{(}\mathrm{d}s_{\mathrm{AdS}_{2}}^{2}+\mathrm{d}s_{M_{8}}^{2}\bigr{)}\,,$
(6.19)

the effective two-dimensional Newton constant is given by

$\frac{1}{G_{(2)}}=\frac{32\pi^{2}}{(2\pi\ell_{s})^{8}}\int\mathrm{e}^{8A-2\Phi}\,\mathrm{vol}(M_{8})\,,$
(6.20)

and therefore the entropy reads

$S=\frac{1}{4G_{(2)}}=\frac{1}{(2\pi\ell)^{8}}\frac{9(3\pi\lambda)^{4}k_{8}^{1/2}}{20g^{8}k_{4}}\,4\pi\kappa(1-\mathrm{g})\,A_{h}\,,$
(6.21)

where $A_{h}$ is the area of the horizon of the four-dimensional black hole
with $L_{\mathrm{AdS}_{4}}=1$, specifically [7]

$A_{h}=\frac{1}{2}(y_{2}-y_{1})\Delta
z=\pi\,\frac{-(n_{+}+n_{-})+\sqrt{2}\sqrt{n_{+}^{2}+n_{-}^{2}}}{n_{+}n_{-}}\,.$
(6.22)

Both the Romans mass $F_{(0)}$ and the part of the four-form flux $F_{(4)}$
along the four-hemisphere $\mathbbl{S}^{4}$ remain unaltered with respect to
(2.36) and (2.37), therefore the quantization of the fluxes in ten dimensions
is unchanged, giving the relations (2.25). The final expression of the entropy
is

$S=\frac{8\kappa(1-\mathrm{g})N^{5/2}(\mathtt{z}^{2}-\kappa^{2})^{3/2}(\sqrt{\kappa^{2}+8\mathtt{z}^{2}}-\kappa)}{5\sqrt{8-N_{f}}(\kappa\sqrt{\kappa^{2}+8\mathtt{z}^{2}}-\kappa^{2}+4\mathtt{z}^{2})^{3/2}}\,A_{h}\,.$
(6.23)

###  6.2 Entropy function

As before, we can deduce the main ingredients of the field theory construction
from the supergravity solution. In particular, the magnetic fluxes
$\mathfrak{s}_{i}$ show that there is a (standard) topological twist on
$\Sigma_{\mathrm{g}}$, while from the form of the $\mathfrak{n}_{i}$ we see
that there is an anti-twist on $\mathbbl{\Sigma}$. Then applying our
conjecture twice, we obtain the following entropy function

$\begin{split}S(\varphi_{i},\epsilon_{1},\epsilon_{2};\mathfrak{n}_{i},\mathfrak{s}_{i})&=-\frac{1}{4\epsilon_{1}\epsilon_{2}}\Big{[}\mathcal{F}_{5}(\varphi_{i}+\mathfrak{n}_{i}\epsilon_{1}+\mathfrak{s}_{i}\epsilon_{2})+\mathcal{F}_{5}(\varphi_{i}-\mathfrak{n}_{i}\epsilon_{1}+\mathfrak{s}_{i}\epsilon_{2})\\\
&\qquad\qquad-\mathcal{F}_{5}(\varphi_{i}+\mathfrak{n}_{i}\epsilon_{1}-\mathfrak{s}_{i}\epsilon_{2})-\mathcal{F}_{5}(\varphi_{i}-\mathfrak{n}_{i}\epsilon_{1}-\mathfrak{s}_{i}\epsilon_{2})\Big{]}\,,\end{split}$
(6.24)

subject to the constraints

$\mathfrak{n}_{1}+\mathfrak{n}_{2}=\frac{n_{+}-n_{-}}{n_{+}n_{-}}\,,\qquad\mathfrak{s}_{1}+\mathfrak{s}_{2}=2(1-\mathrm{g})\,,\qquad\varphi_{1}+\varphi_{2}-\frac{n_{+}+n_{-}}{n_{+}n_{-}}\epsilon_{1}=2\,.$
(6.25)

Extremizing this with respect to $\epsilon_{2}$ sets $\epsilon_{2}=0$ and
after renaming $\epsilon_{1}\mapsto\epsilon$ we obtain

$\begin{split}S(\varphi_{i},\epsilon;\mathfrak{n}_{i},\mathfrak{s}_{i})=\frac{c}{\epsilon}\left[\sqrt{(\varphi_{1}+\mathfrak{n}_{1}\epsilon)(\varphi_{2}+\mathfrak{n}_{2}\epsilon)}(\mathfrak{s}_{1}(\varphi_{2}+\mathfrak{n}_{2}\epsilon)+\mathfrak{s}_{2}(\varphi_{1}+\mathfrak{n}_{1}\epsilon))+(\epsilon\to-\epsilon)\right]\,,\end{split}$
(6.26)

where

$c\equiv\frac{\sqrt{2}\pi}{5}\frac{N^{5/2}}{\sqrt{8-N_{f}}}\,,$ (6.27)

which has to be extremized subject to the constraints (6.25).

So far we have not used the additional input (6.14) given by the supergravity
solution discussed above. This strongly suggests that the extremization of
(6.26), _without_ imposing (6.14), will give the entropy of a more general
supergravity solution181818The recent paper [47] discusses
$\mathrm{AdS}_{2}\times\mathbbl{\Sigma}\times\Sigma_{\mathrm{g}}$ supergravity
solutions and it would be interesting to understand the relationship to our
work.. However, below we will proceed enforcing (6.14). Doing so, it is
convenient to introduce the rescaled variables

$\varphi_{1}\equiv\Delta_{1}^{*}\lambda_{1}\,,\qquad\varphi_{2}\equiv\Delta_{2}^{*}\lambda_{2}\,,$
(6.28)

subject to the constraint

$\Delta_{1}^{*}\lambda_{1}+\Delta_{2}^{*}\lambda_{2}-\frac{n_{+}+n_{-}}{n_{+}n_{-}}\epsilon=2\,,$
(6.29)

in terms of which the entropy function reads

$\begin{split}S(\lambda_{i},\epsilon;\mathtt{z})=\frac{c}{\epsilon}\Bigl{[}\sqrt{\Delta_{1}^{*}\Delta_{2}^{*}(\lambda_{1}+n\epsilon)(\lambda_{2}+n\epsilon)}(\mathfrak{s}_{1}\Delta_{1}^{*}(\lambda_{2}+n\epsilon)+\mathfrak{s}_{2}\Delta_{2}^{*}(\lambda_{1}+n\epsilon))+(\epsilon\to-\epsilon)\Bigr{]}\,.\end{split}$
(6.30)

Extremizing this we find the critical values

$\epsilon^{*}=\eta\frac{2n_{+}n_{-}}{\sqrt{2n_{+}^{2}+2n_{-}^{2}}}\,,\qquad\lambda_{1}^{*}=\lambda_{2}^{*}=1+\eta\frac{n_{+}+n_{-}}{\sqrt{2n_{+}^{2}+2n_{-}^{2}}}\,,$
(6.31)

where the sign ambiguity $\eta=\pm 1$ arises by solving the equations over the
complex numbers191919Generically, in the presence of rotation, we are forced
to work with the complex numbers [20, 19] and we therefore continue to do so
also in the static case.. Inserting these values back into the entropy
function, we compute

$\begin{split}S(\lambda^{*}_{i},\epsilon^{*};\mathtt{z})&=2\frac{\sqrt{2}\pi}{5}\frac{N^{5/2}}{\sqrt{8-N_{f}}}\sqrt{\Delta_{1}^{*}\Delta_{2}^{*}}(\mathfrak{s}_{1}\Delta_{2}^{*}+\mathfrak{s}_{2}\Delta_{1}^{*})\frac{-n_{+}-n_{-}-\eta\sqrt{2n_{-}^{2}+2n_{+}^{2}}}{n_{+}n_{-}}\\\
&=\frac{8\pi\kappa(1-\mathrm{g})N^{5/2}(\mathtt{z}^{2}-\kappa^{2})^{3/2}(\sqrt{\kappa^{2}+8\mathtt{z}^{2}}-\kappa)}{5\sqrt{8-N_{f}}(\kappa\sqrt{\kappa^{2}+8\mathtt{z}^{2}}-\kappa^{2}+4\mathtt{z}^{2})^{3/2}}\frac{-n_{+}-n_{-}-\eta\sqrt{2n_{+}^{2}+2n_{-}^{2}}}{n_{+}n_{-}}\,,\end{split}$
(6.32)

where we need to pick $\eta=-1$ in order to get a positive entropy that agrees
with the one computed from the ten-dimensional supergravity solution.

It should be straightforward to incorporate electric charge and rotation
(along the spindle), by promoting the extremization of (6.30) to the Legendre
transform. The entropy obtained in this way should match that of the
supergravity solution obtained by uplifting to massive type IIA the rotating
AdS${}_{2}\times\mathbbl{\Sigma}$ solution of [7]202020In the rotating
solution, setting $n_{+}=n_{-}=1$ will reduce to a rotating AdS${}_{2}\times
S^{2}_{\epsilon}\times\Sigma_{\mathrm{g}}$ solution in $D=6$, whose entropy
function has been recently discussed in [42]..

##  7 Discussion

The work of [8] opened the way to a novel class of examples of the AdS/CFT
correspondence, by constructing a supergravity solution with an AdS3 factor,
comprising the two-dimensional orbifold $\mathbbl{\Sigma}$, known as the
spindle. This was interpreted as the near-horizon limit of D3-branes wrapped
on the spindle with the corresponding field theory duals being a class of
$d=4$, ${\cal N}=1$ SCFTs compactified on the spindle, with a new type of
supersymmetry-preserving twist, that was dubbed anti-twist. Following this,
over the past year various extensions, including analogous constructions for
M2 and M5-branes, have appeared, realising either the anti-twist [8], or a
global generalization [9] of the standard topological twist. These
constructions have left out the notable class of solutions describing
D4-branes wrapped on the spindle, that we constructed in this paper. We found
these AdS${}_{4}\times\mathbbl{\Sigma}$ solutions in $D=6$ gauged supergravity
and then uplifted them to massive type IIA supergravity. The type of twist
realised in our solutions is the one previously found in [9] for M5-branes,
which was referred to as a global topological twist. Differently from the
standard topological twist, the $R$-symmetry gauge field does not cancel the
spin connection, despite the fact that its integrated flux is equal to the
Euler characteristic of the spindle. As a consequence the Killing spinors are
then non-trivial sections (in particular, they are not simply constant) of the
same bundles that occur in standard topological twist.

There are several aspects of our solution that may be interesting to
investigate in the future. For example, it may be instructive to cast it in
the form of the classification of supersymmetric AdS4 solutions of massive
type IIA supergravity [34], or to study supersymmetric probe D-branes in our
background in order to extract further information about the dual field
theories. Another question that arises from our work is whether there exists a
more general consistent truncation of massive type IIA supergravity to $D=6$,
analogous to the dyonic consistent truncation to $D=4$ supergravity found in
[32]. It is also intriguing to investigate whether there exist supergravity
solutions corresponding to D4-branes wrapped on the spindle with the anti-
twist. Finally, as an important step towards improving control on the dual
field theories, it would be worthwhile computing an appropriately regularised
on-shell action, that should prove the validity of our conjectural off-shell
free energy (4.18). To do this, one may need to know the full “black two-
brane” solution, interpolating between AdS6 asymptotically and
AdS${}_{4}\times\mathbbl{\Sigma}$ in the near-horizon, similarly to [19].
However, it may be possible to employ the strategy of [25] to prove that the
supergravity (Euclidean) on-shell action localises at the poles of the
spindle, which are the fixed points for the canonical Killing vector field
associated to any (Euclidean) supersymmetric solution of the theory [48].

There is also a number of assorted interesting questions in the field theory
side. The most direct one is to study the five-dimensional SCFTs in the
background of $S^{3}\times\mathbbl{\Sigma}$, computing the localized partition
function from first principles, and then showing that in the large $N$ limit
the associated free energy reduces to (4.18). This is indeed an open problem
for SCFTs compactified on spindles in different dimension.

It is compelling that the five-dimensional off-shell free energy that we
conjectured here and the entropy function that was conjectured in [12] fit
into a broader conjecture for the free energies (_i.e._ minus the logarithm of
partition functions) of SCFTs in various dimensions, namely (5.9). These
extend the idea of gravitational blocks put forward in [21] in two directions.
Firstly, from compactifications on smooth manifolds, to the realm of
compactifications on orbifolds. It is remarkable that compactifications on
spindles can be incorporated by a simple modification of the constraint obeyed
by the fugacities (see eq. (5.10)). This depends on the type of twist
performed and it includes the standard topological twist and the no-twist as
special cases. Secondly, we have pointed out that the form (5.9) should hold
also for observables of higher-dimensional theories, beyond the entropy,
associated to AdS2 solutions dual $d=1$ SCFTs.

Using the AdS/CFT correspondence, the free energies (5.9) are generically
expected to arise as gravitational (Euclidean) on-shell actions, and their
structure is suggested by the fact that for supersymmetric solutions, this
form should arise from summing contributions at fixed points of the canonical
Killing vector field, defined as a bilinear in the Killing spinors of the
solution [25, 49]. An immediate issue that would be nice to clarify is a
better justification of the $\pm$ signs in (5.9) and their relationship with
the type of twist. For example, while for field theories in odd dimensions we
have proposed that the gluing sign should be $-\sigma$, for field theories in
even dimensions, the explicit examples indicate that one should always pick
the minus sign. It would interesting to find out whether the functions that
have not been considered so far, may have critical points that correspond to
new gravitational objects.

It is quite clear that the structure of (5.9) will extend to compactifications
of SCFTs on higher-dimensional212121The recent paper [42] discusses related
setups, for compactifications of $d=5$ and $d=6$ SCFTs on various smooth
manifolds, including $S^{2}_{\epsilon}\times\Sigma_{\mathrm{g}}$ and
$S^{2}_{\epsilon_{1}}\times S^{2}_{\epsilon_{2}}$. _orbifolds_ and in general
we expect that the functions to extremize will take the form of a sum of
contributions from fixed points of the canonical Killing vector on the
compactification orbifold $\mathbbl{M}$. Assuming this is toric, for
concreteness, there will be as many equivariant parameters $\epsilon_{I}$ as
the rank of the torus acting on $\mathbbl{M}$. One of the simplest examples of
this type of construction is given by twisted compactifications of the $d=5$
SCFT on $\mathbbl{M}=\mathbbl{\Sigma}\times\Sigma_{\mathrm{g}}$, that we
discussed in section 6. We have shown that the corresponding entropy function
can be obtained iterating our conjecture for the off-shell free energies twice
and extremizing this reproduces the entropy of a corresponding class of
AdS${}_{2}\times\mathbbl{\Sigma}\times\Sigma_{\mathrm{g}}$ supergravity
solutions. More generally, we expect that there should exist solutions of the
type AdS${}_{2}\times\mathbbl{\Sigma}_{1}\times\mathbbl{\Sigma}_{2}$ in $D=6$
supergravity and of the type
AdS${}_{3}\times\mathbbl{\Sigma}_{1}\times\mathbbl{\Sigma}_{2}$ in $D=7$
supergravity, for which the corresponding entropy function and trial central
charge can be obtained following the rules we proposed in this paper.
Interestingly, the former case would be identified with the near-horizon limit
of a novel class of supersymmetric black holes in AdS6, that may be
accelerating. Work along these lines is underway and we hope to report in the
near future.

As we continue to navigate the landscape of supergravity solutions
corresponding to branes wrapping orbifolds, it will be revealing to employ the
approach developed in [50, 51, 52, 53], adapting it to situations with
orbifold singularities. While this is already well developed for solutions
arising from M2 and D3-branes, it is tantalising to think that an analogous
geometric approach may be concocted for studying AdS5 solutions of $D=11$
supergravity or other backgrounds with an AdS factor.

### Acknowledgments

The research of FF is supported by the project “Nonlinear Differential
Equations” of Università degli Studi di Torino. DM would like to thank J.
Gauntlett and J. Sparks for insightful comments and enjoyable collaborations
on related topics.

##  Appendix A More details on the AdS${}_{4}\times\Sigma_{\mathrm{g}}$
solutions

###  A.1 Relation with the Lagrangian of [1]

Here we make contact between the $D=6$ Lagrangian (2.1), that we use in the
paper, and the Lagrangian used in [1], given explicitly in [54]. In order to
minimise confusion we have relabelled some of the quantities in [54] as
follows

$\varphi_{1,2}\mapsto\hat{\varphi}_{1,2}\,,\qquad
F^{3,6}\mapsto\frac{1}{2}\mathcal{F}^{3,6}\,,\qquad m\mapsto m_{1}\,,$ (A.1)

where the $1/2$ factor is due to the unusual definition of the field strengths
as, _e.g._ ,
$\mathcal{F}^{3}_{\mu\nu}=\frac{1}{2}(\partial_{\mu}\mathcal{A}^{3}_{\nu}-\partial_{\nu}\mathcal{A}^{3}_{\mu})$.
Consistency between the equations of motion and the supersymmetry equations
requires $\hat{\varphi}_{1}=0$, and in this case the $D=6$ Lagrangian in [54]
reads

$\begin{split}e^{-1}\mathcal{L}&=\frac{1}{4}R-\hat{V}-|\mathrm{d}\sigma|^{2}-\frac{1}{4}|\mathrm{d}\hat{\varphi}_{2}|^{2}-\frac{1}{16}\mathrm{e}^{-2\sigma}\cosh(2\hat{\varphi}_{2})\,|\mathcal{F}^{3}|^{2}\\\
&-\frac{1}{16}\mathrm{e}^{-2\sigma}\cosh(2\hat{\varphi}_{2})|\mathcal{F}^{6}|^{2}+\frac{1}{16}\mathrm{e}^{-2\sigma}\sinh(2\hat{\varphi}_{2})\mathcal{F}^{3}_{\mu\nu}\mathcal{F}^{6\mu\nu}\,,\end{split}$
(A.2)

where the scalar potential is

$\hat{V}=-g_{1}^{2}\mathrm{e}^{2\sigma}-4g_{1}m_{1}\mathrm{e}^{-2\sigma}\cosh\hat{\varphi}_{2}+m_{1}^{2}\mathrm{e}^{-6\sigma}\,.$
(A.3)

Defining the quantities

$\begin{split}\vec{\varphi}=-\sqrt{2}\,\bigl{(}\hat{\varphi}_{2},2\sigma\bigr{)}\qquad&\iff\qquad
X_{1}=\mathrm{e}^{\sigma+\hat{\varphi}_{2}}\,,\quad
X_{2}=\mathrm{e}^{\sigma-\hat{\varphi}_{2}}\,,\\\
F_{1}=\frac{1}{2}(\mathcal{F}^{3}+\mathcal{F}^{6})\,,&\quad\qquad
F_{2}=\frac{1}{2}(\mathcal{F}^{3}-\mathcal{F}^{6})\,,\\\
g=g_{1}\,,&\quad\qquad m=2m_{1}\,,\end{split}$ (A.4)

we obtain the Lagrangian given in (2.1), divided by 4.

###  A.2 Equivalence with the solutions of [2]

Below we show that the ten-dimensional background (2.34) - (2.37), obtained
from the uplift of the $D=6$ solution of [1], is equivalent to the solution to
massive type IIA supergravity constructed in [2]. Let us start with the metric
(4.14) of [2]

$\mathrm{d}s_{\text{s.f.}}^{2}=\frac{L_{\mathrm{AdS}_{4}}^{2}}{(Hy)^{1/2}\hat{F}_{0}}\biggl{(}\mathrm{d}s_{\mathrm{AdS}_{4}}^{2}+\mathrm{e}^{2\nu}\mathrm{d}s_{\Sigma_{\mathrm{g}}}^{2}+\frac{H}{4}\,\mathrm{d}s_{\mathbbl{S}^{4}}^{2}\biggr{)}\,,$
(A.5)

where we restored the dimension-full $\mathrm{AdS}_{4}$ radius
$L_{\mathrm{AdS}_{4}}$ that was set to one in [2] and we renamed the constant
$F_{0}$ in [2] as $\hat{F}_{0}$. The metric on the squashed hemisphere reads

$\mathrm{d}s_{\mathbbl{S}^{4}}^{2}=\frac{8}{3}\,\mathrm{d}\mu_{0}^{2}+2\bigl{(}\mathrm{d}\hat{\mu}_{1}^{2}+\hat{\mu}_{1}^{2}\eta_{+}^{2}+\mathrm{d}\hat{\mu}_{2}^{2}+\hat{\mu}_{2}^{2}\eta_{-}^{2}\bigr{)}\,,$
(A.6)

where the coordinates $\mu_{0},\hat{\mu}_{1},\hat{\mu}_{2}$ satisfy the
constraint $\mu_{0}^{2}+a_{+}\hat{\mu}_{1}^{2}+a_{-}\hat{\mu}_{2}^{2}=1$, and
the one-forms $\eta_{\pm}$ are
$\eta_{\pm}=\mathrm{d}\phi_{\pm}-m_{\pm}\,\omega_{\mathrm{g}}$. The functions
$H$ and $y$ are given by

$H=\frac{2}{3\mu_{0}^{2}+4\bigl{(}a_{+}^{2}\hat{\mu}_{1}^{2}+a_{-}^{2}\hat{\mu}_{2}^{2}\bigr{)}}\,,\quad\qquad
y^{3}\hat{F}_{0}^{2}=\frac{3}{2}\mu_{0}^{2}\,,$ (A.7)

while $\mathrm{e}^{2\nu}$ is a constant and reads

$\mathrm{e}^{2\nu}=\frac{-\kappa+\sqrt{\kappa^{2}+8\mathtt{z}^{2}}}{4}\,.$
(A.8)

The constants $a_{\pm}$, $m_{\pm}$ can be expressed in terms of $\kappa$ and a
constant parameter $\mathtt{z}$ as

$a_{\pm}=\frac{1\pm\epsilon}{2}\,,\qquad\epsilon=\frac{\kappa\pm\sqrt{\kappa^{2}+8\mathtt{z}^{2}}}{4\mathtt{z}}\,,\qquad
m_{\pm}=\frac{\kappa\pm\mathtt{z}}{2}\,.$ (A.9)

In order to compare the two solutions we take

$k_{2}=\frac{a_{+}}{a_{-}}\,,\qquad\quad k_{8}=\frac{16}{9}a_{+}a_{-}\,,$
(A.10)

which are consistent with the constraint $a_{+}+a_{-}=1$ required by the
definition of $a_{\pm}$, and relate our coordinates $\mu_{1}$, $\mu_{2}$ to
those in [2] as

$\mu_{1}=\sqrt{a_{+}}\hat{\mu}_{1}\,,\qquad\quad\mu_{2}=\sqrt{a_{-}}\hat{\mu}_{2}\,.$
(A.11)

We then identify $\phi_{1}=\phi_{+}$, $\phi_{2}=\phi_{-}$ and

$p_{1}=2m_{+}=\kappa+\mathtt{z}\,,\qquad\quad
p_{2}=2m_{-}=\kappa-\mathtt{z}\,,$ (A.12)

thus having $\sigma_{1}=\eta_{+}$, $\sigma_{2}=\eta_{-}$. As consistency
checks we have that $2(m_{+}+m_{-})=p_{1}+p_{2}=2\kappa$ and that the
definition of $k_{2}$ in (2.31) agrees with (A.10). Lastly, comparing the
overall factors in the metric and the dilaton of the two solutions, we find

$g^{2}=\left(\frac{3}{2}\right)^{3/2}k_{8}^{3/4}L_{\mathrm{AdS}_{4}}^{-2}\,,\quad\qquad\lambda^{2}=\left(\frac{3}{2}\right)^{-1/6}k_{8}^{1/4}\hat{F}_{0}^{-2/3}\,.$
(A.13)

The Romans mass (2.36) then reads
$F_{(0)}=\frac{\hat{F}_{0}}{L_{\mathrm{AdS}_{4}}}$, reducing to the expression
in [2] for $L_{\mathrm{AdS}_{4}}=1$. Writing our four-form flux (2.37) in the
variables used in [2], we find

$\begin{split}F_{(4)}&=\frac{H}{8}\left(\frac{2\mu_{0}}{3\hat{F}_{0}}\right)^{1/3}L_{\mathrm{AdS}_{4}}^{3}\biggl{\\{}W\frac{\hat{\mu}_{1}\hat{\mu}_{2}}{\mu_{0}}\,\mathrm{d}\hat{\mu}_{1}\wedge\mathrm{d}\hat{\mu}_{2}\wedge\eta_{+}\wedge\eta_{-}\\\
&-\mathrm{vol}(\Sigma_{\mathrm{g}})\wedge\biggl{[}\frac{m_{+}}{a_{+}}\,\mathrm{d}\phi_{-}\wedge\bigl{(}3\mu_{0}\hat{\mu}_{2}\,\mathrm{d}\hat{\mu}_{2}-4a_{-}\hat{\mu}_{2}^{2}\,\mathrm{d}\mu_{0}\bigr{)}\\\
&+\frac{m_{-}}{a_{-}}\,\mathrm{d}\phi_{+}\wedge\bigl{(}3\mu_{0}\hat{\mu}_{1}\,\mathrm{d}\hat{\mu}_{1}-4a_{+}\hat{\mu}_{1}^{2}\,\mathrm{d}\mu_{0}\bigr{)}\biggr{]}\biggr{\\}}\,,\end{split}$
(A.14)

where we defined

$W=9H\mu_{0}^{2}+16H\bigl{(}a_{+}^{3}\hat{\mu}_{1}^{2}+a_{-}^{3}\hat{\mu}_{2}^{2}\bigr{)}-12\,.$
(A.15)

This expression agrees with the $F_{(4)}$ given in [2] only
partially222222While the above $F_{(4)}$ satisfies the equations of motion and
the Bianchi identity, we found that the four-form flux given in [2] is not
closed.. A couple of useful identities that we used for comparing the two
solutions are

$\frac{m_{+}}{a_{+}}=-2\mathrm{e}^{2\nu}[1-2(a_{+}-a_{-})]\,,\qquad\frac{m_{-}}{a_{-}}=-2\mathrm{e}^{2\nu}[1-2(a_{-}-a_{+})]\,.$
(A.16)

##  Appendix B $OSp(2|4)$ superalgebra from bilinears

Below we show that the $OSp(2|4)$ superalgebra can be reconstructed from the
Killing spinors of our $\mathrm{AdS}_{4}\times\mathbbl{\Sigma}$ solution, as
expected. The fact that the Killing vectors of AdS can be constructed as
bilinears in the Killing spinors is well known. Here we demonstrate that the
$D=6$ solution captures part of the $R$-symmetry of the dual $d=3$,
$\mathcal{N}=2$ SCFTs. Specifically, it includes the component of the
$R$-symmetry along the spindle isometry. The full $R$-symmetry would arise
from working with the Killing spinors of the uplifted ten-dimensional
solution.

Let us recall the relevant ingredients from section 3.1. The Killing spinors
of the $D=6$ solution are a pair of symplectic-Majorana spinors, taking the
form

$\epsilon^{A}=\vartheta_{+}\otimes\eta^{A}_{+}+\vartheta_{-}\otimes\eta^{A}_{-}\,,$
(B.1)

where $A=1,2$ is a symplectic index. The spinors $\vartheta_{\pm}$ are the
chiral components of a Majorana Killing spinor
$\vartheta=\vartheta(x^{\hat{\mu}})$ in $\mathrm{AdS}_{4}$, obeying

$\vartheta^{*}=\mathcal{B}_{4}\vartheta\,,\qquad\hat{\nabla}_{\hat{\mu}}\vartheta=\frac{1}{2}\gamma_{\hat{\mu}}\vartheta\,,$
(B.2)

thus we have $\vartheta=\vartheta_{+}+\vartheta_{-}$ with
$\gamma_{5}\vartheta_{\pm}=\pm\vartheta_{\pm}$. The spinors
$\eta^{A}_{\pm}=\eta^{A}_{\pm}(y)$ are Dirac spinors defined on the spindle.
Defining the following vector bilinear

$\mathtt{K}^{\mu}=\mathrm{i}\,\overline{\epsilon^{\prime
1}}\Gamma^{\mu}\epsilon^{1}\,,$ (B.3)

where $\epsilon^{1}$ and $\epsilon^{\prime 1}$ are two Killing spinors,
straightforward manipulations lead to

$\mathtt{K}^{\hat{\mu}}=2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,\bar{\vartheta}^{\prime}\gamma^{\hat{\mu}}\vartheta\,,\qquad\mathtt{K}^{y}=0\,,\qquad\mathtt{K}^{z}=2(\xi^{\prime})^{*}\xi\,\bar{\vartheta}^{\prime}\vartheta\,.$
(B.4)

From (B.2) it immediately follows that

$\nabla_{(\hat{\mu}}(\bar{\vartheta}^{\prime}\gamma_{\hat{\nu})}\vartheta)=0=\nabla_{\hat{\mu}}(\bar{\vartheta}^{\prime}\vartheta)\,,$
(B.5)

hence

$\mathtt{K}^{\mu}\partial_{\mu}=2(\xi^{\prime})^{*}\xi\bigl{(}\mathrm{i}\,m\bar{\vartheta}^{\prime}\gamma^{\hat{\mu}}\vartheta\,\partial_{\hat{\mu}}+\bar{\vartheta}^{\prime}\vartheta\,\partial_{z}\bigr{)}$
(B.6)

are Killing vectors for the $D=6$ solution. One can check that
$\overline{\epsilon^{\prime
2}}\Gamma^{\mu}\epsilon^{2}=-(\overline{\epsilon^{\prime
1}}\Gamma^{\mu}\epsilon^{1})^{*}$, while $\overline{\epsilon^{\prime
1}}\Gamma^{\mu}\epsilon^{2}$, $\overline{\epsilon^{\prime
2}}\Gamma^{\mu}\epsilon^{1}$ and $\overline{\epsilon^{\prime
A}}\Gamma^{7}\Gamma^{\mu}\epsilon^{B}$ are not Killing vectors, so there are
no further vector bilinears to consider.

To make contact with the $OSp(2|4)$ superalgebra we need to look in more
detail into the structure of the Killing spinors in AdS4. Writing the unit
radius $\mathrm{AdS}_{4}$ metric as

$\mathrm{d}s_{\mathrm{AdS}_{4}}^{2}=u^{2}\eta_{mn}\mathrm{d}x^{m}\mathrm{d}x^{n}+\frac{\mathrm{d}u^{2}}{u^{2}}\,,$
(B.7)

and defining the coordinates $x^{\hat{\mu}}=(t,x^{1},x^{2},u)\equiv(x^{m},u)$,
the Killing spinors on $\mathrm{AdS}_{4}$ can be written as [55]

$\epsilon_{\mathrm{AdS}_{4}}=\mathrm{e}^{\frac{1}{2}(\ln
u)\,\gamma_{3}}\biggl{[}1+\frac{1}{2u}x^{m}\gamma_{m}(1-\gamma_{3})\biggr{]}\psi\,,$
(B.8)

with $\psi$ a constant spinor and
$\\{\gamma_{m},\gamma_{n}\\}=u^{2}\eta_{mn}$. This can be conveniently split
into two independent Killing spinors as
$\epsilon_{\mathrm{AdS}_{4}}=\epsilon_{+}+\epsilon_{-}$, where

$\epsilon_{+}=u^{1/2}\psi_{+}\,,\qquad\quad\epsilon_{-}=u^{-1/2}\left(1+x^{m}\gamma_{m}\right)\psi_{-}\,,$
(B.9)

with

$\psi_{\pm}\equiv\frac{I_{4}\pm\gamma_{3}}{2}\psi\qquad\implies\qquad\gamma_{3}\psi_{\pm}=\pm\psi_{\pm}\,.$
(B.10)

Note that $\epsilon_{\pm}$ and $\psi_{\pm}$ are not chiral spinors, since
$\gamma_{3}$ is not the chiral matrix. A Majorana condition can be
consistently imposed on $\epsilon_{\mathrm{AdS}_{4}}$, which implies that
$\psi$ and, in turn, $\psi_{\pm}$ are Majorana spinors. We can then identify
$\vartheta$ with $\epsilon_{\mathrm{AdS}_{4}}$ and write the Killing vectors
in (B.4) as

$\displaystyle\mathtt{K}^{\hat{\mu}}$
$\displaystyle=2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\bigl{(}\bar{\epsilon}^{\prime}_{+}\gamma^{\hat{\mu}}\epsilon_{+}+\bar{\epsilon}^{\prime}_{+}\gamma^{\hat{\mu}}\epsilon_{-}+\bar{\epsilon}^{\prime}_{-}\gamma^{\hat{\mu}}\epsilon_{+}+\bar{\epsilon}^{\prime}_{-}\gamma^{\hat{\mu}}\epsilon_{-}\bigr{)}\,,$
(B.11) $\displaystyle\mathtt{K}^{z}$
$\displaystyle=2(\xi^{\prime})^{*}\xi\bigl{(}\bar{\epsilon}^{\prime}_{+}\epsilon_{+}+\bar{\epsilon}^{\prime}_{+}\epsilon_{-}+\bar{\epsilon}^{\prime}_{-}\epsilon_{+}+\bar{\epsilon}^{\prime}_{-}\epsilon_{-}\bigr{)}\,.$

As standard, the two independent $\epsilon_{+}$ spinors are identified with
the real Poincaré supercharges $Q$ and the $\epsilon_{-}$ with the real
superconformal supercharges $S$. For this reason we shall refer to the
different contributions in (B.11) as $\mathtt{K}^{\hat{\mu}}_{QQ}$,
$\mathtt{K}^{\hat{\mu}}_{QS}$, $\mathtt{K}^{\hat{\mu}}_{SQ}$ and
$\mathtt{K}^{\hat{\mu}}_{SS}$, respectively, and likewise for
$\mathtt{K}^{z}$. Specifically, the components along the spindle are

$\displaystyle\mathtt{K}^{z}_{QQ}$ $\displaystyle=0\,,\qquad$
$\displaystyle\mathtt{K}^{z}_{QS}$
$\displaystyle=2(\xi^{\prime})^{*}\xi\,\bar{\psi}^{\prime}_{+}\psi_{-}\,,$
(B.12) $\displaystyle\mathtt{K}^{z}_{SS}$ $\displaystyle=0\,,\qquad$
$\displaystyle\mathtt{K}^{z}_{SQ}$
$\displaystyle=2(\xi^{\prime})^{*}\xi\,\bar{\psi}^{\prime}_{-}\psi_{+}\,,$

and the components along $\mathrm{AdS}_{4}$ are

$\displaystyle\mathtt{K}^{m}_{QQ}$
$\displaystyle=2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,u\,(\bar{\psi}^{\prime}_{+}\gamma^{m}\psi_{+})\,,\quad$
$\displaystyle\mathtt{K}^{u}_{QQ}$ $\displaystyle=0\,,$ (B.13)
$\displaystyle\mathtt{K}^{m}_{QS}$
$\displaystyle=2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,\bigl{[}x^{m}(\bar{\psi}^{\prime}_{+}\psi_{-})+x^{n}(\bar{\psi}^{\prime}_{+}\gamma^{m}_{\
\ n}\psi_{-})\bigr{]}\,,\quad$ $\displaystyle\mathtt{K}^{u}_{QS}$
$\displaystyle=-2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,u\,\bar{\psi}^{\prime}_{+}\psi_{-}\,,$
$\displaystyle\mathtt{K}^{m}_{SQ}$
$\displaystyle=2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,\bigl{[}-x^{m}(\bar{\psi}^{\prime}_{-}\psi_{+})+x^{n}(\bar{\psi}^{\prime}_{-}\gamma^{m}_{\
\ n}\psi_{+})\bigr{]}\,,\quad$ $\displaystyle\mathtt{K}^{u}_{SQ}$
$\displaystyle=2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,u\,\bar{\psi}^{\prime}_{-}\psi_{+}\,,$
$\displaystyle\mathtt{K}^{m}_{SS}$ $\displaystyle=(x\cdot
x+u^{-2})\,b^{m}-2x^{m}(x\cdot b)\,,\quad$ $\displaystyle\mathtt{K}^{u}_{SS}$
$\displaystyle=2u\,(x\cdot b)\,,$

with $b^{m}\equiv
2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,u\,(\bar{\psi}^{\prime}_{-}\gamma^{m}\psi_{-})$
and $x\cdot b=\eta_{mn}x^{m}b^{n}$. We now define the additional parameters

$\displaystyle a^{m}$ $\displaystyle\equiv
2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,u\,(\bar{\psi}^{\prime}_{+}\gamma^{m}\psi_{+})\,,$
(B.14) $\displaystyle\Lambda^{m}_{\ \ n}$ $\displaystyle\equiv
2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,(\bar{\psi}^{\prime}_{+}\gamma^{m}_{\ \
n}\psi_{-}+\bar{\psi}^{\prime}_{-}\gamma^{m}_{\ \ n}\psi_{+})\,,$
$\displaystyle\lambda$ $\displaystyle\equiv
2\mathrm{i}\,m(\xi^{\prime})^{*}\xi\,(\bar{\psi}^{\prime}_{+}\psi_{-}-\bar{\psi}^{\prime}_{-}\psi_{+})\,,$
$\displaystyle\mathtt{r}$ $\displaystyle\equiv
2(\xi^{\prime})^{*}\xi\,(\bar{\psi}^{\prime}_{+}\psi_{-}+\bar{\psi}^{\prime}_{-}\psi_{+})\,.$

It can be proven that in $D=4$, for any pair of generic Majorana spinors
$\Psi_{1}$ and $\Psi_{2}$, we have

$(\bar{\Psi}_{1}\gamma^{\hat{\mu}_{1}}\ldots\gamma^{\hat{\mu}_{n}}\Psi_{2})^{*}=-\bar{\Psi}_{1}\gamma^{\hat{\mu}_{1}}\ldots\gamma^{\hat{\mu}_{n}}\Psi_{2}\qquad\qquad(n\geq
0)\,,$ (B.15)

namely any bilinear constructed with Majorana spinors is pure imaginary. It
follows that the bilinears inside the parentheses in all the parameters are
pure imaginary. This is consistent with the $\mathcal{N}=1$ AdS4 superalgebra,
which is obtained by formally setting $2m(\xi^{\prime})^{*}\xi=1$ in $a^{m}$,
$b^{m}$, $\Lambda^{m}{}_{n}$ and $\lambda$ and $\mathtt{r}=0$. However,
because we have $\mathcal{N}=2$, the parameters are complexified by $\xi$ and
$\xi^{\prime}$. The Killing vector $\mathtt{K}$ can then be expressed as

$\mathtt{K}=\mathtt{K}_{QQ}+\mathtt{K}_{QS}+\mathtt{K}_{SQ}+\mathtt{K}_{SS}\,,$
(B.16)

with

$\displaystyle\mathtt{K}_{QQ}$ $\displaystyle=a^{m}\partial_{m}\equiv P\,,$
(B.17) $\displaystyle\mathtt{K}_{SS}$ $\displaystyle=(x\cdot
x+u^{-2})\,b^{m}\partial_{m}-2(x\cdot
b)(x^{m}\partial_{m}-u\,\partial_{u})\equiv K\,,$
$\displaystyle\mathtt{K}_{QS}+\mathtt{K}_{SQ}$
$\displaystyle=\lambda(x^{m}\partial_{m}-u\,\partial_{u})+\Lambda^{m}_{\ \
n}\,x^{n}\partial_{m}+\mathtt{r}\,\partial_{z}\equiv D+M+R\,,$

which have the formal structure of the anticommutators of the $OSp(2|4)$
superalgebra. Specifically, the left hand side of the equations above may be
identified with the anticommutators $\\{Q,Q\\}$, $\\{S,S\\}$, and $\\{Q,S\\}$,
respectively. On the right hand side we find the bosonic generators of the
$SO(3,2)$ isometry of $\mathrm{AdS}_{4}$, namely translations $P$, Lorentz
transformations $M$, dilatations $D$, and special conformal transformations
$K$. Moreover, we find that $R\equiv\mathtt{r}\,\partial_{z}$ is the generator
of the $R$-symmetry. The superalgebra may be completed by computing the
spinorial Lie derivatives ${\cal L}_{V}\epsilon_{+}$, ${\cal
L}_{V}\epsilon_{-}$ along the above Killing vectors.

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##  1 Introduction

In high-energy particle collisions, hadrons with heavy quark content, are a
uniquely versatile probe of fragmentation dynamics. Their defining feature, a
charm ($\mrm{c}$) or bottom ($\mrm{b}$) flavoured quark, cannot originate from
the hadronization process but must be created either in the hard process or in
the parton shower, both calculable with perturbative techniques.

As opposed to the even heavier top ($\mrm{t}$) quark, hadrons containing
$\mrm{c}$\- and $\mrm{b}$–type quark content, are still understood to fragment
through the same mechanisms as their light counterparts, the $\mrm{u}$,
$\mrm{d}$, and $\mrm{s}$ quarks. When comparing experimental data to theory,
two quite different (and thus complementary) techniques are used: The
factorisation approach and the route taken by Monte Carlo event generators. In
the factorisation approach [Collins:1985gm, Collins:1989gx], the cross-section
is separated into a convolution of three factors: 1) a Parton Distribution
Function (PDF) of the incoming hadron, 2) the parton level hard scattering
cross-section, where state-of-the-art calculations today are next-to-leading-
order (NLO) in the strong coupling ($\alpha_{s}$) (see [Nason:1987xz,
Nason:1989zy, Mangano:1991jk]) often with next-to-leading-log (NLL)
resummation techniques applied as well, such as GM–VFNS [Helenius:2018uul] or
FONLL [Cacciari:1998it, Cacciari:2012ny], and finally 3) fragmentation
functions, analytical expressions fitted to and data [Belle:2005mtx,
Kneesch:2007ey] giving differential probabilities for the charm quark to
fragment to various hadron species. It has been known at least since SPS
[WA82:1993ghz] that the underlying assumption of independent fragmentation
does not hold, but it has generally been assumed that universal fragmentation
functions can be applied across systems, when studying inclusive quantities,
such as total charm hadron yields per event. Recent work by the ALICE
collaboration [ALICE:2020wfu, ALICE:2021dhb, ALICE:2021rzj, ALICE:2023jgm]
has, however, clearly shown that fragmentation functions tuned to and , cannot
describe the fragmentation of charm into baryons in .

In the Monte Carlo event generator approach, as used in [Bierlich:2022pfr],
PDFs are still used to extract the participating partons from the colliding
nucleons. But where the focus in the factorisation approach tends to be more
directed towards formal precision in the calculation of the hard scattering,
the focus in the Monte Carlo generators is more towards coherent modelling of
both perturbative and non-perturbative aspects, such as hadronization. Once
the total amount of charm quarks present in the event is determined by means
of a leading order calculation, plus parton shower [Norrbin:1998bw,
Norrbin:2000zc], the amount of hadron species, is determined by the dynamical
fragmentation model, the Lund string model [Andersson:1983ia], and its
extensions. This makes charm hadrons very well suited for studies of dynamical
hadronization models. For charm baryon production in particular, the so-called
QCD colour reconnection (CR) model [QCDCR] in has gained a lot of attention,
due to its ability to correctly reproduce the $\Lambda^{+}_{c}$ yield and
$\Lambda^{+}_{c}/D^{0}$ ratio as a function of $p_{\perp}$ in $\pp$ collisions
at various collision energies at LHC [ALICE:2020wfu, ALICE:2021rzj,
ALICE:2023jgm]. However, the predicted production rates of $\Xi_{c}$ and
$\Omega_{c}$ baryons are still undershooting data, even with the model
[ALICE:2021bli, ALICE:2021psx, ALICE:2022cop]. Furthermore, the model has,
until recently not been usable for heavy ion collisions.

One of the key aspects of the model is the formation of junction-like
configurations between two or three colour dipoles. These junction systems
contribute to baryon production in addition to the baryons produced during the
string fragmentation in . We have recently improved the junction fragmentation
for the low-energy junction systems and extended the model with a spatial
constraint [Lonnblad:2023stc]. As a result, the model can be used as a global
CR model for heavy-ion collision simulations in the model [Angantyr,
Lonnblad:2023stc].

In this paper, we further improve the junction formation and fragmentation for
the colour dipoles containing heavy quarks. We use collisions to validate the
framework and show for the first time how collisions generated with + , give a
satisfactory description of $\Lambda^{+}_{c}$ production. We show results
primarily for charm baryons, but a similar outcome can be expected for the
bottom quark containing baryons as well. We present the results using the
upgraded model from [Lonnblad:2023stc] with the new changes we have made in
this paper.

We first provide an overview of the model for heavy-ion event simulation in in
the next section. In section 3 we discuss the perturbative production of the
charm quarks, and in section 3.2, we show the non-perturbative aspects of the
charm hadrons production. We also discuss the changes we have made in junction
formation and fragmentation. Finally, results for charm hadron production in
and are shown in section 4.

##  2 Heavy ion collisions with the model

The model [Angantyr, Bierlich:2016smv] is an extension of to simulate heavy-
ion collision events without assuming the creation of a Quark–Gluon plasma. It
uses a modified Glauber model [Glauber:1955qq, Miller:2007ri] to obtain the
number and types (elastic or inelastic (diffractive or non-diffractive)
interactions) of sub-collisions in a heavy-ion collision event. Based on the
number and type of sub-collisions, multiple $\pp$-like collisions are
generated and stacked together to produce the heavy-ion event.

The arrangement of the nucleons inside a nucleus is obtained using the Woods-
Saxon distribution in the GLISSANDO parametrization [Bozek:2019wyr]. When
nuclei collide with each other at relativistic energies, they are Lorentz
contracted. The wave functions of the nucleons inside the nuclei can be
treated as frozen at the time of the collision. This is realized in the so-
called Glauber-Gribov [Heiselberg:1991is, Blaettel:1993ah, Alvioli:2013vk]
formalism for nucleon wave-function fluctuations and extended it to include
cross-section fluctuations in projectile and target nucleons for pA and AA
collisions.

Once the types of nucleon-nucleon (NN) sub-collisions are decided, the model
uses the model for multiparton interactions to generate respectively non-
diffractive, diffractive, and elastic events. Often it occurs that a nucleon
is participating in more than one NN non-diffractive sub-collision. A key
feature of the model is the special treatment for nucleons participating in
multiple non-diffractive interactions. Given a single projectile nucleon
interacting with several target nucleons, the NN pair with the smallest impact
parameter is denoted the “primary” non-diffractive sub-collision. The others
are denoted “secondary”. The primary sub-collision is generated as a normal
non-diffractive collision, whereas the secondaries are generated as a modified
single diffractive collision (see section 5 in reference [Angantyr] for
further explanation). A secondary non-diffractive interaction will be
discarded once sufficient energy is no longer available.

There is no interaction between the partons produced in different sub-
collisions in the default . All multiple sub-collisions are stacked together
at the parton level as colour singlet Lund strings. Later, the Lund strings
are hadronised and produce a heavy-ion collision event.

Recently we have added a global colour reconnection (CR) in
[Lonnblad:2023stc]. We have extended the model [QCDCR], by adding a spatial
constraint on the colour dipoles to be colour reconnected. We stack the colour
dipoles from different sub-collisions and use the spatially constrained model
such that colour dipoles from nearby sub-collisions can undergo CR. In this
work, we continue to use this upgraded set-up to simulate collision events.

##  3 Charm hadron production in

Since the masses of charm ($\approx$1.5 GeV) and bottom ($\approx$4.8 GeV) are
large compared to the light quarks, they will never be produced through the
tunnelling mechanism by which the string breaks, but only in the hard process
and the parton shower. In sub-section 3.1 we will briefly review the formalism
for heavy quark production, and in sub-section 3.2 we give an overview of the
impact of CR. In the following sub-sections, the modifications relevant to
charm production in will be introduced.

###  3.1 Charm quark in hard process and parton shower

Several different QCD processes in $\pp$ collisions in can produce heavy
quarks. The leading order (LO) processes like $qq\rightarrow Q\Bar{Q}$ and
$gg\rightarrow Q\Bar{Q}$ hard scatterings are the primary processes for heavy
quarks production in . Another source of heavy quark production is weak decays
($Z$ and $W^{\pm}$ bosons decays), Higgs decay, and top and bottom quark
decay, though of those, only the latter contributes in any significant amount
when considering total charm production down to low $p_{\perp}$. Furthermore,
parton showers, where initial or final state partons (mostly gluons) produce
the heavy quarks by pair creation, flavour excitation or gluon splitting. This
is a significant source of charm production, in addition to that produced in
the hard scattering. Furthermore, the “hidden charm” from the PDF of one of
the colliding beams, may come on a mass shell due to the scattering. The
interaction is like $Qq\rightarrow Qq$ or $Qg\rightarrow Qg$, but since the Q
is not a valence quark it has to be produced in pairs by a gluon splitting.

The LO processes have the matrix elements containing the heavy quark mass.
Since quark masses are included, full phase space down to
$p_{\perp}\rightarrow 0$ can be populated. For low $p_{\perp}$ production,
however, using the multiparton interaction framework [Sjostrand:1987su], which
introduces a general parameter $p_{\perp 0}$, is more suitable, in particular
when extending to heavy ion collisions. The heavy quark masses are an
important parameter in the perturbative description of their production. In ,
the default values for the charm and bottom quark masses are set to 1.5 GeV
and 4.8 GeV respectively. The masses affect the matrix elements, splitting
kernels, and the phase space of the heavy quarks production cross-section.
These values are fitted to $D$-meson production rates. To better fit
production rates at LHC, we have in this paper reduced the charm and bottom
quark masses to 1.3 GeV and 4.2 GeV respectively.

###  3.2 Colour reconnection and hadronization

After the multiple parton scatterings and parton showers, outgoing quarks and
gluons are connected by strings. We speak of a string connecting a colour and
an anti-colour – either quark and anti-quark or through one or more gluon
“kinks” – as a chain of colour dipoles. These colour connections are
reassigned through colour reconnection (CR) [Sjostrand:1987su, QCDCR] models.
The conventional argument, and indeed the logic behind the default CR model in
, is that while the parton shower generates a colour configuration in the
$N_{c}\rightarrow\infty$ limit, nature has $N_{c}=3$. The choice of specific
colour connections for a single event is ambiguous and should therefore be
corrected. The calculation itself, however, cannot provide any guidance as to
how to do the reconnection, and one must resort to models. A common feature is
a reduction of the so-called $\lambda$-measure, which is an indirect
representation of the rapidity span of the colour dipoles, which is again a
logarithmic sum of the potential energy of the dipoles, and hence a measure of
the number of hadrons produced by the dipoles. Further details in sub-section
3.3.1. The CR in helps to reproduce the charged particle multiplicity and the
increase in $\langle p_{\perp}\rangle$ as a function of $(N_{ch})$
distribution as observed in the experiments.

The model [QCDCR] is developed with the idea of applying SU(3) colour algebra
on non-correlated colour dipoles before calculating the $\lambda$-measure for
the new configurations of the dipoles. Colour algebra allows the formation of
a colour singlet by three colour string pieces being connected to a “junction”
point (see figure 1). The “string system contains a junction” (junction
system) formed by two or three dipoles is not possible in the earlier case of
$N_{c}\rightarrow\infty$ limit. Hence in the model, the two and three dipoles
can have three string pieces that are colour-connected to a single “junction”
point after the CR. A junction system produces at least one baryon per
junction during the hadronization. In the model, junctions are always produced
as junction and anti-junction pairs and conserve the baryon number. These
baryons (and anti-baryons) are additional baryons due to .

After the CR, the colour singlet Lund strings hadronised by sequential
fragmentation. The different flavours of quarks and anti-quarks are produced
according to the Lund string model [LSM]. Parameter values are fixed from the
model tuning with LEP data [Skands:2014pea, QCDCR, Bierlich:2015rha].

The sequence of the string breaks decides if a string piece will form a meson
or a baryon as a primary hadron. The Lund string fragmenting into $q\bar{q}$
pairs will produce mesons. For baryons production, the string has to break
into a diquark–anti-diquark pair, where the consecutive string breaks of a
$q\bar{q}$ pair on either side of the diquarks will produce a baryon and an
anti-baryon. uses the “popcorn mechanism” [Bar:popcorn], which includes a
probability for a meson production between the baryon and the anti-baryon, and
the results of the “popcorn mechanism” are supported by the experiments
[TPCTwoGamma:1985zxy].

A $q\Bar{q}$ pair production rate,

$\mathrm{d}P\simeq\mathrm{d}^{2}p_{\perp}\exp\left(-\pi
m^{2}_{\perp}/\kappa\right),\hskip
11.38109ptm^{2}_{\perp}=p^{2}_{\perp}+m^{2}_{q},$ (1)

where $\kappa$ is string tension, and $m_{\perp}$ is transverse mass of the
quark with mass $m_{q}$ and opposite transverse momenta $p_{\perp}$. The pair
production rate is mass-dependent, and it gives an extremely low probability
for the production of heavy quarks pair (”charm” pairs) during the string
fragmentation. Therefore all of the heavy quarks are produced either in hard
scatterings or in parton showers in as mentioned in section 3.1. In this
paper, we show results for charm baryons only, so we refer to charm quarks as
the heavy quarks for the rest of the paper.

The heavy quarks form mesons or baryons depending on which quarks/diquarks are
produced next to them during string fragmentation. The only way the
hadronization of the heavy quarks can be influenced is either by modifying
fragmentation parameters or by colour reconnection. LEP data constrains the
fragmentation parameters, while in the CR we have some freedom in rearranging
the colour connections among the partons. Moreover, the model allows junction
configurations, which contribute to baryon production. In such a case the type
of partons attached to the junction legs and the choice of junction
fragmentation sequence can influence the baryon production. Moreover, the
probability for junction formation increases in a more dense environment like
high multiplicity $\pp$ or heavy nuclei collisions.

Until now, no special attention was given if a heavy quark is involved during
the junction formation or fragmentation. In this paper, we improve the
junction treatments if one or more heavy quarks are involved. We discuss these
improvements below in detail.

###  3.3 The role of junctions

We have made several improvements in the model in [Lonnblad:2023stc]. A
crucial addition is the impact parameter-dependent constraint on the colour
dipoles to be colour reconnected, which allowed us to have a global CR among
the partons produced in different sub-collisions in heavy-ion event simulation
in . In addition, we implemented a few technical improvements in the
hadronisation of junctions.

In , there may occur situations when a string system in an event cannot be
hadronised properly. There are several reasons for such failures, and often
they involve junctions. If such a failure arises, will throw away the whole
event and generate a new one. In collisions such failures are typically rare,
but in heavy-ion collisions there can be very many strings and the failure
rate per event increases. And since the rate is higher for high multiplicity
events (many strings), there is a risk that the overall multiplicity
distributions may be skewed. With the model, the number of junctions
increases, which also increases the failure rate, and we found in
[Lonnblad:2023stc] that the effect on multiplicity distributions was
substantial in the heavy-ion collision, and even visible in collisions.

A majority of the discarded events are found to have at least one junction
system with a very low invariant mass ($\lesssim$1 GeV). We added a “junction
collapse” mechanism to hadronize the low-mass junction systems, which were not
treated in prior to [Lonnblad:2023stc]. This “junction collapse” mechanism
produces two hadrons from the junction system. These hadrons can be two
baryons or a meson and a baryon depending on the types of partons attached to
the end of every leg in a junction system. We have also introduced an
additional trial if the string fragmentation fails to hadronise a junction
system. In such a scenario, the junction system is fragmented by the special
version of the junction collapse procedure.

As we mentioned at the beginning of the paper, the conflicting results from
and $\pp$ collisions raise questions about the universality of charm hadron
production. In , this applies in particular to charm baryons. We have seen
that adding improves the description for $\Lambda_{c}$, but for heavier
charmed baryons there is still a problem. Since the additional junctions from
the model are responsible for the increased charm baryon rates, we want to
look in more detail into how junctions involving charm quarks are handled
there, and also how they are treated in the subsequent string fragmentation.

####  3.3.1 Junction formation

In collisions, junctions are normally only formed in the treatment of the
proton remnants, when more than one valence quark undergo scattering in the
multiple parton interactions machinery, but such junctions mainly influence
baryon production in the forward rapidity region. The junction formation due
to colour reconnection is unique to the model. The QCD colour algebra-based
reconnection treatment includes the colour connections beyond the leading
colour approximations. This means that besides the case where two uncorrelated
dipoles having the exact same colour state can “swing” so that the coloured
parton in one dipole becomes colour connected with the anti-colour of the
other, and vice versa, there can also be reconnections between dipoles that
have different colour states. In this way, the partons in two or three colour
dipoles can become colour-connected to junction points (as shown in figure 1)
with a certain probability that they are carrying the right colour charges.
Each of the junction legs has to have a different colour charge so that the
junction system becomes a colour singlet.

(-1,0) – (2,0); [gray] (-1,0) circle (2pt) node[left]$q$; [gray] (2,0) circle
(2pt) node[right]$\bar{q}$; (-1,1) – (2,1); [gray] (-1,1) circle (2pt)
node[left]$q$; [gray] (2,1) circle (2pt) node[right]$\bar{q}$;

[-stealth](3.2,0.5) – (4.8,0.5);

(6,0) – (6.5,0.5); [gray] (6,0) circle (2pt) node[left]$q$; (6,1) – (6.5,0.5);
[gray] (6,1) circle (2pt) node[left]$q$;

(6.5,0.5) – (8.5, 0.5);

(9,0) – (8.5,0.5); [gray] (9,0) circle (2pt) node[right]$\bar{q}$; (9,1) –
(8.5,0.5); [gray] (9,1) circle (2pt) node[right]$\bar{q}$; [] at (4,-0.5) (a);

(-2,-1) – (10, -1); (-1,-2) – (2,-2); [gray] (-1,-2) circle (2pt)
node[left]$q$; [gray] (2,-2) circle (2pt) node[right]$\bar{q}$; (-1,-3) –
(2,-3); [gray] (-1,-3) circle (2pt) node[left]$q$; [gray] (2,-3) circle (2pt)
node[right]$\bar{q}$; (-1,-4) – (2,-4); [gray] (-1,-4) circle (2pt)
node[left]$q$; [gray] (2,-4) circle (2pt) node[right]$\bar{q}$;

[-stealth](3.2,-3) – (4.8,-3);

(6,-2) – (7,-3); [gray] (6,-2) circle (2pt) node[left]$q$; (6,-3) – (7,-3);
[gray] (6,-3) circle (2pt) node[left]$q$; (6,-4) – (7,-3); [gray] (6,-4)
circle (2pt) node[left]$q$;

(9,-2) – (8,-3); [gray] (9,-2) circle (2pt) node[right]$\bar{q}$; (9,-3) –
(8,-3); [gray] (9,-3) circle (2pt) node[right]$\bar{q}$; (9,-4) – (8,-3);
[gray] (9,-4) circle (2pt) node[right]$\bar{q}$; [] at (4,-4) (b);

Figure 1: Illustration of colour reconnections forming junctions. Two dipoles
can form a colour connected junction–anti-junction system (a), and three can
form two separate (anti-) junction systems (b).

After all possible reconnections are tabulated, the model will order them so
that the reconnection reduces the string lengths the most, as defined by the
$\lambda$-measure are performed first. For dipoles between two partons the
$\lambda$-measure in the model is given by111The infrared $\lambda$-measure
for a string in Ref. [Andersson:1988ee], is in the model approximated by the
sum of these contributions from the individual dipoles.

$\lambda=\ln\left(1+\frac{\sqrt{2}E_{1}}{m_{0}}\right)+\ln\left(1+\frac{\sqrt{2}E_{2}}{m_{0}}\right),$
(2)

where the energies $E_{i}$ are given in the dipole’s rest frame, and $m_{0}$
is a tunable parameter. For a dipole connecting a parton to a junction, the
model similarly defines the $\lambda$-measure as

$\lambda_{j}=\ln\left(1+\frac{\sqrt{2}E}{m_{j}}\right),$ (3)

where $E$ is the energy of the parton given in the junction rest frame222In
the junction rest frame, the angles between all momenta of the connected
partons are $120^{\circ}$. and $m_{j}$ is a tunable parameter not necessarily
the same as $m_{0}$.

As discussed in the introduction the $\lambda$-measure is an estimate of the
rapidity range for the hadrons in the string breakup. The definitions in
equations (2 and 3) are well motivated for light quarks and massless gluons.
However, for a string piece connected to a heavy quark, these expressions are
not good estimates of the rapidity range. In this case, we instead use the
rapidity of the heavy quark in the rest frame of the junction:

$\lambda_{HQ}=\frac{1}{2}\log\left(\frac{E+p}{E-p}\right).$ (4)

Here $E$ and $p$ are the energy and momentum of the heavy quark in the
junction rest frame. The $\lambda_{HQ}$ will give a lower value than
$\lambda_{j}$ for heavy quarks, especially for small $p$. Hence with this new
change, we enhance the possibility for a heavy quark to be part of a junction
system during CR in . We note that there is no need for the parameter $m_{j}$
to set the scale in $\lambda_{HQ}$ since the quark mass does that for us.
Also, the “1+” in the logarithm, which protects the $\lambda$ from becoming
negative is also not needed.

####  3.3.2 Junction fragmentation

After the Colour Reconnections, the colour strings will undergo string
fragmentation. In the model in , the three junction legs are treated
separately according to the following steps.

  * •

A few attempts are made to move the junction system to the junction rest
frame.

  * •

If the algorithm fails to obtain the junction rest frame, then the junction
system fragments in the centre of the mass frame of the junction system.

  * •

Once the frame is found, the summed energy of the partons on each junction leg
is calculated in that frame and the junction legs are tagged as low-, middle-,
and high-energy legs.

  * •

The low-energy leg is fragmented first. A fictitious particle is assumed on
the opposite side of the junction point for the given junction leg, and the
string fragments from the endpoint towards the junction point until a parton
closest to the junction point is left on the junction leg.

  * •

Similarly the middle-energy leg is fragmented.

  * •

A diquark is formed by combining the flavour and momenta of the two partons
left on the low and middle legs closest to the junction point.

  * •

The diquark is connected at one end of the high-energy leg, the junction no
longer exists and the string is fragmented via the usual string fragmentation
mechanism.

Finding the junction frame for three massless partons is trivial, but as soon
as one or more of them are massive, the process does not always converge to a
stable solution, because such a solution does not exist. (Also in the case
where a junction leg has a long chain of gluons, the proper frame can be
difficult to find.)

[black] (-3,-1) circle (4pt) node[above]HQ; (-3,-1.5) node $\phi$; [line width
= 0.8mm] (-3,-1) – (-3.4, -1.8) circle (2pt) node [left] q; [line width =
0.8mm] (-3,-1) – (-2.6, -1.8) circle (2pt) node [right] q; [-¿] (-3.5, -1.9) –
(-3.6, -2.2); [-¿] (-2.5, -1.9) – (-2.4, -2.2); (-3, -2.5) node (a);

[black] (1,-1) circle (4pt) node[above]HQ; [-¿] (0.7, -1.2) – (0.7, -0.9);
[line width = 0.8mm] (1,-1) – (1,-1.4); [line width = 0.8mm] (1,-1.4) – (0,
-2) circle (2pt) node [left] q; [line width = 0.8mm] (1,-1.4) – (2, -2) circle
(2pt) node [right] q; [-¿] (-0.1, -2.1) – (-0.4, -2.3); [-¿] (2.1, -2.1) –
(2.4, -2.3);

(1, -2.5) node (b);

[black] (5,-1) circle (4pt) node[above]HQ; (5,-1.5) node $\phi>120^{\circ}$;
[line width = 0.8mm] (5,-1) – (4, -1.3) circle (2pt) node [left] q; [line
width = 0.8mm] (5,-1) – (6, -1.3) circle (2pt) node [right] q; [-¿] (3.9,
-1.4) – (3.5, -1.6); [-¿] (6.1, -1.4) – (6.5, -1.6);

(5, -2.5) node (c);

Figure 2: (a) A representation of a junction system with a heavy quark
(denoted with HQ) and two light partons denoted with (q), (the arrows show
their momentum vectors) with an angle $\phi$ between them, (b) a junction rest
frame system for the same configuration, and (c) a scenario where the heavy
quark coincides with the junction point and the angle between two light
partons is greater than $120^{\circ}$.

Figure 2 (a) illustrates a system with a heavy quark and two light colour
charges (quarks or gluons) in the initial rest frame of the heavy quark. If
the angle $\phi$ between the light charges is smaller than $120^{\circ}$,
there is always a frame, in which a junction is at rest as in figure 2 (b).
The massive quark moves more slowly, and the corresponding string piece is
shorter. For $\phi=120^{\circ}$ this length goes to zero, and for
$\phi>120^{\circ}$ the junction coincides with the heavy quark, see figure 2
(c). In this case, we find that it is most natural to hadronize the system in
the rest frame of the heavy quark. We have implemented this in , and one
consequence of this new procedure is that the heavy quark is more likely to be
the lowest energy leg, and will in addition not be able to fragment into a
heavy meson before being joined into a diquark and then ending up in a baryon.
(We note that in this situation the fragmentation system instead hadronizes
three strings connected at the centre in the rest frame of the whole junction
system. This reduces the probability of producing a heavy baryon and
overestimates the number of produced hadrons.)

We have made one more change in to enhance the chance of a heavy quark ending
up in a baryon, and that is to change the ordering of the junction legs.
Instead of taking the leg with the lowest summed energy of the connected
partons, we use the sum of absolute spatial momentum instead. In analogy with
the change in the $\lambda$-measure in the , this will more closely correspond
to how long the actual string is, and will more often put the leg with a heavy
quark among the two legs that are fragmented first. Again this improves the
chances that the heavy quark ends up in a baryon.

Besides changing the actual algorithms in and in the model, we have also
investigated some of the parameters that can affect the production of charmed
baryons. In the diquark formation by combining the two quarks from the low-
and middle-energy legs, the spin assignment is done by a set of
parameters333The parameter used to set these values is
StringFlav:probQQ1toQQ0join. suppressing the expected ratio of 3 spin-1 vs.
spin-0 states. The default values in are 0.5, 0.7, 0.9, and 1, for the cases
where the heaviest quark is u/d, s, c, and b, respectively, but these are not
well constrained by experimental data. In the so-called mode-0 tune for the
model, these were instead all set at 0.0275, which is close to the more well
constrained value used for the diquark–anti-diquark breakups in a normal
string. There is, however, no reason to expect that these parameters should be
the same, since the formation of diquarks in the joining of junction legs is
very different from the breakup in strings. And since we know that the model
has difficulties in describing the production of heavier baryon states, we
have checked the effect of raising the values to the default ones in also when
using .

Since we will here mostly be concerned with charmed baryons that also includes
strange quarks, there are also other effects that can influence the
production. It is well-known that strangeness enhancement is present not only
in heavy-ion collisions but also in high multiplicity collisions (see, ,
[ALICE:2016fzo]). In Lund we have studied the so-called rope hadronisation
model [Bierlich:2014xba, Bierlich:2022oja, Bierlich:2022ned], where
overlapping strings gives an increase in the string tension, $\kappa$. This
results in an increased probability of strange quarks in the string breakups
(, eq. 1) and the results are promising. Our current implementation does not,
however, handle junctions very well, which is why we here have decided to
emulate the effect by increasing the overall relative probability of having
strange quarks in sting breakups444The parameter for this is called
StringFlav:probStoUD. from the default value of 0.217 to 0.4. The number may
seem to be high but since most of the charmed baryons are produced at high
multiplicities, where there are many dipoles that can reconnect, and hence
also many strings can overlap, we do not consider it to be unreasonably high.

The overall charm content in an event is mainly governed by perturbative
effects, and can be gauged by the rate of the most common charmed D-mesons,
which are reasonably well described by the default . With the modifications we
have described here, however, a larger fraction of charm quarks will end up in
baryons, reducing the rate of D-mesons, and we have therefore decided to
compensate for this by increasing the overall charm production by reducing the
charm (and bottom) quark mass i from the default value of 1.5 (4.8) GeV to 1.3
(4.2) GeV.

.25  .25  .25  .25

Figure 3: Integrated prompt $\Xi^{0}_{c}$ cross-section for at $\sqrt{s}=5.02$
TeV, for $|y|<0.5$ using different options for compared to ALICE data
[ALICE:2021psx]. In all cases, we show default and, from the left, we show (a)
(mode-0), and spatially constrained (as SC-CR); (b) SC-CR with corrections in
the heavy quarks junction formation and fragmentation; (c) SC-CR will all the
new changes from this work; and (d) SC-CR (with all changes) with the
$\delta_{b}$ parameter set to 0.8 fm.

To get an indication of the overall effects of the changes we have proposed
here, we show in figure 3 the rate of direct production of $\Xi^{0}_{c}$
baryons in $\pp$ collisions at $\sqrt{s}=5.02$ TeV. The model results are
compared with the ALICE data [ALICE:2021psx] using the Rivet
[Bierlich:2019rhm] routine called ALICE_2021_I1863039. In the left-most
histogram, we show the results of the default (red line), (mode-0) (blue
line), and spatially constrained (green line). Here we see clearly the effect
of introducing the junction reconnections in . Our spatially constrained
version of the gives a slightly reduced rate, mainly because of the
constraint, but also because of differently tuned parameters (see
[Lonnblad:2023stc] for details). In the second to the left histogram, we show
the effect of the changes in junction formation and fragmentation for the SC-
CR case, and find an increase of around 35%. In the third histogram, we have
also added the parameter changes described above and found an additional
increase of almost 60%, giving an almost doubled rate compared to the default
SC-CR, and and a factor 8 more than the default . We are, however, still far
away from the lower bound of the experimental error bar, and a factor almost
three below the central value.

Finally, in the right-most histogram of figure 3, we show that if we allow
reconnection of dipoles farther separated in the transverse plane by
increasing the spatial constraint ($\delta_{b}$) value in the SC-CR model from
0.5 fm to 0.8 fm on top of the other changes we have made, then we can further
enhance the $\Xi^{0}_{c}$ baryon’s production in collision events. However,
since one of the aims of this paper is to compare to data using the model we
will in the following keep the tuned value of 0.5 fm, which we have shown in
[Lonnblad:2023stc] gives a more reasonable description of multiplicities in .

##  4 Results

In this section, we want to look more in detail at the effects of the changes
we made. We will concentrate on the charmed baryons, but will also look at
non-charmed hyperons. We first look at collisions to check that we get
reasonable results there before we extrapolate the models to collisions using
.

###  4.1 Hyperon production in $\pp$ collisions

.42  .42

.45

Figure 4: Top: $p_{\perp}$ differential yield of $\Sigma^{+}$ (left) and
$\Bar{\Sigma}^{-}$ (right). Bottom: $(\Xi^{0}+\Bar{\Xi}^{0})/2$ yield as a
function of $p_{\perp}$. The results are from the ALICE experiment
[ALICE:2014zxz] for $\pp$ collisions at 7 TeV and for mid-rapidity
($\mid$y$\mid$ $<0.5$). The experimental results are compared with default,
(mode-0), and spatially constrained with all the new changes from this work,
which are shown as red, blue, and green lines respectively.

Since we have forcibly increased the overall strangeness rate in the string
fragmentation, it is important to check that what we have done is not
unreasonable. In figure 4 we therefore show $p_{\perp}$ distribution for
$\Sigma^{+}$, $\Bar{\Sigma}^{-}$, and $(\Xi^{0}+\Bar{\Xi}^{0})/2$ baryons
respectively. The ALICE experiment [ALICE:2014zxz] results for $\pp$
collisions at $\sqrt{s}=7$ TeV are used here555The plots were generated using
the ALICE_2014_I1300380 routine in Rivet.. The measurements are reported for
inelastic collisions and for the particles in the mid-rapidity region
($\mid$y$\mid$ $<0.5$).

Comparing the default with and without , it is clear that the junction
reconnections do not contribute much to strange baryon. Instead, the main
production mechanism is diquark breakups in the string fragmentation. We can
therefore conclude that the main effect when looking at the changes we have
done here is the enhancement of strange (di-)quarks in the string breakups. It
can be argued that our enhancement is a bit high, but it is clearly not
completely unreasonable.

###  4.2 Charmed baryon production in $\pp$ collisions

We now turn to the charmed baryons and will start with $\Lambda_{c}$, where we
know that the model does a reasonable job. Looking back at figure 3, we see
that our changes increase the $\Xi^{0}_{c}$ rate substantially, and one can
fear that this is compensated by a decrease of $\Lambda_{c}$.

.5  .5

Figure 5: Prompt $\Lambda^{+}_{c}$ distribution as function of $p_{\perp}$ on
the left. The baryon-to-meson ratio for $\Lambda_{c}^{+}/D^{0}$ as a function
of $p_{\perp}$ on the right. The $\pp$ collision at 5.02 TeV results are from
the ALICE experiment [ALICE:2021psx]. The coloured lines represent the same
setups as in figure 4.

To check this, we show in figure 5 the $p_{\perp}$ distribution of the prompt
$\Lambda^{+}_{c}$ baryons and $\Lambda_{c}^{+}/D^{0}$ ratio is compared with
the ALICE data [ALICE:2020wfu] (using the same rivet routine as in figure 3).
Clearly, we maintain a good description of the $\Lambda^{+}_{c}$ cross section
and $\Lambda_{c}^{+}/D^{0}$ ratio, even after all the new changes we have
introduced in this work.

We note that the , both with and without our changes, gives more enhancement
for $\Lambda_{c}^{+}$ for low $p_{\perp}$, as seen both for the yield and for
the ratio to the $D^{0}$ yield. The reason for this is that most strings in an
event are fairly parallel to the beam, connecting low-$p_{\perp}$ partons
produced by MPI. So the largest chance to get baryons from junction
reconnections is from two or three dipoles from such strings along the beam
direction, which then results in low-$p_{\perp}$ baryons.

The effect is less visible for the strange baryons in figure 4 since the
relative contribution from junction reconnection is smaller but it is still
reflected in a small increase of small $p_{\perp}$ for .

.42  .42

.45

Figure 6:  Top Left: The prompt $\Xi^{0}_{c}$ cross section as a function of
$p_{\perp}$. Top Right: The $\Xi^{0}_{c}/D^{0}$ ratio as a function of
$p_{\perp}$. Bottom: The integrated $\Xi^{0}_{c}/D^{0}$ ratio for all
$p_{\perp}>0$. The data is from ALICE experiment [ALICE:2021psx] $\pp$
collisions at $\sqrt{s}=5.02$ TeV. The coloured lines represent the same
setups as in figure 4.

In figure 6, we then show the corresponding comparison for $p_{\perp}$
distribution of the prompt $\Xi^{0}_{c}$ baryons and the $\Xi^{0}_{c}/D^{0}$
ratio results obtained at the ALICE experiment [ALICE:2021psx] for $\pp$
collisions at $\sqrt{s}$=5.02 TeV. The cross section distribution basically
shows the same thing that we previously showed in figure 3, where the overall
yield for is far below the data while adding brings it closer, and with our
changes even more so.

The $\Xi^{0}_{c}/D^{0}$ ratio is arguably more relevant for assessing our
changes, since the overall (perturbatively modelled) charm rate is factored
out, and only the change in the non-perturbative modelling is important. Both
for the $p_{\perp}$ distribution and the integrated ratio our changes actually
come quite close to the data (note that there is a linear scale for the ratios
here). We note that for the $p_{\perp}$ shape, the data has a tendency to
decrease a bit for the lowest $p_{\perp}$ bin, while the model, with and
without our changes, seems to continue to rise, mirroring the behaviour in the
figure 5.

.5  .5

Figure 7: $p_{\perp}$ differential production cross section of
$\Lambda_{c}^{+}$ on the left and of $\Sigma_{c}^{0,+,++}$ on the right. The
data is from the ALICE experiment for pp collisions at 13 TeV [ALICE:2021rzj].
The coloured lines represent the same setups as in figure 4.

.5  .5

Figure 8: Baryon-to-meson ratio for $\Lambda_{c}^{+}/D^{0}$ on the left and
$\Sigma_{c}^{0,+,++}/D^{0}$ on the right. The data is from the ALICE
experiment for pp collisions at 13 TeV [ALICE:2021rzj]. The red, blue, and
green lines are the same as in the figure 7. The coloured lines represent the
same setups as in figure 4.

.5  .5

Figure 9: Left: $p_{\perp}$ differential production cross-section of
$\Lambda_{c}^{+}$ from $\Sigma_{c}^{0,+,++}$ decays. Right: Ratio of
$\Lambda_{c}^{+}$ from $\Sigma_{c}^{0,+,++}$ decays to the total
$\Lambda_{c}^{+}$ as a function of $p_{\perp}$. The data is from the ALICE
experiment for pp collisions at 13 TeV [ALICE:2021rzj].

Last year, the ALICE collaboration presented results [ALICE:2021rzj] 666The
analysis is implemented in the Rivet routine ALICE_2022_I1868463. also for
$\Sigma_{c}$ baryons in collision, this time using data from the LHC run 2 at
$\sqrt{s}$= 13 TeV. Figure 7 shows a differential production cross-section for
$\Lambda_{c}^{+}$ on the left and $\Sigma_{c}^{0,+,++}$ on the right as a
function of $p_{\perp}$, and in figure 8 the same is shown as a ratio to the
$D^{0}$ cross section. We can clearly see that the modification of the model
done in this paper not only maintains the $\Lambda_{c}^{+}$ description but
also controls the $\Sigma_{c}^{0,+,++}$ production rate in . Finally in figure
9, we show that due to the reduced $\Sigma_{c}^{0,+,++}$ production cross-
section, the fraction of $\Lambda_{c}^{+}$ coming from $\Sigma_{c}^{0,+,++}$
decays, and the ratio to the inclusive $\Lambda_{c}^{+}$ both are improved by
our modifications to the model.

From our changes to the , the one mainly influencing the $\Sigma_{c}$ rate is
the change in the parameter controlling the diquark formation in the joining
of the smallest junction legs in the fragmentation (see section 3.3.2).
Increasing the probability for a charmed diquark to be in a spin-1 rather than
a spin-0 state, means that $\Sigma_{c}^{\star}$ states are favoured over the
$\Sigma_{c}$ ones in the subsequent fragmentation of the largest leg. As
mentioned in section 3.3.2 these parameters were previously completely
unconstrained by data and in [Bierlich:2022pfr], the authors described the
chosen default values as guesswork. In (mode-0) the values were set to the
same, rather low, value for all quark types, but in our change, we decided to
keep the default ones which are higher and dependent on the heavy quark mass.
That the probability should be mass dependent is reasonable since the mass
splitting between the spin-1 and spin-0 state should be smaller when heavier
quarks are involved. (See, , [Bierlich:2022vdf] for a discussion on this).
Thanks to ALICE we now have data [ALICE:2021rzj] that can actually constrain
this parameter. Here also we notice that the $\Lambda_{c}^{+}/D^{0}$ ratio for
low $p_{\perp}$ is increased.

###  4.3 $\pPb$ collisions

.5  .5

Figure 10: Prompt $\Lambda^{+}_{c}$ distribution as a function of $p_{\perp}$
on the left. The baryon-to-meson ratio, $\Lambda_{c}^{+}/D^{0}$, as a function
of $p_{\perp}$ on the right. The red, blue, and green lines are default, new
tune with global CR as SC-CR (def), and the changes we have made in this work
in the global CR as SC-CR (all) respectively. The $pPb$ collision at 5.02 TeV
results are from the ALICE experiment [ALICE:2020wfu]. (Note that the rapidity
region, $-0.96<y<0.04$ is given in the collision rest frame, and corresponds
to the central, $|\eta|<0.5$ region in the laboratory frame.)

With a reasonable description of charmed baryon production in collisions, we
can now use the model to extrapolate our results in heavy-ion collisions. We
have in a previous publication shown that, by introducing a cut in the
transverse distance between dipole below which we allow them to reconnect in .
We can allow for a global colour reconnection between sub-collisions in heavy-
ion collisions, and still retain a reasonable description of hadron
multiplicities. Since the charmed baryon production has been shown to be a
sensitive probe into how the junction reconnections in the model behave, we
can now see in more detail, if our extrapolation to heavy-ion collisions is
reasonable.

In figure 10 we show results from the ALICE experiment at $\sqrt{s_{NN}}$=
5.02 TeV for the $\Lambda_{c}^{+}$ cross section, and the ratio of this w.r.t.
$D^{0}$ cross section, in comparison with model. As expected, the default ,
with colour reconnections only within each sub-collision separately, severely
underestimated the rate of $\Lambda_{c}^{+}$. Adding our spatially constrained
version of the model improves the description of data significantly, although
the $\Lambda_{c}$ cross section is still somewhat underestimated. Adding the
changes introduced in this paper, however, does not influence the result much.
This was not to be expected, since also in the effect on $\Lambda_{c}^{+}$ was
minor.

We can see that for low $p_{\perp}$ the model fails to reproduce the behaviour
of the data. This is best seen in the ratio to $D^{0}$, where our model
completely shows no sign of reduction of the ratio at small $p_{\perp}$. Also
this could be expected, as we had also seen indications of this in collisions
above.

##  5 Conclusion

This paper has shown, for the first time, the effect of applying a modern
colour reconnection model to a heavy ion collision, in order to better
describe baryon yields. We have shown that the production rates of
$\Lambda_{c}^{+}$ are dramatically improved in collisions using the model,
which has previously worked well in collisions. We also show that the diquark
formation in the joining of the junction legs influences the spin-dependent
baryon production, and we require experimental data similar to ALICE
[ALICE:2021rzj] to constrain the parameter in .

Heavy quarks can only be produced in hard scattering or in a parton shower
mechanism in . We show that the application of colour algebra in the model
allows junction formation by connecting three colour dipoles in a junction
point. These junctions contribute significantly to baryon production.

We show that for a heavy quark connected to a junction the $\lambda$-measure
used in the model should be improved. Usually, the $\lambda$-measure
calculates the logarithm of the energy of the dipole in the junction rest
frame. But if the dipole contains a heavy quark then often the invariant mass
of the quark has a non-negligible contribution to the energy of the dipole.
Therefore the rapidity span of the heavy quark from the junction point in the
junction rest frame should be used as the $\lambda$-measure for such a dipole.

Moreover, when a heavy quark dipole is directly connected to the junction
point, the system often fails to obtain a junction rest frame. If the momentum
of the heavy quark is low, it is possible that the string piece between the
junction and the heavy quark collapses to zero. Thus the heavy quark is
directly connected by two strings to the lighter quarks. We show that under
such a scenario fragmenting the junction system in the rest frame of the
heaviest quark is a good choice.

During the fragmentation of the junction system, the convention is to
calculate the energy in every leg and start fragmenting the junction system
from the lowest energy leg. Here again, we show that the choice of the new
$\lambda$-measure should be the scalar value of the momentum instead of the
energy because we should avoid counting the invariant mass of the quarks as
the potential energy available in the junction leg.

We notice that apart from the two modifications in the junction formation
during CR and junction fragmentation during hadronization, we need strange
quarks as many of the heavy baryons contain strange quarks. We have a rope
hadronization model, which contributes to the strangeness enhancement in . The
junction topologies are complex and the string-string interactions in rope
hadronization haven’t been implemented for junction configurations. Hence we
have compensated it by increasing the string fragmentation probability for the
strange quarks.

The charm and bottom quark masses are the other parameters we changed in this
paper. To enhance the charm and bottom quark production in the first place we
decided to use slightly lower mass values within the proposed mass ranges for
the respective quarks.

All these changes together helped us to improve the description for $\Xi$,
$\Sigma$, $\Xi_{c}$, and $\Sigma_{c}$ baryon production rates. We also managed
to keep a good description of the $\Lambda^{+}_{c}$ and
$\Lambda^{+}_{c}/D^{0}$ for different collision energies in $\pp$ collisions.
For the first time, we show the $\Lambda^{+}_{c}$ and $\Lambda^{+}_{c}/D^{0}$
results in $pPb$ collisions. The results are generated with the global CR in
and with the changes we introduced in this paper, and they show a visible
improvement over the default setup.

At this stage, it is also important to note that the increased strange quark
fragmentation should be replaced with the appropriate treatment of the rope
hadronization model. We may be required to retune some of the parameters
because the production of light baryon and other hyperons has not been tracked
against the new changes we have made in this work. Moreover, so far we have
applied the $\lambda$-measure correction only to the junction formation, but a
similar correction should also be applied to the “swing” CR between two
dipoles. We hope that the complete treatment of $\lambda$-measure correction
will affect the Quarkonium production. Hence exploring the possibility of
reproducing Quarkonia suppression in heavy-ion collisions is one of the tasks
for future work.