STag What follows is a brief description of how works, for a more detailed
overview of the architecture the reader is directed to our previous paper
Davison2022.

The tag probabilities are calculated using logistic regression, a function
that rapidly transitions between 0 and 1; given by Equation eq:lr Cramer2005:

$\sigma(z)=\frac{1}{1+e^{-z}}.$ (1)

Here z is as seen in Equation 2:

$z=\beta_{0}+\sum^{N}_{i=1}\beta_{i}x_{i},$ (2)

where $\beta_{0}$ is a normalisation constant, and the $\beta_{i}$ values are
the weights of the flux values $x_{i}$. The beta values (excluding
$\beta_{0}$) form a visual representation of the actual shape of the spectral
feature in question, and as such when Equation 1 is close to 1, the spectrum
has a high probability of showing said feature (and vice versa if Equation 1
is close to 0). Once a spectrum has all the tag probabilities calculated,
these are then passed to a simple ANN which determines the best suited class
using softmax regression, which is given by Equation 3:

$s(x_{i})=\frac{e^{x_{i}}}{\sum\nolimits_{j=1}^{N}e^{x_{j}}},$ (3)

where $x$ is a vector consisting of the weights from the output layer of the
ANN. The output is a vector of class probabilities, which all sum up to 1. The
ANN learns that different classes of SN have distinct tag probabilities and so
is able to accurately classify spectra.

##  Tags

With the introduction of a new dataset, it was necessary to also revisit the
tags and recreate them making use of the new data. Some of the original tags
had long wavelength ranges that potentially included other features (see
Figure 1) and in the case of the hydrogen tag, utilised the whole spectrum. As
such it was unclear exactly what it was using to determine the presence of
hydrogen. This initially led to the recreation of all the original tags with
stricter wavelength limits (see Figure 1), as well as the introduction of tags
for individual hydrogen features.

[]old_si_tag_v2.pdf

Figure : The original $\beta$ values for the $\lambda$6355 tag (originally
labelled as $\lambda$6150 in [Davison2022]), which can be seen to also
possibly include the absorption feature associated with $\lambda$5876\.

[]new_si_tag.pdf

Figure : The $\beta$ values for the $\lambda$6355 tag used in , which now only
contains the relevant feature to avoid potential contamination from other
features.

With the new tags created, the neural network was repeatedly trained and
tested with different combinations of included tags in order to determine
which tags were necessary for the classification of a SN and which were
superfluous. The result of this extensive training and testing was that the
$\lambda$6678 tag (which is characterised by a doublet feature) was causing a
lot of confusion between classifications and as such was removed.
Specifically, since this feature was very seldom seen in spectra outside of
Type IIb SNe, if it had even a $<10\%$ tag probability it was enough to reduce
the softmax probability of the predicted class. This had the knock-on effect
of also necessitating the removal the Type IIb class from ’s functionality as
there was no longer a suitable spectral feature to differentiate it from the
over-arching Type II class. Another change that was borne out of these tests
was that it was decided that an increase in the resolution of the spectra
would help account for smaller changes in the tags. The original value of the
spectral resolution was chosen to be $R=1024$ as this was the default setting
used in DASH, selected so as to be able to distinguish between broad and
narrow features whilst maintaining the desired speed of the classifier
Muthukrishna2019. As such, to accommodate for a greater capability of catching
the more subtle changes of spectral feature whilst also avoiding introducing
too much noise or making drastically more computationally expensive, the
number of points that make up the spectra was increased from 1024 to 1500.
This value was chosen after trialling different spectral resolutions of up to
2048, whereupon it was ultimately decided that the value of 1500 allowed for
the best balance of finer details and computational cost. The tag was also
further reduced, instead of being one tag that encompassed three suspected
features it is now a tag specifically for the $\lambda$5170 feature as it was
found to be the most frequent and readily identifiable. It was also decided
that the distinction between an emission, absorption, and P-Cygni line profile
for the $\lambda$5876 feature was unnecessary and likely causing more
confusion than clarity when reporting on tag probabilities (it was not
uncommon for spectra to have high probabilities for both absorption and
emission tags Davison2022). As a result, only the absorption feature is now
explicitly tagged for and results in a clear identifier of the $\lambda$5876
absorption feature. To further strengthen the classifications, as well as to
help the neural network to make the correct classification in the first place,
a number of new tags were also introduced. These consist of a tag for
$\lambda$4000, H$\alpha$ and a narrow H$\alpha$ $\lambda$6563 (henceforth
referred to as n-H$\alpha$, see Appendix LABEL:app:ixf for a more detailed
description of these two hydrogen lines), and H$\beta$ $\lambda$4861\. Whilst
not requiring a new tag specifically, the spectral features of $\lambda$5876
and D $\lambda$5876 both make use of the tag for at the same wavelength. All
three of these features overlap at approximately the same wavelength so even
with the more restricted ranges of the tags, it would still be impossible to
create separate tags for each of these. We therefore make use of the fact that
the $\lambda$5876 tag is now only for an absorption feature and that both
$\lambda$5876 and D $\lambda$5876 are also absorption features to effectively
create a multi-purpose tag. The final set of tags included in are H$\alpha$,
n-H$\alpha$, and H$\beta$ for Type II SNe, H&K, $\lambda$4000, $\lambda$6355,
and for Type Ia SNe, $\lambda$5876 for Type Ib SNe, and $\lambda$5170 as a
general tag. The differences between the tags included in the two different
versions of can be found in 1.

Tag |  |
---|---|---
H&K | $\checkmark$ | $\checkmark$
$\lambda$4000 | $\times$ | $\checkmark$
H$\beta$ | $\times$ | $\checkmark$
| $\checkmark$ | $\checkmark$
$\lambda$5170 | $\times$ | $\checkmark$
$\lambda$5876 (Emission) | $\checkmark$ | $\times$
$\lambda$5876 (Absorption) | $\checkmark$ | $\checkmark$
$\lambda$5876 (P Cygni) | $\checkmark$ | $\times$
$\lambda$6355 | $\checkmark$ | $\checkmark$
H$\alpha$ | $\times$ | $\checkmark$
H$\alpha$ (Narrow) | $\times$ | $\checkmark$
$\lambda$6678 | $\checkmark$ | $\times$
Table : Table comparing the tags present in the two different versions of .
Note that the tag was present in the first version of as a combination of
three lines rather than just the one in . It should also be noted that there
was a hydrogen tag in the first version of , however it used the full spectrum
and so wasn’t specific to any one feature [Davison2022].

##  Equivalent Width

The equivalent width (W) of a line is defined as in Equation 4:

$W=\int\frac{F_{c}-F_{l}}{F_{c}}d\lambda,$ (4)

where $F_{c}$ is the intensity of the continuum and $F_{l}$ is the intensity
of the spectral line. For an absorption line the equivalent width is positive,
whilst for an emission line it is negative. The equivalent widths in this
paper were calculated using the specutils package nicholasearl2023; which is
itself a package of Astropy astropy:2013,astropy:2018. All tags used in also
have an equivalent width also calculated for the same feature using the same
wavelength range as for the tag, with the exception of the sulphur line as
this is a ‘W’ shaped feature and so it is less obvious what the equivalent
width of this feature would correspond to. In order to see if there was extra
information that could be used between the probability of a given tag and the
equivalent width of the associated spectral feature, we also trialled
including a nonlinear combination of these two values by passing the product
of each pairing as separate inputs to the neural network. Ultimately it was
determined that such nonlinear combinations offered no extra information for
which the neural network could use to improve the classifications, with the
exception of the Si $\lambda$4000 feature. It is unclear why this feature, and
this alone, makes a difference on the accuracy of the classifications, however
its inclusion does lead to a greater accuracy and so it continues to be
utilised. The inclusion of equivalent widths, and the changes to the tags,
meant that the architecture of had to be changed. now consists of an input
layer of 18 nodes, followed by three fully connected layers of 48 nodes each,
then two layers of 96 nodes each, which then connects to two more layers of 48
nodes each, and finally a softmax output layer consisting of 4 nodes. Each of
the hidden layers make use of the Rectified Linear Unit (ReLU) activation
function [nair2010rectified]. This neural network was built using a
combination of both keras [chollet2015keras] and TensorFlow
[tensorflow2015-whitepaper] and has the architecture it does after extensive
testing to find the best-performing number of hidden layers and nodes.

##  rlap Score

In order to give us a further measure of confidence in a classification beyond
just the associated softmax probability, we also made use of what is known as
the rlap score, as devised by Blondin2007. A value of 0 would mean that there
is no correlation between two spectra being considered, whilst it is typically
accepted that a value above $\sim$5-6 suggests a reasonably strong correlation
Blondin2007,Muthukrishna2019. This parameter is built off of first cross-
correlating the spectrum of the SN with that of a template spectrum that is at
$z=0$:

$c(n)=s(n)\star t(n)=t(n)\star t(n-\delta)+a(n).$ (5)

Here $c(n)$ is the cross-correlation, $s(n)$ is the SN spectrum, $t(n)$ is the
template spectrum, $t(n-\delta)$ is the template spectrum shifted by some
amount $\delta$ in logarithmic wavelength space, and $a(n)$ is a function that
distorts the peak of the correlation function Blondin2007. The rlap score is
the product of two values: r, which is the cross-correlation height-noise
ratio Tonry1979 which is given by Equation 6, and lap, which is a measure of
the overlap in wavelength space of the two spectra being cross-correlated and
is defined by Equation 7 Blondin2007.

$r=\frac{h}{\sqrt{2}\sigma_{a}},$ (6)
$\mathrm{lap}=\ln\frac{\lambda_{1}}{\lambda_{0}},$ (7)

where h is the height of the peak of the cross-correlation compared to the rms
of the anti-symmetric component, given by $\sigma_{a}$. $\lambda_{1}$ and
$\lambda_{0}$ are the maximum and minimum wavelength the input spectra
overlaps with the comparison spectra; see Figure 1 for an example of how to
interpret Equation 6 visually. Note that we are able to use z here since we
are working in log wavelength space and so a shift in this space is equivalent
to a linear shift in $1+z$ Blondin2007.

[]cross-correlation.pdf

Figure : The cross-correlation function of a real DESI spectrum with a Type Ia
template spectrum. This is a functional recreation of similar plots found in
Blondin2007 and Muthukrishna2019, Figures 5 and 3 respectively.

By using both the value of r and of lap one has a way of quantifying how
similar a given spectrum is to a template spectrum and so is an extra level of
certainty that our classification is accurate. To streamline the process, we
assume the softmax classification is the best estimation of the class and so
only cross-correlate a given spectrum with templates of the same type, but
with differing phase. This now gives the added functionality of estimating the
phase of a SN, though we stress this it not the primary function of and the
range of phases are used primarily to take care of the fact that SN spectra
change with time and so prevent a dilution of the rlap value due to comparison
with many spectra of the correct type but wrong phases. We make use of 5
different phase bins, which are based on the phase range used to generate the
DESI template spectra (and as such the DESI simulated spectra). The first bin
encompasses all negative phase spectra (with respect to maximum light), the
next 3 bins are separated by 7 days each, and the final bin is for all phases
greater than 21 days since maximum light, up to 30 days. The cross-correlation
process was adapted from that used in DASH Muthukrishna2019, with minor
adjustments made to fit the pipeline of . These changes included, but are not
limited to, adjusting the wavelength points to consider (since we increased
the resolution of from 1024 to 1500 points) and a change to account for the
reading of the different template spectra being used, but functionally it
works the same as in DASH).

##  Redshift Checking

is highly dependent on accurate redshifts for it to be able to classify a
spectra correctly, as such it is important that any redshifts being supplied
are correct. However, is not designed to be something that can estimate a
redshift. As such, we are not interested in producing a comprehensive
framework for calculating the redshift of a SN, though we note that could be
modified to do this by using the rlap score. Instead, we adopt the philosophy
that if a redshift is significantly inaccurate, we accept that will simply be
unable to return a satisfactory classification. We decided that should a
spectrum initially fail the rlap criteria, this may be caused by an incorrect
redshift and so we can then perform a check for a degree of inaccuracy. In the
case of a relatively small inaccuracy it is worth being able to potentially
use a more accurate redshift as this will lead to a better classification, as
well as the possibility of a spectrum passing the rlap cut that it would have
previously failed. The general principle we used was to check a series of
redshifts within 10% of 1+z (where $z$ is the redshift returned by Redrock),
resulting in a check of 5% either side of the given redshift. We chose to look
at just 10 redshifts in the proposed range (balancing speed with enough
redshift values to properly explore values around the given redshift), which
means that each redshift is roughly 1% from its surrounding redshift values.
What we found was that a spectra would pass the final rlap cut if the redshift
was within 1% of the true redshift, whilst the rlap value would quickly fall
off outside of this limit (see Figure 1). Therefore, should none of the 10
alternative redshifts pass the rlap criteria then will not return a
classification and indicate to the user that it is possibly being caused by an
incorrect redshift value.

[]rlap_z_v2.pdf

Figure : The rlap score for the spectra of SN 2021qtc when pre-processed at
different redshifts. The rlap score drops quickly outside 1% of the true
redshift value, meaning only when the redshift is accurate will the rlap score
be able to pass the cutoff point of 6, indicated by the horizontal black
dashed line.


# Hadro-chemistry effects on charm decayed leptons in heavy-ion collisions

Kai Lu  Department of Applied Physics, Nanjing University of Science and
Technology, Nanjing 210094, China  Min He  Department of Applied Physics,
Nanjing University of Science and Technology, Nanjing 210094, China

###### Abstract

Charm-hadrons possess versatile hadro-chemistry as characterized by various
transverse-momentum-dependent ratios between their different species. In
particular, the charm hadro-chemistry may be modified in relativistic heavy-
ion collisions with respect to proton-proton collisions at the same energy, as
caused by novel diffusion and hadronization mechanisms of charm quarks in the
environment of the created quark-gluon plasma (QGP) in the former. Inspired by
recent measurements of leptons from charm-hadron decays (separated from bottom
decays) in Pb-Pb and Au-Au collisions, we investigate the effects of the charm
hadro-chemistry on the leptonic observables. We find that full consideration
of charm hadro-chemistry in both proton-proton and heavy-ion collisions causes
only mild change of charm-leptons’ suppression factor with respect to previous
calculations hadronizing charm quarks into $D$ mesons only, whereas the
resulting change (increase) in the charm-leptons’ elliptic flow turns out to
be more pronounced as a consequence of the larger collectivity of
$\Lambda_{c}$ baryons than $D$ mesons.

Heavy Flavor, Quark Gluon Plasma, Ultrarelativistic Heavy-Ion Collisions

###### pacs:

25.75.Dw, 12.38.Mh, 25.75.Nq

##  I Introduction

Heavy-ion collisions at collider energies produce a novel state of deconfined
strong-interaction matter that is known as quark-gluon plasma (QGP) and
behaves as a strongly-coupled near-perfect fluid Shuryak:2014zxa ;
Busza:2018rrf . Heavy quarks, charm and bottom, have masses much larger than
the intrinsic non-perturbative scale $\Lambda_{\rm QCD}$ as well as the
typical temperatures reached in current heavy-ion collision experiments. Thus
they are produced in the initial stage of the collisions and participate in
the full evolution of the system (yet with the charm/bottom number conserved),
constituting powerful probes of the matter created (for pertinent recent
reviews, see Dong:2019unq ; Dong:2019byy ; Zhao:2020jqu ; He:2022ywp ).

Historically, the traditional observables associated with heavy flavor probes
were non-photonic leptons (electrons and muons) from semileptonic decays of
combined charm and bottom hadrons STAR:2006btx ; PHENIX:2006iih ;
Sakai:2013ata ; ALICE:2015xyt ; ALICE:2016ywm ; ALICE:2020sjb . The strong
suppression and large elliptic flow of heavy flavor leptons provided evidence
of strong coupling of heavy quarks with the medium Rapp:2009my . In recent
years, measurements of charm-hadrons have become accessible, which not only
corroborated the strong collectivity of charm probes in a more direct fashion
ALICE:2015vxz ; ALICE:2013olq ; ALICE:2017pbx ; ALICE:2021rxa ; CMS:2017qjw ;
CMS:2017vhp ; STAR:2018zdy ; STAR:2017kkh ,but also revealed versatile charm
hadro-chemistry as characterized by various transverse-momentum ($p_{T}$)
dependent charm-hadron ratios ALICE:2015dry ; ALICE:2021kfc ; ALICE:2021bib ;
CMS:2019uws ; STAR:2021tte ; STAR:2019ank . Interpretation of the strong
collectivity and differential hadro-chemistry of charm-hadrons requires inputs
of charm quark diffusion in the quark-gluon plasma with a rather small spatial
diffusion coefficient and hadronization in terms of recombination that plays
the dominant role in the low and intermediate $p_{T}$ regime He:2012df ;
He:2019vgs ; Plumari:2017ntm ; Nahrgang:2013xaa ; Ke:2018tsh ; Katz:2019fkc ;
Cao:2019iqs ; Beraudo:2022dpz .

Going beyond the early measurements of comprehensive heavy flavor leptons,
leptons from charm decays have been rather recently separated from those from
bottom decays, both in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV by
ATLAS collaboration ATLAS:2020yxw ; ATLAS:2021xtw  and in Au-Au collisions at
$\sqrt{s_{\rm NN}}=200$ GeV by PHENIX PHENIX:2015ynp  and STAR collaboration
STAR:2021uzu ; Si:2019esg , offering an opportunity to test if differential
charm hadro-chemistry leaves any imprints on the charm leptonic observables.
Indeed in previous studies of heavy flavor leptons vanHees:2005wb ;
Gossiaux:2008jv ; He:2011qa ; He:2014cla ; Song:2016rzw ; Katz:2019fkc , only
$D$ mesons were accounted for in the charm quark hadronization and in the
ensuing charm decayed leptons. However, from the latest charm hadro-chemistry
point of view He:2019vgs ; Andronic:2021erx , charm quarks are hadronized into
different species of charm-hadrons that possess varying $p_{T}$ spectra
(including their anisotropy) and have different kinematics and branching
ratios when decaying into leptons, leading to possible changes in the charm
leptonic observables with respect to the case of hadronizing all charm quarks
into $D$ mesons only.

The aim of the present work is to investigate the quantitative effects of full
charm hadro-chemistry on the suppression and collective flow of charm leptons.
A first attempt in this regard was made in Refs. Sorensen:2005sm ; Martinez-
Garcia:2007mzr , where a schematic charm-baryon-to-meson enhancement in Au-Au
collisions (relative to $pp$ collisions) was assumed without taking account of
realistic energy loss, and an additional moderate suppression of charm
electrons at intermediate $p_{T}$ was estimated as a result of the smaller
branching ratio of $\Lambda_{c}$ baryons decaying to electrons than $D$
mesons. In the present work, we employ the full $p_{T}$ dependent charm hadro-
chemistry computed in a comprehensive charm transport approach recently
developed in He:2019vgs  for heavy-ion collisions. For calculating the charm
leptons spectrum in $pp$ collisions as the baseline to characterize the medium
effect, we use the $p_{T}$ spectra of various charm-hadrons computed in an
extended statistical hadronization model He:2019tik , which successfully
explained the charm hadro-chemistry measured in high-energy $pp$ collisions by
ALICE collaboration ALICE:2020wfu ; ALICE:2021dhb . We find that, with respect
to calculations considering charm quarks hadronizing into only $D$ mesons,
full consideration of charm hadro-chemistry in both $pp$ and heavy-ion
collisions results in only mild change of charm-leptons’ nuclear modification
factor, while the resulting change (increase) in the charm-leptons’ elliptic
flow turns out to be more pronounced, which seems to be supported by the
recent measurement of charm muons by the ATLAS collaboration ATLAS:2020yxw ;
ATLAS:2021xtw .

The organization of our article is as follows. In Sec. II, we briefly recall
the essential ingredients and features of our models developed recently for
calculating the charm hadro-chemistry in $pp$ and heavy-ion collisions and
show the pertinent results in $\sqrt{s_{\rm NN}}=5.02$ TeV semicentral Pb-Pb
collisions from an updated calculation. In Sec. III, we perform the
semileptonic decays of various charm-hadrons and compare the corresponding
observables between charm muons and their parent hadrons for each species. In
Sec. IV, we compare the nuclear modification factor and elliptic flow of total
charm leptons computed from all charm-hadron species with the ones calculated
from considering charm quarks hadronizing into only $D$ mesons. The
quantitative influence of charm hadro-chemistry on charm leptonic observables
is then discussed when confronted with pertinent measurements in Pb-Pb and Au-
Au collisions. We finally summarize in Sec.V.

##  II Charm hadro-chemistry in $pp$ and heavy-ion collisions

Recent measurements of charm-hadrons differential yields in $\sqrt{s}=5.02$
TeV $pp$ collisions at mid-rapidity by the ALICE collaboration demonstrated
that charm fragmentation is non-universal across different colliding systems,
as highlighted by the significant enhancement of the $\Lambda_{c}^{+}/D^{0}$
ratio at low $p_{T}$ relative to $e^{+}e^{-}$ collisions ALICE:2020wfu . This
was successfully explained by an extended statistical hadronization model
calculation He:2019tik , where the PDG list of charm-baryons was augmented
with many more not-yet-observed states predicted from relativist quark model
calculations Ebert:2011kk . The $p_{T}$ spectra of ground state $D^{0}$,
$D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$ were well reproduced and the total
charm cross section per unity rapidity (at mid-rapidity) was fitted to be
$d\sigma^{c{\bar{c}}}/dy\sim 1$ mb, which is consistent with the measured
value reported in ALICE:2021dhb . These $p_{T}$ differential cross sections
for the ground state charm-hadrons will be used to perform semileptonic
decays, in order to get the baseline spectra for calculating the nuclear
modification factors of the charm leptons in $\sqrt{s}=5.02$ TeV Pb-Pb
collisions.

Figure 1: (Color online) Calculated charm-hadron ratios in $\sqrt{s}=5.02$ TeV
Pb-Pb collisions in the 30-50% centrality bin at mid-rapidity in comparison
with ALICE measurements ALICE:2021rxa ; ALICE:2021kfc ; ALICE:2021bib . For
$\Lambda_{c}^{+}/D^{0}$, the band encompasses uncertainty in the branching
raitos of the not-yet-measured excited charm-baryons feeding down to the
ground state $\Lambda_{c}^{+}$, taken to be 50%-100%, cf. He:2019vgs .

For the charm hadro-chemistry in heavy-ion collisions, we conduct a
calculation within our recently developed transport approach He:2019vgs . In
this approach, charm quark diffusion in the hydrodynamically expanding QGP was
simulated with the transport coefficient calculated in the lattice-constrained
$T$-matrix approach Riek:2010fk ; Liu:2018syc . The charm quark hadronization
was modelled by resonance recombination; in particular the three-body
resonance recombination model (RRM) was developed to compute the charm-baryon
formation, taking advantage of the light diquark correlations in the charm-
baryon sector. In addition, a method was devised to incorporate space-momentum
correlations (SMCs) between the phase space distributions of charm quarks and
light quarks that are built up whether through diffusion or from hydrodynamic
flow. In practice, the RRM was implemented on an event-by-event basis in
combination with the relativistic Langevin simulation of charm quark diffusion
in QGP. With the selfconsistely determined recomination probability as a
function of charm quark momentum in the fluid rest frame, this implementation
allows to conserve charm quark number in each event and satisfies both kinetic
and chemical equilibrium limits when using thermal quark distributions as
inputs. These features, together with the inclusion of all charm-hadron
species including augmented charm-baryons (same as in $pp$), are pivotal for
controlled predictions of the $p_{T}$ dependent charm hadro-chemistry.

In Fig. 1, we show the $p_{T}$ dependence of ground state charm-hadron ratios
$D^{+}/D^{0}$, $D_{s}^{+}/D^{0}$ and $\Lambda_{c}^{+}/D^{0}$ in comparison
with ALICE data in 30-50% Pb-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV,
from a calculation within the aforementioned transport model with updated
light diquark masses. More specifically, unlike the degenerate diquark masses
used in He:2019vgs , we here use distinct scalar diquark mass ($\sim 710$ MeV)
for $\Lambda_{c}$ states and axial-vector diquark mass ($\sim 909$ MeV) for
$\Sigma_{c}$ states Ebert:2011kk . Such a refinement yields a better
description of the $p_{T}$ shape of the $\Lambda_{c}^{+}/D^{0}$ than reported
in He:2019vgs . Compared to the corresponding ratios in $pp$ collisions
He:2019tik , $D^{+}/D^{0}$ remains almost the same but $D_{s}^{+}/D^{0}$ is
significantly enhanced in the low and intermediate $p_{T}$ regime as a result
of charm recombination in the presence of strangeness equilibration in QGP
He:2012df . We emphasize here that it is important to correctly reproduce the
$D^{+}/D^{0}$ in the present context of studying the hadro-chemistry effects
on the charm leptons, since the $D^{+}$ semileptnic decay branching ratio is
almost two and half times that of $D^{0}$ ParticleDataGroup:2018ovx . For
$\Lambda_{c}^{+}/D^{0}$, the moderate enhancement relative to the $pp$ case
only shows up at intermediate $p_{T}$ due to recombination incorporating SMCs,
whereas at low $p_{T}$, the value turns out to be compatible with the $pp$
value, implying the integrated ratio may not change ALICE:2021bib . Taken as a
whole, we note that these ratios, when integrated over $p_{T}$, are consistent
with the statistical hadronization model (SHMc) calculations Andronic:2021erx
(albeit a smaller $\Lambda_{c}^{+}/D^{0}$ value found in SHMc because of less
charm-baryons considered therein Andronic:2021erx ), as dictated by the
relative chemical equilibrium reached in the RRM He:2019vgs .

##  III Charm hadrons vs decayed leptons observables

Figure 2: (Color online) Charm muons invariant $p_{T}$ spectra decayed from
absolutely normalized $p_{T}$ spectra of ground state charm-hadrons $D^{0}$,
$D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$ in $\sqrt{s}=5.02$ TeV Pb-Pb
collisions in the 30-50% centrality bin at mid-rapidity. For the case of
$\Lambda_{c}^{+}$, we show only the result corresponding to branching ratios
$\sim 100$% of excited $\Lambda_{c}$ and $\Sigma_{c}$ states decaying to the
ground state $\Lambda_{c}^{+}$. Figure 3: (Color online) $R_{\rm AA}$’s of
ground state charm-hadrons in comparison their corresponding charm muons in
$\sqrt{s}=5.02$ TeV Pb-Pb collisions in the 30-50% centrality bin at mid-
rapidity. ALICE data are taken from ALICE:2021rxa ; ALICE:2021kfc ;
ALICE:2021bib . For $\Lambda_{c}^{+}$, we show only the result corresponding
to branching ratios $\sim 100$% of excited $\Lambda_{c}$ and $\Sigma_{c}$
states decaying to the ground state $\Lambda_{c}^{+}$.

The calculated ground state charm-hadron $p_{T}$ spectra are then taken to
perform semileptonic decays He:2011qa . More specifically, the semileptonic
decays of charm-hadrons are simulated as free quark decays $c\rightarrow
s+l+\bar{\nu}_{l}$, with the decay matrix element taken from the low-energy
$V$-$A$ theory Griffiths : $\overline{|\mathcal{M}|^{2}}\propto(p_{s}\cdot
p_{\bar{\nu}_{l}})(p_{c}\cdot p_{l})$ and branching ratios taken from
ParticleDataGroup:2018ovx . The quark masses are replaced by the corresponding
hadron masses to correctly account for the phase space. The hadronic form
factors have little influence on the lepton energy distribution in the parent
charm-hadron rest frame He:2011qa . The lepton momenta are then boosted into
lab frame.

In Fig. 2, we display the invariant muons spectra decayed from ground state
$D^{0}$, $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$ spectra that have been
normalized to the realistic total charm number $dN_{c\bar{c}}/dy\sim 3.44$
(after taking account of shadowing reduction) in $\sqrt{s}=5.02$ TeV Pb-Pb
collisions in the 30-50% centrality bin at mid-rapidity. One notes that even
if the $D^{+}/D^{0}$ is around $\sim 0.4$, the invariant muons spectrum
decayed from $D^{+}$ finally exceeds that from $D^{0}$, because the branching
ratio of the former ($\sim 16.1\%$) is almost $2.5$ times that of the latter
($\sim 6.5\%$). The charm muons spectra decayed from $D_{s}^{+}$ and
$\Lambda_{c}^{+}$ are significantly softer than those from $D^{0}$ and
$D^{+}$, because of the steeper $p_{T}$ spectra of the former two charm-
hadrons as well as kinematic effect in the decay associated with their larger
masses. This, in combination with the smaller $\Lambda_{c}^{+}$ semileptonic
branching ratio ($\sim 4.5\%$; the $D_{s}^{+}$ branching ratio is roughly the
same as that of $D^{0}$), renders the total charm muons spectrum dominated by
those decayed from two un-stranged charm mesons, which is more so toward high
$p_{T}$; e.g., at $p_{T}\sim 1~{}(6)$ GeV, the latter accounts for $\sim
70\%~{}(90\%)$ of the total. In the present study, we have neglected the
contribution from $\Xi_{c}$ states, since their fraction of total charm cross
section is not significant He:2019tik .

The nuclear modification factor of a charm-hadron is defined as

$R_{\rm AA}(p_{T})=\frac{dN_{\rm AA}/dp_{T}dy}{N_{\rm coll}/\sigma_{\rm
NN}^{\rm in}d\sigma_{\rm pp}/dp_{T}dy},$ (1)

where the numerator is the absolute $p_{T}$ differential yield of the charm-
hadron under consideration in heavy-ion collisions, $d\sigma_{\rm
pp}/dp_{T}dy$ is the charm-hadron $p_{T}$ differential cross section in $pp$
collisions He:2019tik  at the same colliding energy, and $N_{\rm coll}$ and
$\sigma_{\rm NN}^{\rm in}$ are the binary collision number of the considered
centrality bin and the inelastic nucleon-nucleon cross section, respectively.
The nuclear modification factors of each charm-hadron species (compared to
ALICE measurements) and of their corresponding charm muons are shown in Fig.
3. An overall feature is a shift of the “flow bump” in the well reproduced
charm-hadron $R_{\rm AA}$’s toward lower $p_{T}$ as a consequence of the
decay, which is more significant for the muons from $D_{s}^{+}$ and
$\Lambda_{c}^{+}$ decays.

Figure 4: (Color online) $D^{0}$ and $\Lambda_{c}^{+}$ $v_{2}$ in
$\sqrt{s}=5.02$ TeV Pb-Pb collisions in the 30-50% centrality bin at mid-
rapidity, in comparison with those of their decayed muons. The CMS data of
$D^{0}$ $v_{2}$ are taken from CMS:2017vhp .

The elliptic flow coefficient, defined as

$v_{2}(p_{T})=\left\langle\frac{p_{x}^{2}-p_{y}^{2}}{p_{x}^{2}+p_{y}^{2}}\right\rangle,$
(2)

characterizes the momentum anisotropy as a result of charm coupling with the
medium through diffusion and hadronization. The $v_{2}$ of $D^{0}$ (compared
to CMS measurements) and $\Lambda_{c}^{+}$ are shown in Fig. 4 ($D^{+}$ and
$D_{s}^{+}$ $v_{2}$ are similar to that of $D^{0}$), together with the $v_{2}$
of their decayed muons. An important observation here is that the
$\Lambda_{c}^{+}$ $v_{2}$ is significantly larger than that of $D^{0}$ toward
$p_{T}>3$ GeV, as a result of three-body RRM incorporating SMCs that push the
reach of recombination to higher $p_{T}$ He:2019vgs . An immediate consequence
is that the muons decayed from $\Lambda_{c}^{+}$ also have a significantly
greater $v_{2}$ than the $D^{0}$ muons for $p_{T}>2$ GeV, which may finally
help increase the $v_{2}$ of total charm muons.

##  IV Total charm leptonic observables vs data

Figure 5: (Color online) $R_{\rm AA}$ and $v_{2}$ of total charm muons in
$\sqrt{s}=5.02$ TeV Pb-Pb collisions in the 30-50% centrality bin at mid-
rapidity from calculations taking account of full charm hadro-chemistry or
from the scenario of charm quarks hadronizing into only $D$ mesons. ATLAS data
are taken from ATLAS:2020yxw ; ATLAS:2021xtw  for charm muons with
pseudorapidity cut $|\eta|<2.0$.

Having analysed the observables of charm muons decayed from each charm-hadron
species, we are now in a position to combine them together and show the
results for the total charm muons. On the other hand, in previous studies of
charm leptons within transport approaches vanHees:2005wb ; Gossiaux:2008jv ;
He:2011qa ; He:2014cla ; Song:2016rzw ; Katz:2019fkc , only $D$ mesons
(denoting the sum of $D^{0}$ and $D^{+}$) were accounted for in the charm
quark hadronization and therefore in the ensuing charm decayed leptons,
ignoring in particular the role of the $\Lambda_{c}^{+}$ baryons (we have
checked that $D_{s}^{+}/D^{0}$ enhancement as shown in Fig. 1 does not play a
significant role in the observables of the final total charm muons, including
their $v_{2}$, because $D_{s}^{+}$ decayed muons accounts for a rather minor
fraction of the total as seen from Fig. 2 and the $D_{s}^{+}$ $v_{2}$ is
similar to that of the $D$ mesons). Here for the purpose of making out the
pertinent charm hadro-chemistry effects through comparison, we have also
conducted a calculation assuming all charm quarks are hadronized into $D$
mesons only, which are then subject to semileptonic decays to get the total
charm leptons with an average branching ratio of $\sim 9.4$% (obtained from an
average between the branching ratios of $D^{0}$ and $D^{+}$ using the
integrated ratio $D^{+}/D^{0}$ as pertinent weighs). More specifically, we use
the charm quark recombination probability into $D$ mesons, $P_{D}^{\rm
coal}(p_{c}^{*})$, selfconsistently determined from the RRM formalism
He:2019vgs , and renormalize it to unity at charm quark momentum $p_{c}^{*}=0$
(to ensure low momentum charm quarks are hadronized via recombination
He:2019vgs ), with the remaining $1-P_{D}^{\rm coal}(p_{c}^{*})$ identified as
the charm quark fragmentation probability into $D$ mesons. This follows the
procedure done for the calculation with full charm hadro-chemistry, but in the
latter, recombination probabilities of charm quarks into all charm-hadrons
including the augmented charm-baryon excited states are added up and then
renormalized He:2019vgs .

Figure 6: (Color online) $R_{\rm AA}$ and $v_{2}$ of total charm electrons in
$\sqrt{s}=200$ GeV Au-Au collisions in the 20-30% centrality bin at mid-
rapidity from calculations taking account of full charm hadro-chemistry or
from the scenario of charm quarks hadronizing into only $D$ mesons. STAR data
are taken from STAR:2021uzu .

The calculated $R_{\rm AA}$ and $v_{2}$ of the total charm muons are compiled
in Fig. 5 in comparison with the ATLAS measurements for $\sqrt{s_{\rm
NN}}=5.02$ TeV Pb-Pb collisions in the 30-50% centrality bin at mid-rapidity.
For consistency, for calculating the $R_{\rm AA}$ in the scenario of all charm
quarks hadronizing into $D$ mesons only, we have adopted the same hadro-
chemistry in $pp$ collisions as in Pb-Pb collisions; i.e., in $pp$ collisions
all charm quarks are fragmented into only $D$ mesons whose $p_{T}$ spectrum as
baseline is normalized to the realistic total charm cross section. One can
read from the upper panel of Fig. 5, that taking full account of the charm
hadro-chemistry only causes a mild change in the nuclear modification factor
of the total charm muons, relative to the scenario of hadronizing charm quarks
into $D$ mesons only. Both results are within the error bars of the ATLAS data
that is currently limited to $p_{T}>4$ GeV.

A more pronounced change (increase) in the $v_{2}$ of the total charm muons is
seen upon full consideration of charm hadro-chemistry, as demonstrated in the
lower panel of Fig. 5. This is mostly caused by the significantly greater
$v_{2}$ of $\Lambda_{c}^{+}$ and its decayed muons than the $D$-meson
counterparts, cf. Fig. 4. However, this increase is not as large as that seen
in Fig. 4, since the total charm muons are dominated by those decayed from $D$
mesons and the contribution from $\Lambda_{c}^{+}$ accounts only for a minor
fraction at intermediate $p_{T}$, as demonstrated in Fig. 2. The pronounced
increase in the $v_{2}$ of the total charm muons as a result of full account
of the charm hadro-chemistry seems to be supported by the large value of the
ATLAS measurement in the 30-40% centrality, although the present result is
still significantly below the ATLAS data in nearly whole measured $p_{T}$
range.

The same calculations have been performed for $\sqrt{s_{\rm NN}}=200$ GeV Au-
Au collisions in the 20-30% centrality bin as a proxy for the minimum bias
collisions He:2019vgs . The results for the total charm electrons are shown in
Fig. 6. The calculated $R_{\rm AA}$ from full consideration of charm hadro-
chemistry is comparable to the STAR data in 0-80% centrality, and the
corresponding $v_{2}$ is also significantly greater than the result from the
scenario of hadronizing charm quarks into $D$ mesons only.

##  V Summary

In this work, inspired by the recent measurements of charm leptons by ATLAS
and STAR experiments in Pb-Pb and Au-Au collisions, we have investigated the
charm hadro-chemistry effects on the charm leptonic observables within our
recently developed transport approach for charm probes He:2019vgs . Our study
demonstrates that, because the total charm leptons spectrum is still dominated
by the ones decayed from $D$ mesons even after taking into account full charm
hadro-chemistry, the total charm leptons’ nuclear modification factor does not
change much as compared to the scenario of hadronizing all charm quarks into
$D$ mesons only, which has been widely adopted by transport approaches when
computing heavy flavor leptons. Yet the total charm leptons’ elliptic flow
acquires a pronounced increase because of the inclusion of the
$\Lambda_{c}^{+}$ baryons that have significantly greater collectivity as a
result of the three-body recombination, rendering the computed total charm
muons $v_{2}$ closer to ATLAS data.

Going beyond the charm sector, bottom electrons $v_{2}$ has also recently been
measured for the first time by ALICE experiment in Pb-Pb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV ALICE:2020hdw . Calculated results from transport
models Nahrgang:2013xaa ; Song:2016rzw ; Ke:2018tsh  that assumed substantial
interactions of bottom quark with the QGP and hadronized bottom quarks into
$B$ mesons only significantly underestimated the bottom electron $v_{2}$ at
intermediate $p_{T}$ ALICE:2020hdw , signalling the potentially significant
role of $\Lambda_{b}$ baryons (through its semileptonic decays) that are
expected to have larger collectivity than $B$ mesons. This calls for full and
controlled computation of bottom hadro-chemistry in transport approach like
for the charm sector He:2019vgs , which will be addressed in our near-future
work.

###### Acknowledgements.

This work was supported by NSFC grant 12075122.

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