aainstitutetext: Institut für Kernphysik, Universität Münster, Wilhelm-Klemm-
Straße 9, 48149 Münster, Germanybbinstitutetext: Institut für Theoretische
Physik, Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany

# Di-electron production at the LHC: Unravelling virtual-photon and heavy-
flavour contributions

Anton Andronic b , Tomáš Ježo b , Michael Klasen a , Christian Klein-
Bösing b and Alexander Puck Neuwirth<EMAIL_ADDRESS>tomas.jezo@uni-
muenster.de<EMAIL_ADDRESS>christian.klein-boesing@uni-
muenster.de<EMAIL_ADDRESS>
###### Abstract

The production of virtual photons is a very sensitive probe of the properties
of the quark-gluon plasma. As they are experimentally detected by lepton
pairs, they suffer from a large background arising from hadron decays. Light-
flavour hadrons dominate at low invariant masses below $m_{ee}\sim 0.5$ GeV
and heavy flavours above. These contributions must therefore also be taken
into account in experimental analyses at the LHC. In this paper, we calculate
the direct contribution from virtual photons produced in the Drell-Yan process
with an additional jet in POWHEG and find that it is significant at low
invariant masses. We also simulate the background contributions from
$c\bar{c}$ and $b\bar{b}$ production with POWHEG and quantify the theoretical
uncertainties due to variations of the perturbative scales and parton
distribution functions. We find larger relative and absolute uncertainties for
the lighter $c$ quarks than for heavier $b$ quarks.

###### Keywords:

Perturbative QCD, hadron colliders, lepton pairs, Drell-Yan, heavy flavours

††preprint: MS-TP-23-22

##  1 Introduction

One of the key objectives of the Large Hadron Collider (LHC) physics program
is the understanding of high-energy density and temperature conditions in
ultra-relativistic heavy-ion collisions. These collisions lead to the creation
of a quark-gluon plasma (QGP), a system of quarks and gluons in a deconfined
state with restored chiral symmetry Rapp:2009yu  that is believed to have
existed in the early Universe shortly after the Big Bang and which is
predicted by quantum-chromodynamic (QCD) calculations on the lattice
Karsch:1998qj . In the high-temperature and low baryochemical potential region
of the QCD matter phase diagram, this state of matter is reached in a smooth
crossover, while at low temperatures and high baryochemical potentials it is
expected to be reached in a first-order phase transition with a discontinuity
in the thermodynamic variables Philipsen:2019rjq . One of the expected signals
of a QGP is the radiation of “thermal photons”, with a particular transverse
momentum spectrum reflecting its temperature Rapp:2009yu ; Tserruya:2009zt .

Photons are widely regarded as indispensable probes of heavy-ion collisions.
Generated throughout all stages of the collision process, they originate
either from decays of light- and heavy-flavour hadrons or are produced
directly from hard-scattering processes like the quark-gluon Compton
scattering and quark–anti-quark annihilation. In addition to these sources,
the hot medium produced in heavy-ion collisions may emit thermal radiation,
commonly grouped together with the direct production. Unaffected by the strong
interaction, the photons traverse the dense medium without undergoing any
final-state interactions. In contrast to real photons, virtual photons
($\gamma^{*}\to e^{+}e^{-}$) carry a mass ($m_{ee}$), unaltered by the radial
flow, that can act as an approximate clock, allowing for the separation of
different collision stages. The pre-equilibrium effects become evident around
$m_{ee}\gtrsim$2\text{\,}\mathrm{GeV}$$, followed by the expected formation of
the QGP in the range $$1\text{\,}\mathrm{GeV}$\lesssim
m_{ee}\lesssim$2\text{\,}\mathrm{GeV}$$, and finally the hadronic phase at
later emission stages with $m_{ee}\lesssim$1\text{\,}\mathrm{GeV}$$
Rapp:1999ej ; Coquet:2021lca . They can therefore provide valuable insight
into the entire time evolution and dynamics of the hot system. However,
measurements of the thermal photon signal in heavy-ion collisions face some
considerable challenges, such as a small production cross section due to the
electromagnetic coupling and significant combinatorial and physical
backgrounds from hadron decays.

In proton-proton (pp) collisions, the di-electron low and intermediate mass
spectrum can be well described by a combination of expected hadronic sources,
the so-called cocktail PHENIX:2008qav ; PHENIX:2009gyd ; STAR:2012dzw ;
ALICE:2018fvj ; ALICE:2018gev . The low-mass region (LMR,
$m_{ee}<$1.1\text{\,}\mathrm{GeV}$$) is dominated by light-flavoured vector
meson decays, whereas the intermediate-mass region (IMR,
$$1.1\text{\,}\mathrm{GeV}$<m_{ee}<$2.7\text{\,}\mathrm{GeV}$$) receives
contributions mostly from semileptonic decays of charm and bottom hadrons
correlated through flavour conservation. Despite the overwhelming backgrounds,
significant enhancements of direct real or low-mass virtual photons have been
observed in heavy-ion collisions at the SPS, RHIC and LHC CERES:1995vll ;
CERESNA45:1997tgc ; CERES:2006wcq ; CERESNA45:2002gnc ; NA60:2006ymb ;
NA60:2008ctj ; PHENIX:2008uif ; PHENIX:2009gyd ; STAR:2013pwb ; STAR:2015tnn ;
PHENIX:2014nkk ; ALICE:2015xmh . The enhancements are well described by
exponential distributions in $p_{T,\gamma}$ and are thus indicative of thermal
radiation consistent with average (effective) temperatures in the range of
$200-$300\text{\,}\mathrm{MeV}$$ PHENIX:2008uif ; PHENIX:2014nkk ;
ALICE:2015xmh . In the intermediate-mass window, an excess over decay di-
electrons has so far been observed only at the SPS NA60:2006ymb ; NA60:2008ctj
; NA60:2008dcb ; NA60:2007lzy . In order to discern the characteristics of the
subtle thermal photon signal in this range at the LHC, that promises increased
sensitivity to QGP thermal radiation, it is crucial to understand the di-
electron yield originating from the hard production of photons as well as from
heavy-flavour particle decays.

In this publication, we report on theoretical progress in the description of
the di-electron mass spectrum in the intermediate-mass region in setups
similar to those used in ALICE experimental analyses. First, we calculate the
contribution to the direct photon production signal originating from the hard
scattering. This component is usually extracted from the data ALICE:2018ael ,
assuming the invariant mass shape from the Kroll-Wada prescription
Kroll:1955zu  with additional constraints from real direct photon
measurements, and must be subtracted from the total direct photon signal for
reliable temperature estimates Klasen:2013mga ; ALICE:2015xmh , especially in
the intermediate transverse momentum region. Here, for the first time, we
obtain a prediction for the di-electron invariant mass spectrum based on a
first-principles calculation using the Drell-Yan process in association with
an extra jet at next-to-leading order (NLO) QCD accuracy matched to parton
showers. Second, we perform a thorough analysis of the dominant decay
backgrounds with a variation of perturbative scales and parton distribution
functions (PDFs), also taking into account cold nuclear effects through
nuclear PDFs. Such uncertainties can have an appreciable impact on the
systematic uncertainty estimates in the measured di-electron spectrum baseline
in pp as well as heavy-ion collisions. Studying this process is also of
broader interest, as it offers a means to test perturbative QCD (pQCD)
calculations and Monte Carlo (MC) event generators at the limits of
perturbativity. In heavy-ion collisions it can be used in studies of energy
loss, levels of thermalisation of charm and bottom quarks within the medium,
and mechanisms of heavy-quark hadronisation Greco:2003vf ; Andronic:2003zv ;
Baier:1996sk ; Braaten:1991we . The distribution of correlated $e^{+}e^{-}$
pairs stemming from charm-hadron decays provides information about kinematic
correlations between charm and anticharm quarks, shedding light on the
production mechanisms and offering sensitivity to soft heavy-flavour
production LHCb:2015swx ; CMS:2021lab ; ALICE:2021mgk  and thermalization, in
PbPb collisions.

The manuscript is organised as follows. We first present general theoretical
considerations and define our signal and background in the di-electron
observables in the kinematic region of interest in Section 2. We then explain
our simulation setup including our workflow and the event selection in Section
3. In Section 4 we present our results for virtual photon and heavy flavour
production combined into a new prediction for the “hadronic cocktail”.
Finally, we summarise this work in Section 5.

##  2 Signal and background

The intermediate mass window region in measurements of di-electron spectra is
characterised by a lepton pair with a relatively low invariant mass, between
roughly 1 and 3 GeV, as well as relatively low transverse momentum, below 10
GeV. Besides the thermal photon signal, there are two dominant production
modes in this kinematic regime. One of them is the direct photon production
accompanied by a photon conversion into di-electrons. In the following we will
refer to this contribution as the signal. The second contribution is the
heavy-flavoured hadron production followed by a decay into a di-electron
pair.111The light-flavoured hadron production with di-electron decay dominates
the invariant mass spectrum below 1 GeV, but is rather suppressed above. This
contribution, instead, we will refer to as the background.

The signal is the direct production of photons with non-zero virtuality that
decay into di-electrons, that is the Drell-Yan (DY) process. The tree-level DY
process is described by the partonic process $q\bar{q}\to\gamma^{*}\to
l^{+}l^{-}$ and is of the order $\mathcal{O}(\alpha^{2})$ in the perturbative
expansion of the electromagnetic coupling constant. In the absence of any
other particles the $e^{+}e^{-}$ pair is produced back-to-back and the system
carries no transverse momentum, falling out of the kinematic region of our
interest. To get around this problem, one must produce a pair of di-electrons
in association with one jet at order $\mathcal{O}(\alpha^{2}\alpha_{S})$ in
the simultaneous expansions of the electromagnetic and strong couplings
constants. At this order one does not only receive contributions from the
$q\bar{q}$ channel, but also from the $qg$ channel and therefore picks up
sensitivity to the gluon density.222Furthermore, the NLO QCD corrections,
$\mathcal{O}(\alpha^{2}\alpha_{S}^{2})$, can also be gluon-gluon initiated.
Note that this process also receives contributions from the topology with the
photon replaced by the $Z$ boson and the corresponding $\gamma-Z$
interference, which we both include, but their contributions are negligible in
the invariant-mass regime far below the $Z$-boson mass. We simulate our signal
at NLO QCD accuracy, i.e. including higher order real and virtual corrections
up to $\mathcal{O}(\alpha^{2}\alpha_{S}^{2})$, using the Zj process
Alioli:2010qp  from the POWHEG BOX V2 package Nason:2004rx ; Frixione:2007vw ;
Alioli:2010xd , which is now also available in Multi-Scale improved NLO
(MiNLO) Hamilton:2012np  and NNLO (MiNNLO) Monni:2019whf ; Monni:2020nks
accuracies.

In the limit of zero di-electron virtuality one recovers real direct photon
production at $\mathcal{O}(\alpha\alpha_{S})$, in which the photon is produced
on-shell and does not decay into di-electrons. Similarly to the fragmentation
of jets into photons in direct real photon production, contributions from the
fragmentation of jets into di-electrons are possible. In analogy to our
calculation of real direct photons Jezo:2016ypn  within the POWHEG framework,
the Zj process includes this fragmentation contribution, partly generated
through parton shower emissions of the photon.333Contrary to real photons, in
virtual photon production there is no explicit divergence that must be
absorbed into fragmentation functions and, instead, the divergence is
regulated by the photon virtuality Klasen:1997jm ; Klasen:2002xb .
Nevertheless, the resulting logarithms $\ln(m_{ee}/p_{T,ee})$ arising per
order of $\alpha_{S}$ could in principle become large and spoil the
convergence of the perturbative series. We keep track of the $m_{ee}/p_{T,ee}$
ratio and find it ranges from $10^{-1}$ to $10^{1}$ with a sharp peak around
1. We thus expect our predictions to be stable with respect to such higher
order effects in the kinematic region of our interest.

We now move on to the discussion of the background, which can be further split
into the open production of charm and bottom quarks Rapp:1999ej ;
Mohanty:2013rna . While photon-induced or DY-like production of heavy quarks
is possible, due to the ratio of couplings strengths
$\alpha^{2}/\alpha_{S}^{2}\lesssim$0.5\text{\,}\%$$ it is typically sufficient
to only consider the QCD production Bonciani:2015hgv ; Pagani:2016caq . That
is, heavy-flavour partons are only produced from a subset of the QCD Feynman
diagrams of light-flavour production, namely where the heavy quarks in the
final state are connected by a fermion line and thus belong to the same
flavour generation independently of the initial-state partons. The heavy
quarks then hadronise predominantly into $D$ and $B$ mesons, which further
decay, in a correlated manner, into electrons. This results in correlated
electron pairs stemming from two different hadrons originating from the same
hard process. To simulate our background at NLO QCD accuracy
($\mathcal{O}(\alpha_{S}^{3})$), we use the hvq process Frixione:2007nw  from
POWHEG BOX V2.

Both simulations, of the signal and of the background, are further interfaced
to Pythia8. In the case of the signal, it only provides higher order QCD and
QED corrections in the soft-collinear approximation via parton showers,
whereas in the case of the background we rely on it also for the hadronisation
and the heavy-flavoured hadron decays. We use Pythia8 version 8.3.08
Bierlich:2022pfr .

##  3 Simulation setup and event selection

In this manuscript we focus on the production of di-electrons, open charm and
bottom pairs in proton-proton (pp), proton-lead (pPb) and lead-lead (PbPb)
collisions at the LHC with $\sqrt{s}=5.02$ TeV (per nucleon pair). The
experimental pp and pPb data from the corresponding ALICE measurement
ALICE:2020mfy  are available in HEPdata Maguire:2017ypu . The ALICE data in
PbPb have been recently published ALICE:2023jef , but the numerical values are
not yet available in HEPdata. Unless stated otherwise, we use MSHT20nlo
Bailey:2020ooq  PDFs for the proton and nCTEQ15HQ Duwentaster:2022kpv  nuclear
PDFs for lead as provided by LHAPDF6 LHAPDF  with the corresponding value of
$\alpha_{S}(M_{Z})$.

We generate the Les Houches Event (LHE) samples using POWHEG BOX V2 and shower
them using Pythia8 piloted by the main-PYTHIA83-lhef executable, which is in
turn coupled to Rivet 3.1.7 RIVET  based on the Rivet 2 interface originally
developed in Ref. Jezo:2018yaf  and later upgraded for Rivet 3 in Ref.
FerrarioRavasio:2023kjq . The relevant physics input parameters are chosen
from Ref. ParticleDataGroup:2022pth  with

$\displaystyle\alpha$ $\displaystyle=1/128.89,$ $\displaystyle m_{c}$
$\displaystyle=$1.27\text{\,}\mathrm{GeV}$,$ $\displaystyle m_{b}$
$\displaystyle=$4.18\text{\,}\mathrm{GeV}$.$

For the production of heavy quarks, the renormalisation and factorisation
scales are chosen dynamically as $\mu_{r}=\mu_{f}=\sqrt{p_{T}^{2}+m_{q}^{2}}$,
where $p_{T}\neq p_{T,ee}$ is the transverse momentum of the heavy quark and
$q$ labels its flavour. For $c\bar{c}$ a three-flavour scheme is used, while
for $b\bar{b}$ we use a four-flavour scheme. The corresponding running of
$\alpha_{S}$ is calculated with internal POWHEG subroutines. For the Drell-Yan
process instead we set the value of the renormalisation and factorisation
scales to $\mu_{r}=\mu_{f}=\max\left(2\ {\rm
GeV},\sqrt{p_{T,\gamma}^{2}+p_{\gamma}^{2}}\right)$ with its minimum value
frozen at 2 GeV and use a five-flavour scheme. In both cases we calculate
perturbative scale uncertainties using the standard factor-two seven-point
$\mu_{r},\mu_{f}$ scale variation method, excluding relative factors of four.

In Pythia8, we leave parton shower, hadronisation and MPI enabled. Hadron
decays are only enabled for our background predictions, as to avoid decay
photons in our signal. Analogously, in heavy flavour production we disable the
QED shower, but leave it on for the production of virtual photons in order to
generate the fragmentation contribution. To match to POWHEG we employ the
standard procedure in which shower evolution starts unrestricted but shower
emissions are vetoed using PowhegHooks. In Table 1 we list values of selected
Pythia8 parameters for each process.

Table 1:  Parameters used in Pythia for the different processes. Values differing from the default are highlighted in bold.  Pythia setting | Zj | hvq
---|---|---
`PartonLevel:ISR` | on | on
`PartonLevel:FSR` | on | on
`PartonLevel:MPI` | on | on
`PartonLevel:Remnants` | on | on
`PartonShowers:model` | 1 | 1
`SpaceShower:pTmaxMatch` | 2 | 2
`TimeShower:pTmaxMatch` | 2 | 2
`POWHEG:veto` | 1 | 1
`POWHEG:pTdef` | 1 | 1
`SpaceShower:QEDshowerByL` | on | off
`SpaceShower:QEDshowerByQ` | on | off
`TimeShower:QEDshowerByGamma` | on | off
`TimeShower:QEDshowerByL` | on | off
`TimeShower:QEDshowerByOther` | on | off
`TimeShower:QEDshowerByQ` | on | off
`HadronLevel:Hadronize` | on | on
`HadronLevel:Decay` | off | on

Figure 1:  Di-electron mass spectra for $\text{pp}\to c\bar{c}(X)$ (left) and
$\text{pp}\to b\bar{b}(X)$ (right) production, decomposed into open heavy
flavour continuum and the resonance production in the LMR and IMR. All the
contributions were normalised as in ALICE:2020mfy .

Our event selection and histogramming of the observables is performed by a
Rivet analysis prepared for this publication that mimics the analysis of Ref.
ALICE:2020mfy  and which we include in the ancillary files. We select all
electrons and positrons that fulfil the acceptance cuts of $|\eta_{e}|<0.8$,
required due to ALICE’s central barrel geometry, and
$$0.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<p_{T,e}<$10\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
We then use the electron and positron to build the transverse momentum and
virtuality of the pair, $p_{T,ee}$ and $m_{ee}$. As in the ALICE analysis, we
consider the transverse momentum and invariant mass ranges
$$0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<p_{T,ee}<8$ GeV and
$$0.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<m_{ee}<$1.1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
or
$$1.1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<m_{ee}<$2.7\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
for the di-electron pair. For the invariant mass spectra, we also consider the
range
$$0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<m_{ee}<$7\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
which extends upon the ALICE measurement that is only available in the range
$$0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<m_{ee}<$3.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

We remove the combinatorial background using the same-sign approximation, as
in Ref. ALICE:2018fvj , where the raw pair signal $S$ is obtained with the
formula

$S=\underbrace{\left(N_{+-}\right)}_{\text{OS}}-R_{\text{acc}}\underbrace{\left(N_{++}+N_{--}\right)}_{\text{SS}}\
,$ (1)

where we estimate the number of same-sign (SS) pairs with an arithmetic mean,
which is more suitable than a geometric mean when considering a contribution
to the spectrum due to a single process, and where we set the relative
acceptance correction factor, $R_{\text{acc}}$, equal to one. Moreover
$N_{+-}$ is the number of opposite-sign (OS) pairs and $N_{++}$, $N_{--}$ the
numbers of positron-positron, electron-electron pairs in the signal region.
Assuming that the SS pairs are uncorrelated, the subtracted term will
correspond to the uncorrelated OS pairs at the high statistical limit. This
gives the raw pair signal $S$ with unwanted uncorrelated contributions
removed.

In order to isolate the continuum contribution to open heavy flavour
production from the heavy quark POWHEG+Pythia8 sample we classify electrons
according to their origin. In Figure 1, we show di-electron invariant mass
spectra from $\text{pp}\to c\bar{c}(X)$ and $\text{pp}\to b\bar{b}(X)$
production. In both cases, below 1 GeV, the spectrum receives contributions
from heavy to light flavoured hadron cascade decays:
$\pi^{0},\rho^{0},\eta,\omega,\eta^{\prime},\phi$. Above 1 GeV, only the
$J/\psi$ peak is visible at around 3 GeV, whereas in between, in the IMR, the
continuum dominates.

This inspires our definition of the continuum contribution to the hadronic
cocktail as the POWHEG+Pythia8 subsample with electrons from light hadron and
$J/\psi$ decays filtered away. Practically speaking, we require all leptons to
have a parent particle that is either a $b$ or $c$ flavoured hadron by
inspecting the event record in Rivet. It is worth noting that this implies
that the hadron decay chains $b\to c\to e$ are not filtered away. This is
meant to align with the definition used in ALICE in which contributions
removed by filtering based on origin are instead obtained using the Monte
Carlo event generator EXODUS PHENIX:2009gyd . Our predictions, blue
histograms, reproduce the predictions from ALICE extracted from Ref.
ALICE:2020mfy , $c\bar{c}(b\bar{b})$ ALICE-MC, which were obtained with
POWHEG+Pythia6 and CTEQ6.6 PDFs, red curves, very well. We note that our
predictions were normalised such that the area under the curve of our “open c”
and “open b” prediction match the area under the “$c\bar{c}$ ALICE-MC” and
“$b\bar{b}$ ALICE-MC” curves, respectively.

##  4 Our predictions

In Refs. ALICE:2020mfy ; ALICE:2018fvj ; ALICE:2018gev  the production of di-
electron pairs ($e^{+}e^{-}$) in proton-proton collisions at
$\sqrt{s}=(5.02,7,13)$ TeV was measured using the ALICE detector at the LHC.
The studies focused on the invariant mass and transverse momentum
distributions of the $e^{+}e^{-}$ pairs. The data was compared to a hadronic
cocktail composed of expected di-electron distributions of known three
dominant hadronic sources: light flavour decays, the $J/\psi$ resonance, and
open heavy flavour decays. Good agreement between data and the hadronic
cocktail was observed over the entire mass range ($m_{ee}<3.5$ GeV). However,
the extracted total cross sections for charm and beauty strongly depend on
details of theoretical modelling (e.g. Pythia6 vs. POWHEG+Pythia6). The same
measurement was also carried out in proton-lead collisions at
$\sqrt{s_{\text{NN}}}=5.02$ TeV ALICE:2020mfy . Again good agreement with the
cocktail was found assuming heavy flavour cross sections scale with the atomic
mass number $A$ of the lead nucleus in pPb-collisions with respect to the pp
reference. The measured ratio $R_{\text{pPb}}$, corrected for such $A$
scaling, is consistent with unity within uncertainties in the IMR where most
of the $e^{+}e^{-}$ pairs originate from correlated open heavy flavour hadron
decays. Thus, the uncertainties on the measured $p_{T,ee}$ dependence of
$R_{\text{pPb}}$ are still too large to draw conclusions on the nuclear
modification of heavy flavour production in pPb collisions. In PbPb
collisions, ALICE measured the $e^{+}e^{-}$ spectra at
$\sqrt{s_{\text{NN}}}=(2.76,5.02)$ TeV ALICE:2018ael ; ALICE:2023jef , but the
former measurement is statistically limited. Also here the cocktail with an
additional scaling by $R^{b,c\to e}_{\text{PbPb}}$ due to final state medium
effects is in reasonable agreement with the data, despite neglecting the cold
nuclear matter effects.

In this section we study two predictions that enter those analyses. First we
present our predictions for the prompt component of the di-electron spectrum
due to direct virtual photon production in Section 4.1. This contribution has
so far not been considered in the ALICE measurements. Second we show our
predictions for the open charm and bottom production, including a careful
analysis of uncertainties from scale and PDF variations in Section 4.2. In
Section 4.3 we analyse all our predictions together and assemble them into a
new prediction for the hadronic cocktail with realistic uncertainties.
Finally, Section 4.4 is dedicated to our predictions for PbPb collisions and a
discussion on initial- and final-state matter effects.

###  4.1 The signal: direct virtual photon production

Figure 2:  Di-electron pair invariant mass spectrum from virtual photon
production in association with at least one jet at NLO+PS for pp (left) and
pPb (right) obtained with POWHEG+Pythia8 using MSHT20nlo and nCTEQ15HQ PDFs
for p and Pb. The uncertainty band combines scale and PDF uncertainties in
quadrature. The second and third ratio panels respectively show the relative
scale and PDF uncertainties independently.

Figure 3:  Di-electron pair transverse momentum spectrum from virtual photon
production in association with at least one jet at NLO+PS for pp (left) an pPb
(right) obtained with POWHEG+Pythia8 using MSHT20nlo and nCTEQ15HQ PDFs for as
p and Pb. The uncertainty band combines scale and PDF uncertainties in
quadrature. The second and third ratio panels respectively show the relative
scale and PDF uncertainties independently.

Figure 2 shows the invariant mass spectra of the di-electron system
originating from virtual photons produced in association with at least one jet
in 5.02 TeV pp and pPb in collisions at NLO+PS in red and at the NLO LHE
level444NLO LHE corresponds to an NLO prediction supplemented by the Sudakov
form factor of the NLO emission. in grey. Going below an invariant mass of
$m_{ee}=1$ GeV, the cross section becomes less reliable due to the photon pole
at $m_{ee}=0$. It is therefore not surprising that the invariant mass spectrum
increases continuously towards lower masses, see the top panels. The total
uncertainties combined in quadrature are dominated by the variation of scales
and are about $\pm$$25\text{\,}\%$ across the whole invariant mass range, but
up to $\pm$$50\text{\,}\%$ in the low tail of the transverse momentum spectra
(see below). In contrast, the PDF uncertainties are roughly
$\pm$$10\text{\,}\%$ for proton-proton collisions and only a few percent more
for proton-lead, see the third and the fourth panels. Note that the PDF
uncertainty band beyond $m_{ee}>3$ GeV, saturated by the Monte Carlo errors
due to limited size of our event sample, likely overestimates the true
uncertainty. The difference between the sizes of the cross section of the
different initial states is a relative increase by $A=208$ from pp to pPb. For
the $p_{T}$ spectrum in Figure 3 we observe that the parton shower emissions
beyond $\mathcal{O}(\alpha_{S})$ soften the low tail and harden the high tail
of the $p_{T,ee}$ spectrum (NLO LHE vs. NLO+Py8). Such a transformation into a
more physical distribution is expected from matching to parton showers.

The virtual photon production is a subleading but not a negligible
contribution to the di-electron pair production in this kinematic regime. It
reaches fractions up to 45% in the range of 1.2 to 3 GeV of the invariant mass
spectrum, where the depletion at 3 GeV is due to the $J/\psi$ contribution in
the measured spectrum. Similarly, in the transverse momentum spectrum it
contributes up to 40% in the intermediate $p_{T}$ range of 1 to 4 GeV. Note
that our scale choice, appropriate for this kinematic regime, enhances the
virtual photon production cross section appreciably. With the (inappropriate)
choice $\mu_{f}=\mu_{r}=m_{Z}$ this contribution would be only at about 5%
with very similarly shaped distributions.

In Figure 4 we show the ratio of pPb and pp predictions for the virtual photon
production. Both the invariant mass and the transverse momentum spectra lie
below one by up to 20-30% with an appreciable kinematic dependence. The
uncertainty band represents the usual 7-point scale variation band obtained by
taking the envelope of ratios of pPb over pp predictions with the same scale
choice. Contrary to the usual expectation for the scale uncertainty to cancel
out in the $R_{\text{pPb}}$ ratio, it remains relatively large at about 25%
across the whole $m_{ee}$ and $p_{T,ee}$ range as compared to roughly $\pm
15\%$ to $\pm 40\%$ in the absolute predictions.

Figure 4:  Di-electron pair ratios $R_{\text{pPb}}$ of the invariant mass
(left) and transverse momentum distributions (right) for the Drell-Yan
process. The uncertainty bands are correlated scale uncertainties.

###  4.2 The background: open charm and bottom production

Figure 5:  Invariant di-electron mass spectra in open charm production
$\text{pp, pPb}\to c\bar{c}(X)$ obtained with POWHEG+Py8 and MSHT20nlo and
nCTEQ15HQ PDFs for p and Pb. The second row shows the scale and the third row
the PDF uncertainties.

Figure 6:  Di-electron transverse momentum spectra in open charm production
$\text{pp, pPb}\to c\bar{c}(X)$ obtained with POWHEG+Py8 and MSHT20nlo and
nCTEQ15HQ PDFs for p and Pb. The second row shows the scale and the third row
the PDF uncertainties.

Figure 7:  Invariant di-electron mass spectra in open bottom production
$\text{pp, pPb}\to b\bar{b}(X)$ obtained with POWHEG+Py8 and MSHT20nlo and
nCTEQ15HQ PDFs for p and Pb. The second row shows the scale and the third row
the PDF uncertainties.

Figure 8:  Di-electron transverse momentum spectra in open bottom production
$\text{pp, pPb}\to b\bar{b}(X)$ obtained with POWHEG+Py8 and MSHT20nlo and
nCTEQ15HQ PDFs for p and Pb. The second row shows the scale and the third row
the PDF uncertainties.

Figure 9:  Di-electron pair ratios $R_{\text{pPb}}$ of the invariant mass
(left) and transverse momentum distributions (right) for the $c\bar{c}$
process. The uncertainty bands are correlated scale uncertainties.

Figure 10:  Di-electron pair ratios $R_{\text{pPb}}$ of the invariant mass
(left) and transverse momentum distributions (right) for the $b\bar{b}$
process. The uncertainty bands are correlated scale uncertainties.

Next, we present our predictions for open charm and bottom production, again
in pp and pPb collisions at
$\sqrt{s}=$5.02\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. The first rows of
Figures 5 and 6 respectively show the absolute predictions for the invariant
mass and the transverse momentum spectra of di-electrons from open charm at
NLO+PS in green. In the second rows we show the perturbative scale variations
and generally observe very large uncertainties, presumably due to large
logarithms related to the charm quark mass, which can exceed a factor of three
regardless of the collision system. For di-electrons from open bottom
production the situation is under much better control and the scale
uncertainties drop under ${}^{+$60\text{\,}\%$}_{-$40\text{\,}\%$}$, see
Figures 7 and 8. The third rows in Figures 5, 6, 7 and 8 show that the PDF
uncertainty is small relative to the scale band and under $\pm$$20\text{\,}\%$
($\pm$$10\text{\,}\%$) for the open charm (bottom) channel for both pp and pPb
collisions throughout. Comparing the pp and pPb PDF uncertainties they
increase slightly from pp to pPb in open bottom but not in the open charm
production. This suggests that very low-$x$ gluons in the nuclear PDF fit
(nCTEQ15HQ) have a smaller uncertainty than in the proton PDF fit (MSHT20nlo).
A realistic PDF uncertainty estimate in pPb collisions should of course
include both the proton as well as the nuclear uncertainty dimension, which is
however not available in the nCTEQ15HQ fit.

We also compare our predictions to the POWHEG+Pythia6 (ALICE-MC) calculations,
in blue, and data from ALICE ALICE:2020mfy , in black. The heavy flavour
production is subleading in the LMR, but in the region in $m_{ee}\in[1,2]$ GeV
the open charm production almost saturates the data. The two predictions
generally agree well. There are differences both in normalisation and shape,
but they agree within the scale uncertainty band of our prediction for both
the open charm and open bottom channel, except for the tails of $m_{ee}$. The
remaining differences in the shapes of the invariant mass distributions can be
explained by the different hadronisation model in Pythia version 6 vs. version
8 and the use of different, in the case of our predictions more modern, PDF
sets.

In the measurements of the di-electron spectra, the total $c\bar{c}$ and
$b\bar{b}$ production cross section is determined from the data. The
normalisation component of the uncertainty must thus not necessarily be taken
into account. If we were to fix the normalisation of each of the predictions
in the scale or PDF variation bands, we would expect the uncertainty due to
normalisation to drop out and the uncertainty bands to shrink. To what extent
this happens is investigated in the next section.

In Figures 9 and 10 we show the ratio of pPb and pp predictions for charm and
bottom production respectively. In the case of open charm the ratios as
functions of both $m_{ee}$ and $p_{T,ee}$ are up to fluctuations consistent
with one within the scale uncertainty which is significantly reduced in the
ratio, a few tens of percent, as compared to the absolute pp and pPb
predictions, up to factor three. For open bottom production, the scale
uncertainties vanish almost completely and the $R_{pPb}$ ratios are up to
fluctuations within 10% of unity, except in the high tail of $m_{ee}$ where it
reaches the value of 0.8.

We find that Pythia8 mainly hadronises the charm quarks to D±
($50\text{\,}\%$), D0 ($39\text{\,}\%$) and D${}_{s}^{\pm}$ ($7\text{\,}\%$)
hadrons, whereas the immediate parent particles of the electrons in open
bottom production are mainly D± ($23\text{\,}\%$), D0 ($23\text{\,}\%$), B0
($22\text{\,}\%$), B± ($20\text{\,}\%$), D${}^{\pm}_{s}$ ($7\text{\,}\%$) and
B${}_{s}^{\pm}$ ($4\text{\,}\%$) hadrons.

###  4.3 Combined analysis of the signal and background

Figure 11:  Comparison of the cocktail used by ALICE against our new cocktail
including the Drell-Yan process with an extra jet. The shaded band is the
scale shape uncertainty. The blue shaded band on the cocktails is the
uncertainty given by ALICE collaboration on their cocktail. The red shaded
band combines scale shape and ALICE uncertainties.

In this section we consider our new prediction for the virtual photon and the
revised predictions for open heavy-flavour production together. The first rows
on both panels in Figure 11 show our absolute predictions for virtual photon
production in purple, $b\bar{b}$ in yellow and $c\bar{c}$ in green, whereas
the third rows show their ratio to the largest channel ($c\bar{c}$). The
normalisation of the heavy flavour continuum is typically determined via
double differential fits in $m_{ee}$ and $p_{T,ee}$ in the intermediate mass
regime ALICE:2020mfy  or via the measured and extrapolated total cross section
ALICE:2018gev ; Averbeck:2011ga . In this section, thus, we also normalise our
predictions to the same cross sections that ALICE does, ALICE:2020mfy , by
multiplying by the two overall normalisations factors, $f_{c\bar{c}}$ and
$f_{b\bar{b}}$, for open charm and open bottom production respectively. We
determine those factors to be $f_{c\bar{c}}=2.5$ and $f_{b\bar{b}}=0.75$.
Similarly, we normalise the virtual photon contribution by a factor
$f_{\gamma/Z}=0.06$. This value will be explained in what follows. Such a
normalisation procedure also impacts our predictions for uncertainties. In
fact, one needs to normalise each of our predictions in the seven-point scale
or PDF variations individually. This means that the uncertainty due to
normalisation cancels out and only that due to change of shape remains. We
observed that the different error sets of the PDF do not give a different
shape and thus nearly vanish when normalised such that PDF uncertainties can
be neglected. The $c\bar{c}$ contribution dominates across the whole $m_{ee}$
and $p_{T,ee}$ range, with $b\bar{b}$ coming in second reaching up to 30% in
the bin at 2 GeV in the invariant mass and up to 100% in the high tail of the
transverse momentum spectrum. The virtual photon contributes under 10% across
the whole range. The $c\bar{c}$ production yields an appreciably softer
transverse momentum spectrum, dominating the low $p_{T}$ region, as compared
to the $b\bar{b}$ which starts to contribute significantly at around 2 GeV and
eventually matches the $c\bar{c}$ channel. The shapes of the invariant mass
spectra for the two heavy flavour production modes, instead, are relatively
similar, except below $m_{ee}=1.5$ GeV, where the bottom spectrum flattens out
whereas the charm does not. Both the invariant mass and the transverse
momentum spectra of the virtual photon contribution are relatively flat, as
compared to those of open charm production. Thanks to our normalisation
procedure, the uncertainty bands are now considerably reduced, compared to
Figures 5, 6, 7 and 8. However, the remaining shape uncertainty, dominated by
the scale variation, is still considerable especially in the $c\bar{c}$
channel where it goes from about $\pm$10% at $m_{ee}=m_{c}$ to about
${}^{+150\%}_{-50\%}$ in the high tail. This shape uncertainty is in fact
significantly larger than the statistical and systematic uncertainty as well
as the uncertainty on the branching ratios into electrons for $c\bar{c}$
production, respectively, which were found to be 11%, 5% and 22% in a previous
study by ALICE.

We now want to combine our predictions for virtual photon and open heavy
flavour production into a new prediction for the hadronic cocktail. To this
end we start by isolating the contributions from light flavour (LF) and
$J/\psi$ from ALICE cocktail from Ref. ALICE:2020mfy , simply by subtracting
the $c\bar{c}$ and $b\bar{b}$ ALICE-MC predictions bin-by-bin. We then sum all
of the contributions together, optionally including virtual photons, with a
normalisation for each contribution left as a free parameter, and perform
several fits to all publicly available ALICE data in which we extract the
values of $f_{c\bar{c}}$, $f_{b\bar{b}}$, $f_{\text{LF}+J/\psi}$ and
$f_{\gamma/Z}$. Our fit results are summarised in Table 2. In the first column
we report the results of the original ALICE fit, using the ALICE-MC
predictions. The fit determines $f_{c\bar{c}}=2.5$ and $f_{b\bar{b}}=0.75$ and
describes the data well with $\chi^{2}/\text{ndf}=0.9$. We repeat this fit
twice, first with all the data points in the LMR and in the IMR (i.e. one
invariant mass spectrum in its full range and two transverse momentum spectra
for each pp and pPb collisions) we had access to (npts = 89 as compared to
npts = 123 in the ALICE fit) and only our predictions for the background, and
then the second time only in the IMR (npts = 53, i.e. one invariant mass
spectrum in a reduced range and one transverse momentum spectrum for each pp
and pPb collisions) but now also with our prediction for the virtual photons.
Each time we offer two alternatives, one in which the shape uncertainties are
not considered (“central”, $\chi^{2}=\sum_{i}(x_{i}-m_{i})^{2}/\Delta
m_{i}^{2}$) and one in which they are considered in a fully uncorrelated
manner (“shape”, $\chi^{2}=\sum_{i}(x_{i}-m_{i})^{2}/(\Delta m_{i}^{2}+\Delta
x_{i}^{2})$). The uncertainties on the normalisation coefficients $f$,
reported in round brackets in columns 2 to 5 of Table 2, are the Hesse errors
obtained from the $\chi^{2}$ minimisation by iminuit MINUIT ; IMINUIT . The
uncertainty on $\sigma_{c\bar{c}}$ and $\sigma_{b\bar{b}}$ in the first
column, also reported in round brackets, are instead obtained by combining all
uncertainties from Table 3 of Ref. ALICE:2020mfy  in quadrature.
Correspondingly, the uncertainties on $\sigma_{c\bar{c}}$ and
$\sigma_{b\bar{b}}$ in columns 2 to 5 are obtained by rescaling the
uncertainty in column 1 by the ratio of the newly extracted cross section to
that of column 1. In our first fit in columns 2 and 3, we get a very good data
description with $\chi^{2}/\text{ndf}$ equal to 0.95 and 0.74, respectively.
The normalisation coefficient $f_{c\bar{c}}$ increases which in turn leads to
an increase of extracted values of the $c\bar{c}$ cross sections from 756
$\mu$b $\to$ 788 $\mu$b and to 888 $\mu$b, respectively. The normalisation
coefficient $f_{b\bar{b}}$ is instead slightly decreased. The relative fit
uncertainties on the normalisation coefficients are small ($\sim$ 5%) as
compared to the uncertainty on the total extracted cross section reported by
ALICE ($\sim$ 25%) for the $c\bar{c}$ channel, but not for $b\bar{b}$ ($\sim$
18-25% vs. $\sim$ 14%). In our second fit in columns 4 and 5, the addition of
the virtual photon contribution pulls the fit back in the direction of the
original fit in column 1 reducing the factor $f_{c\bar{c}}$ and increasing
$f_{b\bar{b}}$. The fit prefers a smaller value of the virtual photon
normalisation of 23% and 6% that is however still compatible with 1.0 within
$3\sigma$, in particular in view of the fact that in this kinematic region
with very small values of photon virtuality and transverse momentum, the
dependence of the cross section on the choice of the factorisation and
renormalisation scales is significant. Thus the sensitivity to the virtual
photon contribution is limited and strongly dependent on the uncertainties of
the background contributions. All our fits prefer a reduced fraction of the
light flavour and $J/\psi$ contributions, more notably in fits that include
the virtual photon contribution. Ultimately, however, the fact whether the
shape scale uncertainty is included in the fit or not plays a more important
role. All in all, the light flavour and $J/\psi$ contribution is compatible
with 1.0 within 2 standard deviations and in agreement with previous studies.
Owing to the large branching ratio uncertainty the extracted charm and bottom
cross sections are also compatible within one sigma across all five fits. In
general, the inclusion of shape uncertainties increases these uncertainties
for every fitting parameter, but most significantly, as expected, for
$f_{c\bar{c}}$. The best description of the data, i.e. the smallest
$\chi^{2}/\text{ndf}$, is obtained in the fit which takes both the virtual
photon and the shape uncertainties into account.

In Figure 11, we compare our best prediction for the hadronic cocktail
(corresponding to fit 2) denoted Total and shown in red, to the hadronic
cocktail from ALICE, in blue, and to data in black. Our prediction for the
hadronic cocktail agrees with the previous prediction from ALICE and describes
the measurement well. Our central fit describes the data less well than the
central fit from ALICE, deviating from the data by up to 30%, however only one
point is more than 1$\sigma$ away, not considering the theoretical
uncertainties. However, our hadronic cocktail deviates from the one by ALICE
considerably when it comes to uncertainties. Whereas ALICE’s uncertainties are
well below $\pm$5% in the low tail increasing to about $\pm$10% in the high
tail, our uncertainties reach up to $\pm$50% and almost never drop under
$\pm$10%. This striking difference is due to the perturbative scale shape
uncertainties in the $c\bar{c}$ production, which were not considered in the
ALICE analysis. The inclusion of the virtual photon contribution does not
change the shape of the total result appreciably relative to the data
uncertainties.

Table 2:  Different normalisation parameters $f$, $\chi^{2}/\text{ndf}$ and heavy-quark cross sections for the different combinations of processes contributing to di-electron production obtained from a fit with iminuit MINUIT ; IMINUIT .  | ALICE cocktail | LF+$J/\psi$+$b\bar{b}$+$c\bar{c}$ | LF+$J/\psi$+$b\bar{b}$+$c\bar{c}$+$\gamma/Z$
---|---|---|---
dataset | pp & pPb | pp & pPb | pp & pPb (no LMR)
uncertainty | central | central | shape | central | shape
$f_{c\bar{c}}$ | 2.50 | $2.61(16)\text{\,}$ | $2.93(19)\text{\,}$ | $2.35(24)\text{\,}$ | $2.86(31)\text{\,}$
$f_{b\bar{b}}$ | 0.75 | $0.60(11)\text{\,}$ | $0.48(12)\text{\,}$ | $0.63(13)\text{\,}$ | $0.57(15)\text{\,}$
$f_{\text{LF}+J/\psi}$ | 1.0 | $0.96(3)\text{\,}$ | $0.93(3)\text{\,}$ | $0.91(6)\text{\,}$ | $0.88(8)\text{\,}$
$f_{\gamma/Z}$ | - | - | - | $0.23(17)\text{\,}$ | $0.06(20)\text{\,}$
$\chi^{2}/\text{ndf}$ | $110.9/123$ | $84.2/89$ | $65.9/89$ | $47.7/49$ | $33.3/49$
| $0.90$ | $0.95$ | $0.74$ | $0.97$ | $0.68$
$\differential{\sigma}_{c\bar{c}}/\differential{y}|_{y=0}$ [$\mu\mathrm{b}$] | $756(188)\text{\,}$ | $788(196)\text{\,}$ | $888(221)\text{\,}$ | $709(176)\text{\,}$ | $866(215)\text{\,}$
$\differential{\sigma}_{b\bar{b}}/\differential{y}|_{y=0}$ [$\mu\mathrm{b}$] | $28(5)\text{\,}$ | $22(3)\text{\,}$ | $22(3)\text{\,}$ | $24(3)\text{\,}$ | $21(3)\text{\,}$

Figure 12:  $R_{\text{pPb}}$ in invariant mass (left) and transverse momentum
(right) for the total cocktail. For the Total curve the uncertainty band are
correlated scale uncertainties of the $c\bar{c}$,$b\bar{b}$ and $Zj$ process.
The uncertainty band of the ALICE cocktail is computed solely from the lead
uncertainty given by ALICE in the nominator of the ratio.

Finally, in Figure 12, we show the ratios our pPb and pp compared to those
from ALICE and the data. Both ratios were obtained by simply dividing the pPb
prediction by the pp one after appropriate rebinning. The $R_{pPb}$
predictions agree very well across the whole range of $m_{ee}$ and $p_{T,ee}$
spectra except in the first transverse momentum bin where our prediction is
closer to the data. In the intermediate range of the invariant mass spectrum,
our prediction features appreciably larger uncertainties as a result of the
open charm production scale shape uncertainties not completely canceling in
the ratio.

###  4.4 Initial- and final-state nuclear effects in PbPb collisions

In this section we focus on PbPb collisions at 5.02 TeV. Our predictions for
the invariant mass and transverse momentum spectra per collision of open heavy
flavour production are presented in Figures 13 and 14, respectively. The
predictions are qualitatively very similar to the pp and pPb cases with very
large scale variation uncertainties of up to a factor four in the case of
$c\bar{c}$ and ${}^{+60\%}_{-40\%}$ for $b\bar{b}$. The PDF uncertainties, as
expected, are now instead larger reaching up to $\pm 30\%$ and $\pm 20\%$ for
$c\bar{c}$ and $b\bar{b}$ respectively. Even though a measurement of both
spectra in PbPb collisions is now available in Ref. ALICE:2023jef , we weren’t
able to compare our predictions to data since it is not yet available in
HEPdata.

Figure 13:  Invariant mass spectrum to open charm production $\text{PbPb}\to
c\bar{c}(X),b\bar{b}(X)$ using POWHEG and Pythia with nCTEQ15HQ as PDFs. The
second row is scale uncertainty and the third row is PDF uncertainty.

Figure 14:  Transverse momentum spectrum in the intermediate invariant mass
region to open charm production $\text{PbPb}\to c\bar{c}(X),b\bar{b}(X)$ using
POWHEG and Pythia with nCTEQ15HQ as PDFs. The second row is scale uncertainty
and the third row is PDF uncertainty.

The nCTEQ15HQ Duwentaster:2022kpv  nPDFs improve on the previous nCTEQ15WZSIH
Duwentaster:2021ioo  fit by including the heavy quark (HQ) data, both the open
HQ hadrons and quarkonia. It is thus very interesting to see what impact the
inclusion of heavy quark data in the PDF has on virtual photon and heavy quark
production in the kinematic regime considered here. Figure 15 shows the
$p_{T,ee}$ spectra of the production of virtual photons on the left,
$c\bar{c}$ in the middle and $b\bar{b}$ on the right in simulated
$5.02\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ PbPb collisions. The inclusion
of the HQ data reduces the virtual photon production cross section by about
20% and hardens the spectrum of open charm production gently but leaves the
open bottom spectrum mostly unaffected. Because the uncertainties are large,
however, all the predictions are still consistent within their uncertainties.
Curiously though, we note that the inclusion of the HQ data reduces the
uncertainties in the virtual photon and in the open heavy flavour production
predictions only marginally. This is counterintuitive. We would expect the
open heavy flavour dominated by gluon initiated channels to profit from the
reduction of uncertainties in the gluon distribution of the nCTEQ15HQ fit in
low-$x$ as compared to nCTEQ15WZSIH and the mostly quark initiated virtual
photon production to not change. In order to understand this one has to also
consider the momentum fractions relevant for our kinematic regime: the heavy
quark process receives most of its contribution at $x\sim 10^{-3}$ and the
virtual photon production at $x\sim 10^{-4}$, where sea quarks that are
correlated with the gluon dominate.

Figure 15:  Comparison of PbPb nuclear PDF uncertainties between nCTEQs WZSIH
and HQ PDF sets. Left: Drell-Yan with one jet. Middle: $c\bar{c}$-production.
Right: $b\bar{b}$-production. The shaded band is the PDF uncertainty. The
vertical bars are the statistical uncertainty of the simulation.

There is another important correction to the processes $\text{PbPb}\to
c\bar{c}(X),b\bar{b}(X)$, namely in Ref. ALICE:2019nuy  a suppression of
hadronic decays involving transitions $c\to e$ and $b\to e$ was measured
relative pp collisions, i.e. $R_{AA}^{c/b\to e}(p_{T})$ Song:2018xca . This
correction depends on the centrality and can give a reduction of up to
$50\text{\,}\%$ in collisions of centrality $0\text{\,}\%10\text{\,}\%$ and
$30\text{\,}\%50\text{\,}\%$ and the effects disappear in collisions of
centrality $60\text{\,}\%80\text{\,}\%$. The inclusion of this final state
effect is multiplicative and due to the dependence on $p_{T}$ it doesn’t only
affect the normalisation but also the shape of the distribution. Similar
effects for $R_{pA}$ are possible, but have not yet been measured with
sufficient precision ALICE:2019nuy . Figure 16 shows the effect of including
this nuclear modification factor $R^{b,c\to e}_{AA}$ on $b\bar{b}$ production
in $0\text{\,}\%10\text{\,}\%$ centrality collisions. The correction by a
factor of 2 to 3 is well above the largest source of uncertainty so far, which
is the scale uncertainty, as we have seen in the PbPb case in Figure 14. The
inclusion of such a factor and its uncertainty is therefore necessary to
obtain a reliable prediction for the $b\bar{b}$ production in PbPb collisions,
especially if the normalisation is to be determined by a fit.

Figure 16:  Open bottom production $\text{PbPb}\to b\bar{b}(X)$ with and
without including a $R^{b,c\to e}_{AA}$ factor. The lead nuclear PDF in use is
nCTEQ15HQ.

##  5 Conclusion

In this paper we presented a new calculation of the production of virtual
photons in pp, pPb and PbPb collisions and compared it with ALICE
measurements. We used POWHEG BOX V2 to calculate the Drell-Yan process in
association with at least one jet through to NLO QCD and matched it to
Pythia8. The produced photon converts into an electron-positron pair, which is
then detected by the ALICE detector. This signature is in the invariant mass
range of
$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$
much more likely to be produced by the correlated semileptonic decay of open
heavy charm and bottom hadrons. Therefore, we recalculated predictions for
those too with more recent PDFs and nuclear PDFs, and performed a thorough
study of the theoretical uncertainties. As the spectra’s normalisations are
determined from a fit to the experimental data we introduced shape
uncertainties in which the large scale uncertainties associated with this
kinematic regime are significantly reduced and the PDF uncertainties become
negligible. By repeating the ALICE analysis and fitting the normalisations of
the heavy quark, light flavour and $J/\psi$ contributions, we have found that
the light flavour contributions reported by ALICE are compatible with their
fixed normalisation of 1. Unsurprisingly, the remaining large shape scale
uncertainties result in relaxed constraints on the normalisation of the heavy
quark contributions. The addition of the virtual photon contribution to the
fit results in a small reduction of the normalisation of the other
contributions. The photon normalisation is reduced significantly, but still
compatible with 1 within $3\sigma$ as it constitutes the smallest part of the
total combined prediction. With the new virtual photon contribution and a more
rigorous treatment of theoretical uncertainties our prediction for the
hadronic cocktail represents a more solid basis for ALICE to extract the long
sought QGP contribution from thermal photon radiation. All our predictions are
available in the YODA format as ancillary files. The cold nuclear matter
effects were included throughout via nuclear PDFs and we also showed the
impact of the nuclear modification factor $R^{b,c\to e}_{PbPb}$ in PbPb
collisions, which is likely to be necessary for comparing theoretical
predictions to data. Moreover we investigated the impact of including heavy
quark data in the PDF fit by switching between nCTEQ15HQ and nCTEQ15WZSIH nPDF
sets and found the effect to be marginal.

It would be very interesting to explore the possibility of using the MiNLO
method for the Drell-Yan+jet process to get a better handle on the scale
uncertainties and extend our predictions into the low invariant mass region
below 1 GeV where it could eventually be compared to the Kroll-Wada
approximation. We look forward to upcoming ALICE publications on dileptons in
pp at $13.6$ TeV and PbPb at $5.36$ TeV.

###### Acknowledgements.

T.J. and A.P.N. are grateful to Raphaelle Bailhache and Jerome Jung for
insightful discussions. This work has been supported by the BMBF under
contract 05P21PMCAA and by the DFG through the Research Training Network 2149
“Strong and Weak Interactions - from Hadrons to Dark Matter”.

#### Open Access.

This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY-4.0), which permits any use, distribution and
reproduction in any medium, provided the original authors(s) and source are
credited.

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