EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2023-142 LHCb-PAPER-2023-006 September 5, 2023

Measurement of prompt $D^{+}$ and $D^{+}_{s}$ production in $p\mathrm{Pb}$
collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$$\mathrm{\,Te\kern-2.07413ptV}$

The LHCb collaboration†††Authors are listed at the end of this paper.

The production of prompt $D^{+}$ and $D^{+}_{s}$ mesons is studied in proton-
lead collisions at a centre-of-mass energy of
$\sqrt{s_{\mathrm{NN}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$. The data sample
corresponding to an integrated luminosity of $(1.58\pm 0.02)\mathrm{nb}^{-1}$
is collected by the LHCb experiment at the LHC. The differential production
cross-sections are measured using $D^{+}$ and $D^{+}_{s}$ candidates with
transverse momentum in the range of $0<p_{\mathrm{T}}<14\,\mathrm{GeV}/c$ and
rapidities in the ranges of $1.5<y^{*}<4.0$ and $-5.0<y^{*}<-2.5$ in the
nucleon-nucleon centre-of-mass system. For both particles, the nuclear
modification factor and the forward-backward production ratio are determined.
These results are compared with theoretical models that include initial-state
nuclear effects. In addition, measurements of the cross-section ratios between
$D^{+}$, $D^{+}_{s}$ and $D^{0}$ mesons are presented, providing a baseline
for studying the charm hadronization in lead-lead collisions at LHC energies.

Submitted to JHEP

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

Under extremely high energy densities, a state of nuclear matter having quarks
and gluons as its fundamental constituents can be formed [1]. The production
of such a quark-gluon plasma (QGP) and investigation of its properties are
among the primary goals of high energy heavy-ion collision experiments. Heavy
quarks are sensitive and effective probes for QGP transport properties, as
they are produced in pairs in the early stage of heavy-ion collisions by hard
scatterings and experience the entire evolution of the fireball prior to the
hadronization process. The measured $D$-meson nuclear modification factors
($R_{\mathrm{AA}}$) in nucleus-nucleus (AA) collisions at the RHIC [2, 3] and
the LHC [4, 5, 6] colliders show a strong suppression at high transverse
momentum ($p_{\mathrm{T}}$) and demonstrate a significant charm-quark energy
loss in the medium [7, 8, 9, 10]. This modification factor is calculated as
the ratio of $D$-meson cross-sections from AA to proton-proton ($pp$)
collisions, and is scaled by the binary collision number. The production ratio
of strange ${D}^{+}_{s}$ to non-strange ${D}^{0}$ mesons is found to be
enhanced in ${\sqrt{s_{\mathrm{NN}}}}\,=\,$200$\mathrm{\,Ge\kern-1.00006ptV}$
gold-gold collisions [11] and in
${\sqrt{s_{\mathrm{NN}}}}\,=\,5.02$$\mathrm{\,Te\kern-1.00006ptV}$ lead-lead
collisions [12, 13], compared with the results from $pp$ collisions data [14,
15] and Pythia $pp$ simulation [16, 17]. This is attributed to the enhanced
production of strange quarks in the hot dense medium and charm quark
hadronization via the coalescence mechanism [18, 19, 20, 21]. To fully
understand the experimental measurements for nucleus-nucleus collisions, it is
essential to characterize the cold nuclear matter (CNM) effects due to the
involvement of heavy nuclei like lead in the colliding system.

Multiple CNM effects could modify the heavy-flavor hadron production and
kinematic distributions. One way to investigate these effects is to study
proton-nucleus ($p$A) collisions, where it is assumed that QGP effects are not
dominant. In the initial state, the nuclear environment influences the parton
distribution functions (PDFs) of the bound nucleons, and this modification
depends on the parton momentum fraction (Bjorken-$x$), the momentum transfer
squared ($Q^{2}$), and the nucleus mass number (A) [22, 23]. At LHC energies
and at forward rapidity (corresponding to Bjorken-$x\approx 10^{-6}-10^{-5}$),
the most relevant effect on PDFs is nuclear shadowing [24]. If the gluon phase
space is saturated then the $D$-meson yield, which may be significantly
affected at low $p_{\mathrm{T}}$, can be described by the Colour Glass
Condensate (CGC) effective theory [25, 26, 27, 28].

Measurements of lepton production from semileptonic decays of heavy flavour
hadrons in deuteron-gold collisions at the RHIC [29, 30] and in proton-lead
($p\mathrm{Pb}$) collisions at the LHC [31, 32, 33] suggest that CNM effects
are present. These can be further studied with measurements of decays of charm
hadrons in ${p\mathrm{Pb}}$ collisions at the LHC [34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49]. Recently, the LHCb experiment measured
the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ [34], $\psi{(2S)}$ [35],
${{D}^{0}}$ [36] mesons and ${\mathchar 28931\relax}^{+}_{c}$ [37] baryon
production in ${p\mathrm{Pb}}$ collisions at
${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$ at forward and
backward rapidities. The ALICE collaboration also studied the $D$-meson
production in $p\mathrm{Pb}$ collisions [38, 41, 43, 45] at the same centre-
of-mass energy in the rapidity interval $-0.96<y^{*}<-0.04$, where $y^{\ast}$
is the rapidity of the $D$ mesons in the nucleon-nucleon centre-of-mass frame.
Extensive measurements of charm-quark production at low $p_{\mathrm{T}}$ have
greatly constrained in the PDFs [50, 51], in particular the ${D}^{0}$ results
in LHCb $p\mathrm{Pb}$ collisions [51].

The highly debated yet possible existence of QGP droplets in small systems
like ${p\mathrm{Pb}}$ [52] could modify the hadronization of charm quarks,
i.e. charmed hadrons could be created via the coalescence of charm and light
quarks, similar to the case in nucleus-nucleus collisions. In $pp$ and
$p\mathrm{Pb}$ collisions, the production ratios of multi-strange hadrons with
respect to pions are found to increase significantly with the particle
multiplicity [53], attaining similar values in lead-lead collisions where the
QGP is formed. Studies of the possible enhancement of the ${D}^{+}_{s}$ meson
yields relative to non-strange charmed hadrons like ${D}^{+}$ and ${D}^{0}$ in
high-multiplicity $p\mathrm{Pb}$ events could be beneficial to explore the
mechanism of QGP formation.

This paper reports measurements of production cross-sections, nuclear
modification factors and forward-backward production ratios for prompt
${D}^{+}$ and ${D}^{+}_{s}$ mesons produced directly in proton-lead
interactions or from excited charmed hadron decays. The measurement is
performed at ${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$
with the LHCb detector [54], which is able to cover two different acceptance
regions in the nucleon-nucleon rest frame. In the “forward” (“backward”)
configuration, the region is $1.5<y^{*}<4.0$ ($-5.0<y^{*}<-2.5$), and the
positive direction is defined with respect to the direction of the proton
beam. The measurement of the production cross-section is performed over the
range of ${{D}^{+}}$ and ${{D}^{+}_{s}}$ transverse momentum
$0<\mbox{$p_{\mathrm{T}}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, in both
backward and forward configurations. Finally, the production ratios between
${D}^{+}$, ${D}^{+}_{s}$ and ${D}^{0}$ mesons in $p\mathrm{Pb}$ collisions are
presented as a function of $p_{\mathrm{T}}$ and $y^{*}$, and compared with the
measurements in $pp$ [55, 15] and $p\mathrm{Pb}$ collisions [38] at same
nucleon-nucleon centre-of-mass energy at the LHC.

##  2 Detector and data samples

The LHCb detector [54, 56] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing charm or beauty quarks. The detector includes a high-precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$pp$ interaction region (VELO), a large-area silicon-strip detector (TT)
located upstream of a dipole magnet with a bending power of about
$4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors (IT) and
straw drift tubes (OT) placed downstream of the magnet. The tracking system
provides a measurement of momentum, $p$, of charged particles with a relative
uncertainty that varies from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary interaction vertex (PV), the impact parameter (IP), is measured with
a resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers
[57]. The online event selection is performed by a trigger [58], which
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction.

This analysis uses the $p\mathrm{Pb}$ data sample collected by the LHCb
detector in early 2013, corresponding to integrated luminosities of ($1.06\pm
0.02$)$\mbox{\,nb}^{-1}$ and ($0.52\pm 0.01$)$\mbox{\,nb}^{-1}$ for the
forward and backward collisions, respectively [34]. The instantaneous
luminosity during the data taking period was about 5$\times 10^{27}$ cm-2 s-1,
resulting in event rates three orders of magnitude lower than for typical LHCb
$pp$ interactions. Consequently, the hardware trigger only rejected empty
events, while the software trigger accepted all events with at least one track
reconstructed in the VELO.

Simulated samples of $p\mathrm{Pb}$ collisions at
${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$ are used to
determine detector efficiencies. In the simulation, $D$ mesons in $pp$
collisions are generated using Pythia [16, 59], embedded into minimum bias
$p\mathrm{Pb}$ events from the EPOS generator [60] and calibrated with LHC
data [61]. Hadronic decays are generated using EvtGen [62], in which final-
state radiation is simulated by Photos [63]. The interaction of the generated
particles with the LHCb detector, and its response, are implemented using the
Geant4 toolkit [64, *Agostinelli:2002hh, 66].

##  3 Cross-section determination

The prompt $D$-meson double-differential production cross-section in a given
$(\mbox{$p_{\mathrm{T}}$},y^{*})$ kinematic bin is defined as

$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\mbox{$p_{\mathrm{T}}$}\mathrm{d}y^{*}}=\frac{N(\mbox{$p_{\mathrm{T}}$},y^{*})}{\mathcal{L}{\varepsilon_{\mathrm{tot}}}{\mathcal{B}}\Delta\mbox{$p_{\mathrm{T}}$}\Delta
y^{*}}.$ (1)

In the formula, $N(\mbox{$p_{\mathrm{T}}$},y^{*})$ is the signal yield of
prompt $D$-meson candidates, ${\varepsilon_{\mathrm{tot}}}$ is the total
detection efficiency in a specific bin of $(\mbox{$p_{\mathrm{T}}$},y^{*})$,
$\mathcal{L}$ is the integrated luminosity, ${\mathcal{B}}$ is the branching
fraction of the corresponding $D$-meson decay and
$\Delta\mbox{$p_{\mathrm{T}}$}$ and $\Delta y^{*}$ are the bin widths of the
$p_{\mathrm{T}}$ and rapidity, respectively. The $D$-meson candidates are
reconstructed through the ${{D}^{+}}\rightarrow K^{-}\pi^{+}\pi^{+}$ or
${{D}^{+}_{s}}\rightarrow K^{-}K^{+}\pi^{+}$ decay channels111In this paper
charge-conjugated processes are implied unless stated otherwise., where the
mass of the $K^{+}K^{-}$ pair in the ${D}^{+}_{s}$ meson candidate is required
to fall within $\pm 20$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
mass of the $\phi(1020)$ meson. The corresponding branching fractions are
${\mathcal{B}}=(9.38\pm 0.15)\%$ for the ${{D}^{+}}\rightarrow
K^{-}\pi^{+}\pi^{+}$ decay obtained from Ref. [67], and
${\mathcal{B}}=(2.24\pm 0.13)\%$ for the ${{D}^{+}_{s}}\rightarrow
K^{-}K^{+}\pi^{+}$ decay, obtained from Ref. [68].

The total cross-section over a specific kinematic range is determined by
integrating the double-differential cross-section. The nuclear modification
factor, $R_{p{\rm Pb}}$, i.e. the normalized ratio of the $D$-meson production
cross-section in $p\mathrm{Pb}$ collisions to that in $pp$ interactions at the
same nucleon-nucleon centre-of-mass energy is defined as

$R_{p{\rm Pb}}(\mbox{$p_{\mathrm{T}}$},y^{*})\equiv\frac{1}{A}\frac{{\rm
d}^{2}\sigma_{p{\rm Pb}}(\mbox{$p_{\mathrm{T}}$},y^{*})/{\rm
d}\mbox{$p_{\mathrm{T}}$}{\rm d}y^{*}}{{\rm
d}^{2}\sigma_{pp}(\mbox{$p_{\mathrm{T}}$},y^{*})/{\rm
d}\mbox{$p_{\mathrm{T}}$}{\rm d}y^{*}},$ (2)

where $A$=208 is the mass number of the lead nucleus. The forward-backward
production ratio is defined as

$R_{\rm FB}(\mbox{$p_{\mathrm{T}}$},y^{*})\equiv\frac{{\rm d}^{2}\sigma_{\rm
forward}(\mbox{$p_{\mathrm{T}}$},|y^{*}|;y^{*}>0)/{\rm
d}\mbox{$p_{\mathrm{T}}$}{\rm d}y^{*}}{{\rm d}^{2}\sigma_{\rm
backward}(\mbox{$p_{\mathrm{T}}$},|y^{*}|;y^{*}<0)/{\rm
d}\mbox{$p_{\mathrm{T}}$}{\rm d}y^{*}},$ (3)

in the common rapidity region $2.5<|y^{*}|<4.0$.

The $D$-meson candidates are selected using similar selection criteria to
those for the open charm production measurements in $pp$ collisions at
$\sqrt{s}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$ [55],
7$\mathrm{\,Te\kern-1.00006ptV}$ [69] and 13$\mathrm{\,Te\kern-1.00006ptV}$
[70] at LHCb. The trajectories of kaons and pions from the $D$-meson
candidates are required to be of good quality and originate from a common
vertex. In order to improve the signal purity, more stringent particle
identification (PID) requirements than in $pp$ interactions are exploited.

|
---|---
Figure 1: Distributions of the simultaneous fits to the (left) $\text{M}(K^{-}\pi^{+}\pi^{+})$ and (right) $\log_{10}\chi^{2}_{\text{$\rm IP$}}$ for ${{D}^{+}}$ mesons in the forward data sample in the kinematic bin of $2<\mbox{$p_{\mathrm{T}}$}<3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $3.0<y^{*}<3.5$. |
---|---
Figure 2: Distributions of the simultaneous fits to the (left) $\text{M}(K^{-}\pi^{+}\pi^{+})$ and (right) $\log_{10}\chi^{2}_{\text{$\rm IP$}}$ for ${{D}^{+}}$ mesons in the backward data sample in the kinematic bin of $2<\mbox{$p_{\mathrm{T}}$}<3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $-4.5<y^{*}<-4.0$. |
---|---
Figure 3: Distributions of the simultaneous fits to the (left) $\text{M}(K^{-}K^{+}\pi^{+})$ and (right) $\log_{10}\chi^{2}_{\text{$\rm IP$}}$ for ${{D}^{+}_{s}}$ mesons in the forward data sample in the kinematic bin of $2<\mbox{$p_{\mathrm{T}}$}<3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $3.0<y^{*}<3.5$. |
---|---
Figure 4: Distributions of the simultaneous fits to the (left)
$\text{M}(K^{-}K^{+}\pi^{+})$ and (right) $\log_{10}\chi^{2}_{\text{$\rm
IP$}}$ for ${{D}^{+}_{s}}$ mesons in the backward data sample in the kinematic
bin of $2<\mbox{$p_{\mathrm{T}}$}<3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$-4.5<y^{*}<-4.0$.

The prompt and non-prompt (i.e. from $b$-hadron decays) $D$-meson yields in
intervals of the $p_{\mathrm{T}}$ and $y^{*}$ are determined from simultaneous
extended maximum-likelihood fits to the unbinned invariant-mass and
$\log_{10}\chi^{2}_{\text{$\rm IP$}}$ distributions. Here,
$\chi^{2}_{\text{$\rm IP$}}$ is the difference in vertex-fit $\chi^{2}$ of a
given PV reconstructed with and without consideration of the signal
candidates. The simultaneous fits are performed to the $D$-meson candidates
whose invariant mass lies within $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of their known mass [67]. The
signal shape in the $M(K^{-}\pi^{+}\pi^{+})$ or $M(K^{-}K^{+}\pi^{+})$
distributions is modeled by the sum of a Crystal Ball (CB) function [71] and a
Gaussian with the same mean. The widths of CB function and Gaussian vary
freely in the fit, and the fraction and the tail parameters of the CB function
are fixed to values obtained from simulation. The combinatorial background is
described by a linear function. The $\log_{10}\chi^{2}_{\text{$\rm IP$}}$
shapes for the prompt and non-prompt $D$-meson signal candidates are estimated
using simulated events and modeled with an asymmetric Bukin curve [72] with
tails described by Gaussian functions. In the forward and backward cases for
both prompt and non-prompt components, the width, asymmetry and tail
coefficients of the Bukin function are fixed to the values obtained from
simulated events. Additionally, the Bukin peak location for non-prompt $D$
mesons is fixed to the value obtained from simulated events. The Bukin
parameters are determined from the fit to the whole forward or backward
simulation sample. The combinatorial background $\log_{10}\chi^{2}_{\text{$\rm
IP$}}$ distribution is modeled by a kernel density estimate function [73],
which is created in the side-band interval outside their $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window. The total model is
the sum of the contributions from the prompt and non-prompt signals and the
combinatorial background, where each component is the product of the
corresponding mass and $\log_{10}\chi^{2}_{\text{$\rm IP$}}$ distributions.
The simultaneous fits are carried out independently in each
$(\mbox{$p_{\mathrm{T}}$},y^{*})$ bin of the ${{D}^{+}}$ or ${{D}^{+}_{s}}$
mesons. Figs. 1–4 show the invariant mass and $\log_{10}\chi^{2}_{\text{$\rm
IP$}}$ distributions, for two typical bins of $y^{*}$ in the forward and
backward regions.

The total efficiency, ${\varepsilon_{\mathrm{tot}}}$, in Eq. 1 is the product
of the geometric acceptance, trigger, reconstruction, selection, and PID
efficiencies. The geometric acceptance efficiency is estimated using simulated
$pp$ events to eliminate the effect of the spatial acceptance of the LHCb
detector. The analysis uses a minimum activity trigger whose efficiency for
events containing a ${D}^{+}$ or ${D}^{+}_{s}$ meson is found to be 100%. The
reconstruction and selection efficiencies are calculated with simulated
$p\mathrm{Pb}$ samples at 5.02$\mathrm{\,Te\kern-1.00006ptV}$, and corrected
for known differences in tracking efficiency between data and simulation [74].
The PID efficiency is estimated using a sample of ${D}^{0}$ meson decays
selected from data without PID criteria [56], and collected during the same
period as the $p\mathrm{Pb}$ sample used in the analysis. The PID selection
efficiency is calculated using the kaon and pion single-track efficiencies
from calibration data, and averaging them based on the kinematic distributions
observed in the simulated events in each $(\mbox{$p_{\mathrm{T}}$},y^{*})$
bin. The reconstruction and selection efficiency and the PID efficiency are
sensitive to differences between the track multiplicity distributions in
$p\mathrm{Pb}$ data and those observed in the simulation and PID-calibration
samples. This effect is corrected for by re-weighting the latter to match the
distributions seen in data.

##  4 Systematic uncertainties

The systematic uncertainties affecting the cross-section measurements are
listed in Table 1. The uncertainties of the forward and backward samples were
evaluated separately, unless stated otherwise.

Table 1: Summary of systematic and statistical uncertainties on the $D$-meson cross-section measurements ($\%$). Source | ${{D}^{+}}\rightarrow K^{-}\pi^{+}\pi^{+}$ | ${{D}^{+}_{s}}\rightarrow K^{-}K^{+}\pi^{+}$
---|---|---
Forward | Backward | Forward | Backward
Uncorrelated between bins |  |  |  |
Prompt yield determination | 0.1–1.4 | 0.1–0.9 | 0–3.4 | 0–6.5
Simulation sample size | 0–10 | 0–8 | 1–8 | 1–6
Correlated between bins |  |  |  |
Multiplicity correction | 0.4–2.3 | 1.8–4.8 | 0.4–2.5 | 2.1–5.0
Hadronic interactions | 3.9 | 3.9 | 3.6 | 3.6
PID efficiency | 0–18 | 1–13 | 1–12 | 1–17
Luminosity | 1.9 | 2.1 | 1.9 | 2.1
Branching fraction | 1.6 | 1.6 | 5.8 | 5.8
Statistical uncertainty | 1–20 | 1–28 | 3–48 | 4–52

The systematic uncertainty related to the determination of the prompt signal
yield has contributions both from the assumed fit model and the fixed
parameters in the simultaneous fits, and from the fit method itself. The
uncertainty associated to the fit model is assigned using double CB functions
for the signal component and an exponential function for the background
component to fit the invariant-mass distributions, and using a Gaussian for
non-prompt components to fit the $\log_{10}\chi^{2}_{\text{$\rm IP$}}$
distributions. Parameters in the fit model are allowed to vary from their
nominal values within uncertainties estimated from simulated events. The
standard deviation of the nominal and the alternative fits is taken as the
uncertainty on the prompt signal yield determination. The resulting
uncertainty is estimated to be less than 6.5% for most bins.

The systematic uncertainty related to the multiplicity correction of the
reconstruction and selection efficiencies is due to several contributions. One
source of uncertainty is caused by the choice of the variables used to
represent the detector occupancy for weighting the distribution. The number of
charged tracks, the number of long tracks, the number of hits in the VELO, the
number of hits in the IT and the OT, and the number of hits in the TT, are all
considered separately. The standard deviation of the efficiencies weighted by
each variable is taken as systematic uncertainty. The effects are summed in
quadrature, yielding a total uncertainty on the ${{D}^{+}}$ (${{D}^{+}_{s}}$)
meson multiplicity correction of $0.4-2.3\%$ ($0.4-2.5\%$) and $1.8-4.8\%$
($2.1-5.0\%$) for the forward and backward collision samples, respectively.

The difference in tracking efficiencies between data and simulation is studied
with muons. An additional uncertainty of $1.1\%$ ($1.4\%$) is assigned for
each kaon (pion) present in the final state, due to the incomplete modeling of
the hadronic interactions of these particles with the material of the LHCb
detector [36]. This effect is dominated by the uncertainty on knowledge of the
amount of material present within the detector and hence can be assumed to be
fully correlated for kaons and pions. The total uncertainty associated to this
effect is $3.9\%$ ($3.6\%$) for ${{D}^{+}}$ (${{D}^{+}_{s}}$) mesons.

The finite size of the calibration sample used to determine the particle
identification efficiency also contributes to the systematic uncertainty. This
contribution is evaluated by varying the pion and kaon PID efficiencies
obtained from the calibration sample within their statistical uncertainties.
The uncertainty from the PID binning scheme is studied by using alternative
binning schemes for the single track efficiency. The total PID systematic
uncertainty is a quadratic sum of these two sources and it ranges between 0
and 18% depending on the kinematic region and the collision sample.

The relative uncertainty of the integrated luminosity is nearly $2\%$ for both
forward and backward samples [34]. The relative uncertainty of the branching
fraction is $1.6\%$ [67] and $5.8\%$ [68] for
${\mathcal{B}}({{D}^{+}}\rightarrow K^{-}\pi^{+}\pi^{+})$ and
${\mathcal{B}}({{D}^{+}_{s}}\rightarrow K^{-}K^{+}\pi^{+})$, respectively. The
finite size of the simulation sample introduces uncertainties on the
efficiencies which are then propagated to the cross-section measurements. This
effect is negligible for the central rapidity region, while becomes
significant in the region close to the boundaries of $p_{\mathrm{T}}$ and
$y^{*}$, ranging between 0 and 10%. The total systematic uncertainty varies
gradually from 5% in the central rapidity interval to about 20% in the
boundary interval.

##  5 Results

###  5.1 Production cross-sections

The measured values of the double-differential cross-section of prompt
${{D}^{+}}$ and ${{D}^{+}_{s}}$ mesons in proton-lead collisions for forward
and backward rapidities, as a function of $p_{\mathrm{T}}$ and $y^{*}$, are
displayed in Figs. 5 and 6 and summarized in Tables
A–LABEL:tab:CrossSection2DDsbac of appendix A.

The one-dimensional differential prompt ${{D}^{+}}$ and ${{D}^{+}_{s}}$ meson
cross-sections as a function of $p_{\mathrm{T}}$ or $y^{*}$ are shown in Figs.
7 and 8, and are also summarized in appendix A. The measurements are also
shown as a function of $p_{\mathrm{T}}$ integrated over $y^{*}$ for the common
rapidity range $2.5<|y^{*}|<4.0$.

Figure 5: Double-differential cross-section of prompt ${{D}^{+}}$ mesons in
${p\mathrm{Pb}}$ collisions for the (left) forward and (right) backward
rapidities. The error bars are the statistical uncertainty and the boxes are
the systematic uncertainty, both of which are smaller than the symbol size.
The value in a particular rapidity interval is scaled by a multiplicative
factor $10^{-m}$, where the factor $m$ increase as the rapidity rises.

Figure 6: Double-differential cross-section of prompt ${{D}^{+}_{s}}$ mesons
in ${p\mathrm{Pb}}$ collisions for the (left) forward and (right) backward
rapidities. The error bars are the statistical uncertainty and the boxes are
the systematic uncertainty, both of which are smaller than the symbol size.
The value in a particular rapidity interval is scaled by a multiplicative
factor $10^{-m}$, where the factor $m$ increase as the rapidity rises.

Figure 7: Differential cross-section of prompt ${{D}^{+}}$ meson production in
${p\mathrm{Pb}}$ collisions as a function of (left) $p_{\mathrm{T}}$ and
(right) $y^{*}$ in the forward and backward collision samples. The error bars
are the statistical uncertainty while the boxes are the systematic
uncertainty.

Figure 8: Differential cross-section of prompt ${{D}^{+}_{s}}$ meson
production in ${p\mathrm{Pb}}$ collisions as a function of (left)
$p_{\mathrm{T}}$ and (right) $y^{*}$ in the forward and backward collision
samples. The error bars are the statistical uncertainty while the boxes are
the systematic uncertainty.

The integrated production cross-sections of prompt ${{D}^{+}}$ and
${{D}^{+}_{s}}$ mesons in ${p\mathrm{Pb}}$ forward collisions in the full and
common fiducial regions are

$\sigma({{D}^{+}})_{\mathrm{forward}}(0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},1.5<|y^{*}|<4.0)=79.8\pm
0.7\pm 5.1\mathrm{\,mb},$
$\sigma({{D}^{+}})_{\mathrm{forward}}(0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.5<|y^{*}|<4.0)=43.8\pm
0.2\pm 2.8\mathrm{\,mb},$
$\sigma({{D}^{+}_{s}})_{\mathrm{forward}}(1<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},1.5<|y^{*}|<4.0)=27.5\pm
0.4\pm 2.4\mathrm{\,mb},$
$\sigma({{D}^{+}_{s}})_{\mathrm{forward}}(1<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.5<|y^{*}|<4.0)=14.4\pm
0.3\pm 1.1\mathrm{\,mb}.$

The first uncertainties are statistical while the second are systematic. The
integrated production cross-sections of prompt ${{D}^{+}}$ and ${{D}^{+}_{s}}$
mesons in $p\mathrm{Pb}$ backward collisions in the full and common fiducial
regions are

$\sigma({{D}^{+}})_{\mathrm{backward}}(0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.5<|y^{*}|<5.0)=77.6\pm
0.9\pm 4.9\mathrm{\,mb},$
$\sigma({{D}^{+}})_{\mathrm{backward}}(0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.5<|y^{*}|<4.0)=53.1\pm
0.7\pm 3.5\mathrm{\,mb},$
$\sigma({{D}^{+}_{s}})_{\mathrm{backward}}(1<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.5<|y^{*}|<5.0)=28.7\pm
0.8\pm 3.0\mathrm{\,mb},$
$\sigma({{D}^{+}_{s}})_{\mathrm{backward}}(1<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2.5<|y^{*}|<4.0)=21.9\pm
0.6\pm 2.3\mathrm{\,mb}.$

###  5.2 Nuclear modification factors

The values of the $D$-meson production cross-section in $pp$ collisions at
5.02$\mathrm{\,Te\kern-1.00006ptV}$, which are necessary for the measurement
of the nuclear modification factor $R_{{p\mathrm{Pb}}}$, are taken from Ref.
[55]. Systematic uncertainties associated with the hadronic interaction length
and branching fractions of Ref. [55] are fully correlated with the
measurements presented here, while other uncertainties are assumed to be
uncorrelated. The nuclear modification factors for prompt ${{D}^{+}}$ and
${{D}^{+}_{s}}$ production, displayed in Fig. 9 in bins of $p_{\mathrm{T}}$
and Fig. 10 in bins of $y^{*}$, show a slight increase as a function of
$p_{\mathrm{T}}$, implying that the suppression may decrease with increasing
$p_{\mathrm{T}}$. The values of $R_{{p\mathrm{Pb}}}$ for ${{D}^{+}}$ and
${{D}^{+}_{s}}$ mesons are given in appendix B.

Figure 9: Nuclear modification factors $R_{{p\mathrm{Pb}}}$ as a function of
$p_{\mathrm{T}}$ for prompt ${{D}^{+}}$ and ${{D}^{+}_{s}}$ meson production
in the (left) forward data and (right) backward data. The error bars are the
statistical uncertainty and the boxes are the systematic uncertainty. The CGC
[75, 76, 77] predictions are only available in the forward region. Previous
results on ${D}^{0}$ mesons [36] from LHCb are also shown.

Figure 10: Nuclear modification factors $R_{{p\mathrm{Pb}}}$ as a function of
$y^{*}$ for prompt ${{D}^{+}}$ and ${{D}^{+}_{s}}$ meson production,
integrated up to
$\mbox{$p_{\mathrm{T}}$}=10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The error
bars are the statistical uncertainty and the boxes are the systematic
uncertainty. The CGC [75, 76, 77] predictions are only available in the
forward region. Previous results on ${D}^{0}$ mesons [36] from LHCb are also
shown.

The measurements are compared with previous ${{D}^{0}}$ results by the LHCb
collaboration at
${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$ [36], as well
as with the HELAC-Onia generator [78, 79, 80] using EPPS16 and nCTEQ15 nPDFs
[81, 82], where the nPDFs are re-weighted with the ${{D}^{0}}$ cross-section
[36]. At forward rapidity, the ${{D}^{+}}$ and ${{D}^{+}_{s}}$ results agree
very well with the $R_{p\mathrm{Pb}}$ of ${{D}^{0}}$ mesons and the nPDF
calculations within uncertainties. At backward rapidity, the ${{D}^{+}_{s}}$
results are consistent with the ${{D}^{0}}$ data and the nPDF calculations
within uncertainties. However, the ${{D}^{+}}$ $R_{p\mathrm{Pb}}$ seems to be
systematically lower than ${{D}^{0}}$ and ${{D}^{+}_{s}}$ mesons across the
whole $p_{\mathrm{T}}$ range. The ${D}^{+}$ $R_{p\mathrm{Pb}}$ results at
backward rapidity are lower than nPDF predictions with a significance of about
$1-3$ standard deviations, suggesting possible changes in charm hadronization
where the event multiplicity is large. As some components of ${{D}^{+}}$ and
${{D}^{0}}$ mesons come from the decay of their excited charm resonances
$D^{*+}$ mesons [67, 83], it may be possible to further understand this
phenomenon by investigating the production of $D^{*+}$ mesons in
$p\mathrm{Pb}$ collisions.

In Figs. 9 and 10 the measurements are also compared with predictions in the
CGC framework, which include the effect of the saturation of partons at small
Bjorken-$x$. For CGC1 [75, 76], the cross-section of the $D$ mesons is
obtained with the optical Glauber mechanism correlating the initial state of
the nucleon with that of the proton, and for CGC2 [77] it is derived by
convolving the charm-quark fragmentation function in a transverse momentum-
dependent factorization framework. The CGC models are found to be able to
describe the trend of prompt $D$-meson nuclear modifications as a function of
$p_{\mathrm{T}}$ and of rapidity in the forward regions.

###  5.3 Forward-backward ratio

In the forward-backward production ratio $R_{\mathrm{FB}}$, the uncertainties
from common sources between the forward and backward measurements cancel out.
The uncertainties due to branching fraction and to hadronic interactions with
the detector are considered fully correlated, while other uncertainties are
uncorrelated. The measured $R_{\mathrm{FB}}$ values for ${D}^{+}$ and
${D}^{+}_{s}$ mesons are shown in Fig. 11, as a function of $p_{\mathrm{T}}$
integrated over the range $2.5<|y^{*}|<4.0$, and as a function of $y^{*}$
integrated up to
$\mbox{$p_{\mathrm{T}}$}=14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
$R_{\mathrm{FB}}$ values in different kinematic bins are tabulated in appendix
C. The $R_{\mathrm{FB}}$ values for ${D}^{+}$ and ${D}^{+}_{s}$ mesons are
also compared to the measurements of ${{D}^{0}}$ mesons [36] and ${\mathchar
28931\relax}^{+}_{c}$ baryons [37] by the LHCb collaboration at
${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$ in Fig. 11. The
$R_{\mathrm{FB}}$ of all open charm hadrons deviates further from unity as
$|y^{*}|$ increases. This behavior is consistent with the expectations from
the QCD calculations, suggesting that the asymmetry becomes more pronounced in
the large rapidity region. The $R_{\mathrm{FB}}$ of ${{D}^{+}_{s}}$ mesons as
a function of $y^{*}$ shows reasonable agreement with ${{D}^{0}}$ and
${\mathchar 28931\relax}^{+}_{c}$ results and with the EPPS16 and nCTEQ15
nPDFs [81, 82] within uncertainties. The $R_{\mathrm{FB}}$ of ${{D}^{+}}$
mesons is slightly larger than other charm hadrons and model predictions, due
to the overall suppression of ${{D}^{+}}$ production at backward rapidities.
At low $p_{\mathrm{T}}$, the $R_{\mathrm{FB}}$ of ${{D}^{+}}$ and
${{D}^{+}_{s}}$ mesons are consistent with the $R_{\mathrm{FB}}$ of
${{D}^{0}}$ mesons and ${\mathchar 28931\relax}^{+}_{c}$ baryons, and agree
with nPDFs calculations. However, the more precise ${{D}^{+}}$ data show a
clear increasing trend with increasing $p_{\mathrm{T}}$ and saturates at unity
at high $p_{\mathrm{T}}$ ($>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) within
uncertainties, which deviates from the almost $p_{\mathrm{T}}$ independent
nPDF calculations. This discrepancy derives from the suppression of $D^{+}$
production as a function of $p_{\mathrm{T}}$ in the backward configuration,
which is also pronounced in Fig. 9.

###  5.4 Production ratios

The measured prompt ${D}^{+}_{s}$ and ${D}^{+}$ production cross-sections
enable a calculation of ${{D}^{+}_{s}}$ to ${{D}^{+}}$ ratios. The LHCb
measurement of ${{D}^{0}}$ cross-sections in ${p\mathrm{Pb}}$ collisions at
${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$ [36] also
enables a calculation of the ratio of the production cross-sections of
${{D}^{+}}$ to ${{D}^{0}}$ and ${{D}^{+}_{s}}$ to ${{D}^{0}}$ mesons. The
uncertainty due to hadronic interactions with the detector is considered
partially correlated because of different numbers of kaon and pion tracks
between the numerators and denominators of $R_{{{D}^{+}}/{{D}^{0}}}$,
$R_{{{D}^{+}_{s}}/{{D}^{0}}}$ or $R_{{{D}^{+}_{s}}/{{D}^{+}}}$. The
uncertainties from the luminosity are fully correlated, while the remaining
uncertainties are uncorrelated. Figure 12 illustrates the
$R_{{{D}^{+}}/{{D}^{0}}}$, $R_{{{D}^{+}_{s}}/{{D}^{0}}}$ and
$R_{{{D}^{+}_{s}}/{{D}^{+}}}$ ratios as a function of $p_{\mathrm{T}}$
integrated over total rapidity range $1.5<y^{*}<4.0$ for forward and
$-5.0<y^{*}<-2.5$ for backward. Figure 13 illustrates the
$R_{{{D}^{+}}/{{D}^{0}}}$, $R_{{{D}^{+}_{s}}/{{D}^{0}}}$ and
$R_{{{D}^{+}_{s}}/{{D}^{+}}}$ ratios as a function of $y^{*}$ integrated over
the $p_{\mathrm{T}}$ range up to $10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
measured production ratios are tabulated in appendix D.

The results of relative cross-section ratios between $D$ mesons in LHCb
${p\mathrm{Pb}}$ collisions show a mild $p_{\mathrm{T}}$ and $y^{*}$
dependence, which are consistent with the results from the LHCb collaboration
in $pp$ [55], and from the ALICE collaboration in ${p\mathrm{Pb}}$ [38] and in
$pp$ [15] collisions. The $R_{{{D}^{+}_{s}}/{{D}^{0}}}$ and
$R_{{{D}^{+}_{s}}/{{D}^{+}}}$ ratios are found not to be enhanced in neither
forward nor backward rapidities in $p\mathrm{Pb}$ collisions at
${\sqrt{s_{\mathrm{NN}}}}=5.02$$\mathrm{\,Te\kern-1.00006ptV}$.

Figure 11: Forward-backward ratios $R_{\mathrm{FB}}$ for prompt ${{D}^{+}}$
and ${{D}^{+}_{s}}$ meson production (left) as a function of $p_{\mathrm{T}}$;
(right) as a function of $y^{*}$. The error bars are the statistical
uncertainty while the boxes are the systematic uncertainty. Previous results
on ${D}^{0}$ [36] mesons and ${\mathchar 28931\relax}^{+}_{c}$ [37] baryons
from LHCb are also shown.

Figure 12: Production ratios as a function of $p_{\mathrm{T}}$ in LHCb
${p\mathrm{Pb}}$ collisions. The error bars give the statistical uncertainty
while the boxes give the systematic uncertainty. The uncertainties related to
the branching fractions are not shown in the figure. The measurements are also
compared with other results of $pp$ [55, 15] and $p\mathrm{Pb}$ [38]
collisions at the same centre-of-mass energy from LHCb and ALICE.

Figure 13: Production ratios as a function of $y^{*}$ in LHCb ${p\mathrm{Pb}}$
collisions. The error bars give the statistical uncertainty while the boxes
give the systematic uncertainty. The uncertainties related to the branching
fractions are not shown in the figure. The measurements are also compared with
the results of $pp$ collisions at the same centre-of-mass energy from LHCb
[55].

##  6 Conclusion

The prompt ${{D}^{+}}$ and ${{D}^{+}_{s}}$ production cross-sections have been
measured with LHCb proton-lead collision data corresponding to an integrated
luminosity of $(1.58\pm 0.02)\mbox{\,nb}^{-1}$ collected at
${\sqrt{s_{\mathrm{NN}}}}=5.02\mathrm{\,Te\kern-1.00006ptV}$. The measurement
is performed in the range of the $D$-meson transverse momentum
$0<\mbox{$p_{\mathrm{T}}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, in both
backward and forward collisions covering the rapidity ranges $1.5<y^{*}<4.0$
and $-5.0<y^{*}<-2.5$. This is the first measurement in this rapidity region
down to zero transverse momentum of the ${{D}^{+}}$ and ${{D}^{+}_{s}}$
mesons. Nuclear modification factors and forward-backward production ratios
are also measured in the same kinematic region. A large asymmetry in the
forward-backward production is observed, which is consistent with the
expectations from nuclear parton distribution functions, and colour glass
condensate calculations. The $R_{{{D}^{+}_{s}}/{{D}^{0}}}$ and
$R_{{{D}^{+}_{s}}/{{D}^{+}}}$ ratios show no enhancement of the yield of
strange hadrons at high transverse momentum or rapidity, and may therefore be
used as a reference for experimental measurements related to investigations of
the QGP in nucleus-nucleus collisions.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MCID/IFA (Romania); MICINN
(Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom);
DOE NP and NSF (USA). We acknowledge the computing resources that are provided
by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF
(Netherlands), PIC (Spain), GridPP (United Kingdom), CSCS (Switzerland), IFIN-
HH (Romania), CBPF (Brazil), Polish WLCG (Poland) and NERSC (USA). We are
indebted to the communities behind the multiple open-source software packages
on which we depend. Individual groups or members have received support from
ARC and ARDC (Australia); Minciencias (Colombia); AvH Foundation (Germany);
EPLANET, Marie Skłodowska-Curie Actions, ERC and NextGenerationEU (European
Union); A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes
(France); Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS
CCEPP, Fundamental Research Funds for the Central Universities, and Sci. &
Tech. Program of Guangzhou (China); GVA, XuntaGal, GENCAT, Inditex, InTalent
and Prog. Atracción Talento, CM (Spain); SRC (Sweden); the Leverhulme Trust,
the Royal Society and UKRI (United Kingdom).

## Appendices

##  Appendix A Numerical values of the ${{D}^{+}}$ and ${{D}^{+}_{s}}$ mesons
cross-sections

Tables A–LABEL:tab:CrossSection2DDsbac give the numerical results for the
double-differential cross-sections. Tables LABEL:tab:CrossSection1DPTtot–A
give the numerical results for the one-dimensional differential cross-
sections.

Table 2: Double-differential cross-section (mb) for prompt ${{D}^{+}}$ mesons
as functions of $p_{\mathrm{T}}$ and $y^{*}$ in the forward regions. The first
uncertainty is statistical, the second is the component of the systematic
uncertainty that is uncorrelated between bins and the third is the fully
correlated component.

Forward |  |  |  |  |
---|---|---|---|---|---
$\mbox{$p_{\mathrm{T}}$}\,[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $1.5<y^{*}<2.0$ | $2.0<y^{*}<2.5$ | $2.5<y^{*}<3.0$ | $3.0<y^{*}<3.5$ | $3.5<y^{*}<4.0$
$[0,1]$ | $6.835\pm 1.131\pm 0.590\pm 0.528$ | $7.069\pm 0.207\pm 0.104\pm 0.766$ | $7.953\pm 0.164\pm 0.085\pm 0.659$ | $7.527\pm 0.185\pm 0.091\pm 0.393$ | $7.132\pm 0.270\pm 0.134\pm 0.468$
$[1,2]$ | $11.748\pm 0.481\pm 0.282\pm 0.848$ | $12.768\pm 0.157\pm 0.092\pm 1.079$ | $12.198\pm 0.122\pm 0.069\pm 0.806$ | $10.733\pm 0.130\pm 0.071\pm 0.609$ | $8.238\pm 0.187\pm 0.091\pm 0.487$
$[2,3]$ | $8.377\pm 0.165\pm 0.102\pm 0.527$ | $8.348\pm 0.067\pm 0.041\pm 0.671$ | $7.552\pm 0.054\pm 0.032\pm 0.704$ | $6.096\pm 0.054\pm 0.034\pm 0.523$ | $4.476\pm 0.075\pm 0.039\pm 0.307$
$[3,4]$ | $4.383\pm 0.069\pm 0.044\pm 0.274$ | $4.067\pm 0.032\pm 0.018\pm 0.500$ | $3.678\pm 0.027\pm 0.016\pm 0.539$ | $2.873\pm 0.027\pm 0.015\pm 0.379$ | $2.040\pm 0.037\pm 0.019\pm 0.178$
$[4,5]$ | $2.103\pm 0.034\pm 0.020\pm 0.156$ | $2.015\pm 0.018\pm 0.010\pm 0.338$ | $1.704\pm 0.016\pm 0.009\pm 0.321$ | $1.238\pm 0.015\pm 0.008\pm 0.216$ | $0.805\pm 0.021\pm 0.009\pm 0.081$
$[5,6]$ | $1.108\pm 0.020\pm 0.012\pm 0.099$ | $0.999\pm 0.012\pm 0.006\pm 0.204$ | $0.808\pm 0.010\pm 0.005\pm 0.183$ | $0.601\pm 0.010\pm 0.005\pm 0.124$ | $0.348\pm 0.015\pm 0.006\pm 0.033$
$[6,7]$ | $0.550\pm 0.012\pm 0.007\pm 0.059$ | $0.494\pm 0.008\pm 0.004\pm 0.115$ | $0.412\pm 0.007\pm 0.003\pm 0.103$ | $0.294\pm 0.007\pm 0.003\pm 0.064$ | $0.170\pm 0.013\pm 0.005\pm 0.017$
$[7,8]$ | $0.318\pm 0.009\pm 0.005\pm 0.038$ | $0.267\pm 0.005\pm 0.003\pm 0.067$ | $0.231\pm 0.005\pm 0.002\pm 0.061$ | $0.144\pm 0.005\pm 0.002\pm 0.030$ | $0.073\pm 0.015\pm 0.005\pm 0.009$
$[8,9]$ | $0.188\pm 0.006\pm 0.004\pm 0.025$ | $0.154\pm 0.004\pm 0.002\pm 0.041$ | $0.133\pm 0.004\pm 0.002\pm 0.037$ | $0.084\pm 0.004\pm 0.002\pm 0.016$ | $-$
$[9,10]$ | $0.111\pm 0.005\pm 0.002\pm 0.017$ | $0.094\pm 0.003\pm 0.001\pm 0.026$ | $0.072\pm 0.003\pm 0.001\pm 0.021$ | $0.052\pm 0.004\pm 0.002\pm 0.008$ | $-$
$[10,11]$ | $0.066\pm 0.003\pm 0.002\pm 0.011$ | $0.056\pm 0.002\pm 0.001\pm 0.016$ | $0.043\pm 0.002\pm 0.001\pm 0.012$ | $0.028\pm 0.004\pm 0.002\pm 0.004$ | $-$
$[11,12]$ | $0.043\pm 0.003\pm 0.001\pm 0.007$ | $0.040\pm 0.002\pm 0.001\pm 0.011$ | $0.027\pm 0.002\pm 0.001\pm 0.007$ | $0.026\pm 0.004\pm 0.003\pm 0.005$ | $-$
$[12,13]$ | $0.022\pm 0.002\pm 0.001\pm 0.004$ | $0.022\pm 0.002\pm 0.001\pm 0.007$ | $0.019\pm 0.002\pm 0.001\pm 0.005$ | $-$ | $-$
$[13,14]$ | $0.023\pm 0.002\pm 0.001\pm 0.004$ | $0.014\pm 0.001\pm 0.001\pm 0.004$ | $0.012\pm 0.001\pm 0.001\pm 0.003$ | $-$ | $-$

Table 3: Double-differential cross-section (mb) for prompt ${{D}^{+}}$ mesons
as functions of $p_{\mathrm{T}}$ and $y^{*}$ in the backward regions. The
first uncertainty is statistical, the second is the component of the
systematic uncertainty that is uncorrelated between bins and the third is the
fully correlated component.

Backward |  |  |  |  |
---|---|---|---|---|---
$\mbox{$p_{\mathrm{T}}$}\,[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $-3.0<y^{*}<-2.5$ | $-3.5<y^{*}<-3.0$ | $-4.0<y^{*}<-3.5$ | $-4.5<y^{*}<-4.0$ | $-5.0<y^{*}<-4.5$
$[0,1]$ | $6.430\pm 1.127\pm 0.371\pm 0.647$ | $7.921\pm 0.340\pm 0.115\pm 0.835$ | $9.220\pm 0.321\pm 0.106\pm 0.607$ | $8.069\pm 0.438\pm 0.115\pm 0.571$ | $8.054\pm 0.750\pm 0.195\pm 0.715$
$[1,2]$ | $13.372\pm 0.641\pm 0.321\pm 0.966$ | $13.751\pm 0.252\pm 0.099\pm 1.162$ | $12.931\pm 0.221\pm 0.073\pm 0.855$ | $11.049\pm 0.283\pm 0.073\pm 0.627$ | $7.973\pm 0.497\pm 0.088\pm 0.471$
$[2,3]$ | $8.325\pm 0.210\pm 0.101\pm 0.524$ | $8.064\pm 0.102\pm 0.040\pm 0.649$ | $7.087\pm 0.087\pm 0.030\pm 0.661$ | $5.434\pm 0.092\pm 0.030\pm 0.467$ | $3.603\pm 0.138\pm 0.032\pm 0.247$
$[3,4]$ | $4.058\pm 0.086\pm 0.040\pm 0.254$ | $3.616\pm 0.046\pm 0.016\pm 0.445$ | $2.965\pm 0.039\pm 0.013\pm 0.435$ | $2.017\pm 0.038\pm 0.011\pm 0.266$ | $1.081\pm 0.052\pm 0.010\pm 0.094$
$[4,5]$ | $1.824\pm 0.042\pm 0.018\pm 0.135$ | $1.533\pm 0.024\pm 0.008\pm 0.257$ | $1.194\pm 0.021\pm 0.006\pm 0.225$ | $0.775\pm 0.020\pm 0.005\pm 0.135$ | $0.338\pm 0.026\pm 0.004\pm 0.034$
$[5,6]$ | $0.857\pm 0.024\pm 0.009\pm 0.077$ | $0.702\pm 0.015\pm 0.004\pm 0.144$ | $0.492\pm 0.012\pm 0.003\pm 0.111$ | $0.294\pm 0.011\pm 0.002\pm 0.060$ | $0.094\pm 0.017\pm 0.002\pm 0.009$
$[6,7]$ | $0.433\pm 0.015\pm 0.006\pm 0.046$ | $0.340\pm 0.009\pm 0.003\pm 0.079$ | $0.227\pm 0.008\pm 0.002\pm 0.057$ | $0.128\pm 0.008\pm 0.001\pm 0.028$ | $-$
$[7,8]$ | $0.223\pm 0.009\pm 0.003\pm 0.026$ | $0.159\pm 0.006\pm 0.002\pm 0.040$ | $0.106\pm 0.005\pm 0.001\pm 0.028$ | $0.053\pm 0.005\pm 0.001\pm 0.011$ | $-$
$[8,9]$ | $0.120\pm 0.007\pm 0.002\pm 0.016$ | $0.087\pm 0.004\pm 0.001\pm 0.023$ | $0.055\pm 0.004\pm 0.001\pm 0.015$ | $0.024\pm 0.004\pm 0.001\pm 0.005$ | $-$
$[9,10]$ | $0.065\pm 0.005\pm 0.001\pm 0.010$ | $0.051\pm 0.003\pm 0.001\pm 0.014$ | $0.030\pm 0.003\pm 0.001\pm 0.009$ | $0.014\pm 0.004\pm 0.001\pm 0.002$ | $-$
$[10,11]$ | $0.042\pm 0.004\pm 0.001\pm 0.007$ | $0.030\pm 0.003\pm 0.001\pm 0.009$ | $0.014\pm 0.002\pm 0.000\pm 0.004$ | $-$ | $-$
$[11,12]$ | $0.029\pm 0.003\pm 0.001\pm 0.005$ | $0.014\pm 0.002\pm 0.000\pm 0.004$ | $0.008\pm 0.001\pm 0.000\pm 0.002$ | $-$ | $-$
$[12,13]$ | $0.021\pm 0.002\pm 0.001\pm 0.004$ | $0.008\pm 0.001\pm 0.000\pm 0.002$ | $-$ | $-$ | $-$
$[13,14]$ | $0.013\pm 0.002\pm 0.001\pm 0.002$ | $0.006\pm 0.001\pm 0.000\pm 0.002$ | $-$ | $-$ | $-$

Table 11: Differential cross-section for prompt ${{D}^{+}_{s}}$ mesons as a function of $|y^{*}|$ integrated over $1<\mbox{$p_{\mathrm{T}}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the forward and backward regions, respectively. The first uncertainty is statistical, the second is systematic.  Forward |
---|---
$y^{*}$ |  $\frac{\mathrm{d}\sigma}{\mathrm{d}y^{*}}$ [$\mathrm{\,mb}$ ]
$[1.5,2.0]$ | $13.37\pm 0.58\pm 1.42$
$[2.0,2.5]$ | $12.93\pm 0.26\pm 1.26$
$[2.5,3.0]$ | $13.04\pm 0.25\pm 1.10$
$[3.0,3.5]$ | $9.38\pm 0.26\pm 0.73$
$[3.5,4.0]$ | $6.52\pm 0.45\pm 0.51$
Backward |
$y^{*}$ |  $\frac{\mathrm{d}\sigma}{\mathrm{d}y^{*}}$ [$\mathrm{\,mb}$ ]
$[-3.0,-2.5]$ | $15.71\pm 0.89\pm 2.07$
$[-3.5,-3.0]$ | $14.25\pm 0.45\pm 1.12$
$[-4.0,-3.5]$ | $13.88\pm 0.47\pm 1.20$
$[-4.5,-4.0]$ | $9.88\pm 0.58\pm 1.14$
$[-5.0,-4.5]$ | $3.76\pm 0.92\pm 0.94$

##  Appendix B Nuclear modification factor $R_{{p\mathrm{Pb}}}$

Tables B–B give the numerical results for the nuclear modification factor as a
function of $p_{\mathrm{T}}$. Tables B–B give the numerical results for the
nuclear modification factor as a function of $y^{*}$.

Table 12: Nuclear modification factor $R_{{p\mathrm{Pb}}}$ for prompt
${{D}^{+}}$ meson production in different $p_{\mathrm{T}}$ intervals, for the
forward and backward rapidity regions. The first uncertainty is statistical,
the second is systematic.  Table 13: Nuclear modification factor
$R_{{p\mathrm{Pb}}}$ for prompt ${{D}^{+}_{s}}$ meson production in different
$p_{\mathrm{T}}$ intervals, for the forward and backward rapidity regions. The
first uncertainty is statistical, the second is systematic.  Table 14: Nuclear
modification factor $R_{{p\mathrm{Pb}}}$ for prompt ${{D}^{+}}$ meson
production in different $y^{*}$ intervals, integrated up to
$\mbox{$p_{\mathrm{T}}$}=10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic.  Table 15: Nuclear
modification factor $R_{{p\mathrm{Pb}}}$ for prompt ${{D}^{+}_{s}}$ meson
production in different $y^{*}$ intervals, integrated up to
$\mbox{$p_{\mathrm{T}}$}=10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic.

##  Appendix C Forward-backward production ratios of ${{D}^{+}}$ and
${{D}^{+}_{s}}$

Tables C–C give the numerical results for the forward-backward production
ratios.

Table 16: Forward-backward production ratios of ${{D}^{+}}$ mesons as a
function of $p_{\mathrm{T}}$, integrated over the common rapidity range
$2.5<|y^{*}|<4.0$; and as a function of $y^{*}$, integrated over
$0<\mbox{$p_{\mathrm{T}}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic. Table 17: Forward-
backward production ratios of ${{D}^{+}_{s}}$ mesons as a function of
$p_{\mathrm{T}}$, integrated over the common rapidity range $2.5<|y^{*}|<4.0$;
and as a function of $y^{*}$, integrated over
$0<\mbox{$p_{\mathrm{T}}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic.

##  Appendix D Production ratios between ${{D}^{+}}$, ${{D}^{+}_{s}}$ and
${{D}^{0}}$

Tables D–D give the numerical results for the cross-section ratios between
mesons as a function of $p_{\mathrm{T}}$. Tables D–D give the numerical
results for the cross-section ratios between mesons as a function of $y^{*}$.

Table 18: Measured of $R_{{{D}^{+}}/{{D}^{0}}}$ as a function of
$p_{\mathrm{T}}$ in LHCb ${p\mathrm{Pb}}$ collisions at
5.02$\mathrm{\,Te\kern-1.00006ptV}$ in the forward and backward regions,
integrated over $2.5<|y^{*}|<4.0$. The first uncertainty is statistical, the
second is systematic. Table 19: Measured of $R_{{{D}^{+}_{s}}/{{D}^{0}}}$
ratio as a function of $p_{\mathrm{T}}$ in LHCb ${p\mathrm{Pb}}$ collisions at
5.02$\mathrm{\,Te\kern-1.00006ptV}$ in the forward and backward regions,
integrated over $2.5<|y^{*}|<4.0$. The first uncertainty is statistical, the
second is systematic. Table 20: Measured of $R_{{{D}^{+}_{s}}/{{D}^{+}}}$
ratio as a function of $p_{\mathrm{T}}$ in LHCb ${p\mathrm{Pb}}$ collisions at
5.02$\mathrm{\,Te\kern-1.00006ptV}$ in the forward and backward regions,
integrated over $2.5<|y^{*}|<4.0$. The first uncertainty is statistical, the
second is systematic. Table 21: Measured of $R_{{{D}^{+}}/{{D}^{0}}}$ as a
function of $|y^{*}|$ in LHCb ${p\mathrm{Pb}}$ collisions at
5.02$\mathrm{\,Te\kern-1.00006ptV}$ in the forward and backward regions,
integrated over
$0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic. Table 22: Measured of
$R_{{{D}^{+}_{s}}/{{D}^{0}}}$ as a function of $|y^{*}|$ in LHCb
${p\mathrm{Pb}}$ collisions at 5.02$\mathrm{\,Te\kern-1.00006ptV}$ in the
forward and backward regions, integrated over
$0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic. Table 23: Measured
$R_{{{D}^{+}_{s}}/{{D}^{+}}}$ as a function of $|y^{*}|$ in LHCb
${p\mathrm{Pb}}$ collisions at 5.02$\mathrm{\,Te\kern-1.00006ptV}$ in the
forward and backward regions, integrated over
$0<\mbox{$p_{\mathrm{T}}$}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The first
uncertainty is statistical, the second is systematic.

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Zunica