# Precision identified-hadron spectrum analysis for 5 TeV $\bf p$-$\bf Pb$
collisions – Part II

Thomas A. Trainor  University of Washington, Seattle, WA 98195

###### Abstract

This is the second part of a two-part article on precision modeling of
identified-hadron (PID) $p_{t}$ spectra from 5 TeV $p$-Pb collisions. In Part
I a revised two-component (soft + hard) model (TCM) of PID spectra was
introduced, an apparent detection inefficiency for protons was corrected and
jet-related spectrum hard components isolated via an improved method were
compared to a fixed TCM serving as reference. In the present article (Part II)
the TCM is further elaborated to describe centrality variation of spectrum
hard components (and therefore entire spectra) within data statistical
uncertainties as determined by a standard statistical measure (Z-scores). The
completed PID TCM is then used to investigate properties of spectrum and yield
ratios (e.g. $p/\pi$ ratios) and ensemble-mean $\bar{p}_{t}$ data. Systematic
differences are observed between meson and baryon hard components. With
increasing $p$-Pb centrality meson hard components shift to lower $p_{t}$
while baryon hard components shift to higher $p_{t}$. Those trends explain
quantitatively the main features and centrality variation of spectrum ratios.
Ensemble-mean $\bar{p}_{t}$ trends are also predicted at the level of data
statistical uncertainties, and details of the $\bar{p}_{t}$ centrality trends
are explained quantitatively.

###### pacs:

12.38.Qk, 13.87.Fh, 25.75.Ag, 25.75.Bh, 25.75.Ld, 25.75.Nq

††preprint: version 1.2

##  I Introduction

This article presents Part II of a two-part study of identified-hadron (PID)
$p_{t}$ spectra from 5 TeV $p$-Pb collisions aliceppbpid . In Part I pidpart1
a PID spectrum two-component (soft + hard) model (TCM) introduced in Ref.
ppbpid  is extended in several ways, including incorporation of a resonance
contribution to the pion soft-component model. The proton spectra are
corrected for an apparent detection inefficiency. Coefficients $z_{si}(n_{s})$
and $z_{hi}(n_{s})$ are inferred directly from spectra and modeled by simple
functions of TCM hard/soft ratio $x\nu$. Minimum-bias data hard components
$H_{i}(y_{t},n_{s})$ (for hadron species $i$) are derived from published
spectra with a more precise technique. Evolution of data hard components with
centrality is determined relative to the fixed PID TCM as a reference. And
comparison is made between the revised TCM and the preliminary version
described in Ref. ppbpid .

In the present study PID TCM hard-component models $\hat{H}_{0i}(y_{t},n_{s})$
are modified to accommodate variation of data hard components with charge
multiplicity $n_{ch}$. Systematic differences between meson and baryon trends
are revealed. The resulting accuracy of the overall TCM spectrum model is
evaluated with the Z-score statistic. The PID TCM is employed to predict
spectrum and yield ratios compared with corresponding data. The comparison
provides jet-related explanations for critical spectrum ratio features. The
PID TCM is employed to predict ensemble-mean $\bar{p}_{t}$ centrality trends.
Certain small-amplitude systematic features are explained by variation of jet-
related TCM hard components. Several aspects of those results support the
validity of a correction for proton detection inefficiency derived in Part I.
Some theoretical analysis is also consistent with the possibility of
significant proton detection inefficiency stoeckerstatmodel ; thermalprotons .

The overall two-part study responds to a tendency in recent years to interpret
similarities between certain data features derived from smaller collision
systems and corresponding features in A-A data as indicating formation of a
quark-gluon plasma (QGP) in the smaller systems based on argument from
analogy: Assuming QGP formation in larger collision systems, and given data
similarities between smaller and larger systems, then QGP may be formed in
smaller systems. As noted in Part I, because PID spectrum data are
conventionally used to buttress such arguments it is imperative to establish
the fullest possible understanding of spectrum composition and its response to
A-B centrality variation, especially as regards contributions from minimum-
bias jets. That is the principal purpose for developing an extended PID TCM
capable of exhaustively characterizing PID $p_{t}$ spectra.

This article is arranged as follows: Section II introduces the basic PID
spectrum TCM developed in Part I and reviews 5 TeV $p$-Pb PID spectrum data.
Section III describes a newly-developed variable TCM wherein hard-component
model functions are adjusted to accommodate spectrum data. Section IV presents
TCM analysis of PID spectrum and yield ratios with explanations for certain
prominent features. Section V describes application of the variable PID TCM to
$p$-Pb ensemble-mean $\bar{p}_{t}$ data. Section VI reviews systematic
uncertainties. Sections VII and VIII present discussion and summary.

##  II $\bf p$-$\bf Pb$ PID Spectrum $\bf TCM$

Part I of this study, reported in Ref. pidpart1 , addressed several issues:
(a) A proton detection inefficiency is quantitatively evaluated and corrected.
(b) The centrality dependence of PID TCM coefficients $z_{si}(n_{s})$ and
$z_{hi}(n_{s})$ is obtained directly from PID spectra. (c) That centrality
dependence is incorporated into the PID TCM and $p_{t}$ spectra are reanalyzed
with greater accuracy. This section presents the basic PID spectrum TCM from
Part I including those elements and introduces the PID spectrum data that are
the subject of analysis.

###  II.1 p-Pb spectrum TCM for identified hadrons

To establish a TCM for A-B PID $p_{t}$ spectra it is assumed that (a) N-N
(i.e. nucleon-nucleon) parameters $\alpha$, $\bar{\rho}_{sNN}$ and
$\bar{\rho}_{hNN}$ have been inferred from unidentified-hadron (nonPID) data
and (b) geometry parameters $N_{part}$, $N_{bin}$, $\nu$ and $x$ are a common
property (relating to centrality) of a specific A-B collision system
independent of specific hadron species. Quantity
$n_{s}=\Delta\eta\bar{\rho}_{s}$ (referring to the soft component of total
charge density $\bar{\rho}_{0}=\bar{\rho}_{s}+\bar{\rho}_{h}$) is employed as
a centrality index. Transverse rapidity
$y_{ti}\equiv\ln[(p_{t}+m_{ti})/m_{i}]$ is define for hadron species $i$. For
pion rapidity $y_{t\pi}$ the correspondence with $p_{t}$ is 1 vs 0.16 GeV/c, 2
vs 0.5 GeV/c, 2.67 vs 1 GeV/c, 4 vs 3.8 GeV/c and 5 vs 10.4 GeV/c. $y_{t\pi}$
is the default rapidity for all spectrum plots appearing in this study.

Given the A-B spectrum TCM for unidentified-hadron spectra a corresponding TCM
for identified hadrons can be defined by assuming that each hadron species $i$
comprises certain fractions of soft and hard TCM components denoted by
$z_{si}(n_{s})$ and $z_{hi}(n_{s})$ and assumed independent of $y_{t}$. The
PID spectrum TCM is then expressed as

$\displaystyle\bar{\rho}_{0i}(y_{t})$ $\displaystyle=$ $\displaystyle
S_{i}(y_{t},n_{s})+H_{i}(y_{t},n_{s})$ $\displaystyle\approx$ $\displaystyle
z_{si}(n_{s})\bar{\rho}_{s}\hat{S}_{0i}(y_{t})+z_{hi}(n_{s})\bar{\rho}_{h}\hat{H}_{0i}(y_{t})$
$\displaystyle\frac{\bar{\rho}_{0i}(y_{t})}{z_{si}(n_{s})\bar{\rho}_{s}}$
$\displaystyle\equiv$ $\displaystyle X_{i}(y_{t})$ $\displaystyle\approx$
$\displaystyle\hat{S}_{0i}(y_{t})+\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})\hat{H}_{0i}(y_{t}),$

where $\tilde{z}_{i}(n_{s})\equiv z_{hi}(n_{s})/z_{si}(n_{s})$ and unit-
integral model functions $\hat{S}_{0i}(y_{t})$ and $\hat{H}_{0i}(y_{t})$
depend on hadron species $i$. The nonPID factors are defined in terms of N-N
quantities by $\bar{\rho}_{s}=(N_{part}/2)\bar{\rho}_{sNN}$ and
$\bar{\rho}_{h}=N_{bin}\bar{\rho}_{hNN}$ with
$\bar{\rho}_{hNN}\approx\alpha\bar{\rho}_{sNN}^{2}$ for linear superposition
of N-N collisions in A-B collisions and with
$\bar{\rho}_{h}/\bar{\rho}_{s}=x(n_{s})\nu(n_{s})$. Soft fractions
$z_{si}(n_{s})$ can be expressed within the TCM by

$\displaystyle z_{si}(n_{s})$ $\displaystyle=$
$\displaystyle\left[\frac{1+x(n_{s})\nu(n_{s})}{1+\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})}\right]{z_{0i}},$
(3)

with $z_{hi}(n_{s})=\tilde{z}_{i}(n_{s})z_{si}(n_{s})$ and
$\bar{\rho}_{0i}\equiv z_{0i}\bar{\rho}_{0}$ defining $z_{0i}$. Thus, if
$\tilde{z}_{i}(n_{s})$ and $z_{0i}$ are specified for relevant hadron species
then the remainder of the PID TCM is determined. Model functions
$\hat{S}_{0i}(y_{t})$ are defined on proper $m_{ti}$ for a given hadron
species $i$ and then transformed to $y_{t\pi}$. $\hat{H}_{0i}(y_{t})$ are
always defined on $y_{t\pi}$. The basis for the $\hat{S}_{0i}(y_{t})$ model
definitions is the limit of normalized data spectra $X_{i}(y_{t})$ as
$n_{ch}\rightarrow 0$. Soft-component models are thus always derived from data
spectra. That description relates to a fixed TCM in which model function
$\hat{H}_{0i}(y_{t})$ does not vary with $n_{ch}$. Elaboration to a varying
TCM that accommodates specific data is described in Sec. III.

Normalized data spectra $X_{i}(y_{t})$ can be combined with soft-component
model function $\hat{S}_{0i}(y_{t})$ per Eq. (II.1) to extract spectrum data
hard components in the form

$\displaystyle Y_{i}(y_{t})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})}\left[X_{i}(y_{t})-\hat{S}_{0i}(y_{t})\right]~{}~{}~{}$
(4)

that may then be compared directly with hard-component model functions
$\hat{H}_{0i}(y_{t})$.

###  II.2 TCM model parameters

Table 1 presents TCM geometry parameters for 5 TeV $p$-Pb collisions inferred
from the analysis in Ref. tomglauber . Those geometry parameters, derived from
$p$-Pb $p_{t}$ spectrum and $\bar{p}_{t}$ data for unidentified hadrons
alicempt ; tommpt ; tomglauber , are assumed to be valid also for each
identified-hadron species and were used unchanged to process $p$-Pb PID
spectrum data in Ref. ppbpid . The $\bar{\rho}_{0}=n_{ch}/\Delta\eta$ charge
densities are measured quantities inferred from Fig. 16 of Ref. aliceglauber .
Specifically, charge density distributions from that figure were averaged over
$|\eta_{\text{lab}}|<0.5$. Relations $N_{part}=N_{bin}+1$ and
$\nu=2N_{bin}/N_{part}$ involve the number of nucleon N participants and N-N
binary collisions. $\bar{\rho}_{sNN}$ is the mean soft-component charge
density per participant pair averaged over all pairs.
$x\equiv\bar{\rho}_{hNN}/\bar{\rho}_{sNN}\approx\alpha\bar{\rho}_{sNN}$ is the
hard/soft density ratio. That approximation is based on TCM results from
$p$-$p$ $p_{t}$ spectra ppprd ; ppquad  and is applied here also to $p$-Pb
collisions assuming they consist of linear superpositions of N-N collisions.
Columns $\sigma^{\prime}/\sigma_{0}$ and $N_{bin}^{\prime}$ present nominal
centralities (bin centers) and binary-collision numbers quoted by Ref.
aliceppbpid  in connection with measured charge densities $\bar{\rho}_{0}$,
whereas columns $\sigma/\sigma_{0}$ and $N_{bin}$ present values inferred in
Ref. tomglauber  (see Ref. tomexclude  for a possible mechanism for such large
differences). The remaining table values are a result of the latter analysis.

Table 1: TCM fractional cross sections $\sigma/\sigma_{0}$ (bin centers) and geometry parameters, midrapidity charge density $\bar{\rho}_{0}$, N-N soft component $\bar{\rho}_{sNN}$ and TCM hard/soft ratio $x(n_{s})$ used for 5 TeV $p$-Pb PID spectrum analysis tomglauber . Centrality parameters are from Ref. tomglauber . $\sigma^{\prime}/\sigma_{0}$ values are from Table 1 of Ref. aliceppbpid .  $n$ | $\sigma^{\prime}/\sigma_{0}$ | $N_{bin}^{\prime}$ | $\sigma/\sigma_{0}$ | $N_{bin}$ | $\nu$ | $\bar{\rho}_{0}$ | $\bar{\rho}_{sNN}$ | $x(n_{s})$
---|---|---|---|---|---|---|---|---
1 | 0.025 | 14.7 | 0.15 | 3.20 | 1.52 | 44.6 | 16.6 | 0.188
2 | 0.075 | 13.0 | 0.24 | 2.59 | 1.43 | 35.9 | 15.9 | 0.180
3 | 0.15 | 11.7 | 0.37 | 2.16 | 1.37 | 30.0 | 15.2 | 0.172
4 | 0.30 | 9.36 | 0.58 | 1.70 | 1.26 | 23.0 | 14.1 | 0.159
5 | 0.50 | 6.42 | 0.80 | 1.31 | 1.13 | 15.8 | 12.1 | 0.137
6 | 0.70 | 3.81 | 0.95 | 1.07 | 1.03 | 9.7 | 8.7 | 0.098
7 | 0.90 | 1.94 | 0.99 | 1.00 | 1.00 | 4.4 | 4.2 | 0.047

Table 2 presents hard-component model parameters based on optimized
descriptions of spectrum hard components for the most-central ($n=1$) $p$-Pb
event class as described in Ref. pidpart1 . That condition plus the more-
differential format in Part I leads to improved precision of inferred
parameter values relative to the preliminary study reported in Ref. ppbpid .
TCM soft-component parameters remain unchanged from those reported in Table II
of Part I. That table plus Tables 1 and 2 reported here define a fixed PID TCM
established as a reference.

Table 2: Revised PID TCM hard-component model parameters $(\bar{y}_{t},\sigma_{y_{t}},q)$ for identified hadrons from 5 TeV $p$-Pb collisions derived from the differential spectrum analysis in Rev. pidpart1 . Parameter $z_{0i}$ values, inferred as limiting values of $z_{si}(n_{s})$ centrality trends, are also included. Uncertainties are determined as one half the change that would produce an obvious variation in $z_{hi}(y_{t},n_{s})$ ratios. Values with no uncertainties are duplicated from a related particle type.  | $\bar{y}_{t}$ | $\sigma_{y_{t}}$ | $q$ | $z_{0i}$
---|---|---|---|---
$\pi^{\pm}$ | $2.46\pm 0.005$ | $0.575\pm 0.005$ | $4.1\pm 0.5$ | 0.82 $\pm$0.01
$K^{\pm}$ | $2.655$ | $0.568$ | $4.1$ | 0.128 $\pm$0.002
$K_{\text{S}}^{0}$ | $2.655\pm 0.005$ | $0.568\pm 0.003$ | $4.1\pm 0.1$ | 0.064 $\pm$0.002
$p$ | $2.99\pm 0.005$ | $0.47\pm 0.005$ | $5.0$ | 0.065 $\pm$0.002
$\Lambda$ | $2.99\pm 0.005$ | $0.47\pm 0.005$ | $5.0\pm 0.05$ | 0.034 $\pm$0.002

###  II.3 5 TeV $\bf p$-Pb PID spectrum data

The identified-hadron spectrum data reported in Ref. aliceppbpid  and adopted
for the present analysis were produced by the ALICE collaboration at the LHC.
The event sample for charged hadrons is 12.5 million non-single-diffractive
(NSD) collisions and for neutral hadrons 25 million NSD collisions. Collision
events were divided into seven charge-multiplicity $n_{ch}$ or $p$-Pb
centrality classes based on yields in a VZERO-A counter subtending
$2.8<\eta_{lab}<5.1$ in the Pb direction. Hadron species include charged pions
$\pi^{\pm}$, charged kaons $K^{\pm}$, neutral kaons $K^{0}_{\text{S}}$,
protons $p,~{}\bar{p}$ and Lambdas $\Lambda,~{}\bar{\Lambda}$. Spectra for
charged vs neutral kaons and particles vs antiparticles are reported to be
statistically equivalent. Note that a later presentation of PID spectrum data
from the same collision system alicepidnew  extends the $p_{t}$ acceptance to
20 GeV/c for pions, charged kaons and protons, but the $p/\pi$ ratio results
in its Fig. 10 are essentially the same as in Fig. 2 of Ref. aliceppbpid
suggesting that the proton inefficiency is equivalent.

Figure 1 shows PID spectrum data (densities on $p_{t}$) from Ref. aliceppbpid
(points) plotted vs logarithmic variable $y_{t}$. That format (log-log with
respect to $p_{t}$) provides detailed access to low-$p_{t}$ structure (where
most jet fragments appear) and clearly shows power-law trends at higher
$p_{t}$. The curves are TCM parametrizations derived in Sec. III below and
demonstrated there to describe spectrum data within their statistical
uncertainties. As in Ref. aliceppbpid  the spectra are scaled up by powers of
2 according to $2^{n-1}$ where $n\in[1,7]$ is the centrality class index and
$n=7$ is most central (following the usage in Ref. aliceppbpid ). In this
article $n=7$ denotes the least-central data as in Table 1. Even with the
improvement of introducing logarithmic independent variable $y_{t}$ this
method of presenting data provides no significant information on the
substantial variation of spectrum structure with $p$-Pb centrality.

(a)(b)
(c)(d)

Figure 1:  $p_{t}$ spectra for identified hadrons from 5 TeV $p$-Pb
aliceppbpid  plotted vs pion transverse rapidity $y_{t}$ (default) for: (a)
pions, (b) neutral kaons, (c) corrected protons, (d) Lambdas. Solid curves
represent the PID spectrum TCM from Sec. III.

##  III Variable TCM adapted to data

In this section the precision PID data spectrum hard components isolated in
Part I are described accurately with a variable TCM. Factorization of data
hard components $H_{i}(y_{t},n_{s})$ is modified to minimize bias. Centrality-
dependent model functions $\hat{H}_{0}(y_{t},n_{s})$ are inferred from the
minimally-biased data hard components. The revised model parametrization
presents the opportunity to interpret modification of jet-related hard
components in terms of physical mechanisms. The variable TCM appears to
exhaust all information carried by $p$-Pb PID $p_{t}$ spectra.

###  III.1 TCM hard-component algebra

In Part I of this study, as reported in Ref. pidpart1 , PID spectrum hard
components were reported in the form

$\displaystyle Y^{\prime}_{i}(y_{t},n_{s})$ $\displaystyle=$
$\displaystyle\frac{\bar{\rho}_{0i}(y_{t},n_{s})-z_{si}(n_{s})\bar{\rho}_{s}\hat{S}_{0i}(y_{t})}{\bar{\rho}_{h}\hat{H}_{0i}(\bar{y}_{t})}$
$\displaystyle\equiv$ $\displaystyle
z_{hi}(y_{t},n_{s})\frac{\hat{H}_{0i}(y_{t})}{\hat{H}_{0i}(\bar{y}_{t})}$

where $\bar{y}_{t}$ is in this case the mode of the $\hat{H}_{0i}(y_{t})$
model function, in which case $Y^{\prime}_{i}(\bar{y}_{t},n_{s})\approx
z_{hi}(\bar{y}_{t},n_{s})$. An advantage of the first line is that spectrum
hard components are isolated with minimum bias, being scaled only by universal
centrality parameter $\bar{\rho}_{h}=N_{bin}\alpha\bar{\rho}_{sNN}^{2}$
(parameters from Table 1) and fixed amplitude $\hat{H}_{0i}(\bar{y}_{t})$. The
second line can be solved for $z_{hi}(y_{t},n_{s})$ to reveal deviations of
spectrum data from fixed model function $\hat{H}_{0i}(y_{t})$. Those results
were reported in Sec. VI of Part I.

In the present study the intent is to vary TCM model functions so as to
accommodate details of PID hard-component evolution with $p$-Pb centrality. As
noted, the hard component of spectrum data is best estimated via

$\displaystyle H_{i}(y_{t},n_{s})$ $\displaystyle\approx$
$\displaystyle\bar{\rho}_{0i}(y_{t},n_{s})-z_{si}(n_{s})\bar{\rho}_{s}\hat{S}_{0i}(y_{t}),$
(6)

where $z_{si}(n_{s})$ is determined as accurately as possible, individually
for each centrality, at low $y_{t}$ where $H_{i}(y_{t},n_{s})$ is negligible.
$z_{si}(n_{s})\bar{\rho}_{s}\hat{S}_{0i}(y_{t})$ is then defined as the
asymptotic limit of $\bar{\rho}_{0i}(y_{t},n_{s})$ as $n_{s}\rightarrow 0$ so
as to determine $\hat{S}_{0i}(y_{t})$. That differential procedure is most
accurate for high-mass hadrons where the hard/soft ratio is greatest (see Fig.
8 of Part I). The question then becomes how to best factorize
$H_{i}(y_{t},n_{s})$, a peaked distribution with possibly variable centroid
and shape, to characterize it.

The initial factorization strategy, defined in Eq. (1) of Ref. ppbpid , was
$H_{i}(y_{t},n_{s})\approx
z_{hi}(n_{s})\bar{\rho}_{h}(n_{s})\hat{H}_{0i}(y_{t})$ with
$\hat{H}_{0i}(y_{t})$ a fixed model function, the default for the TCM. That
factorization is insufficient because the shapes of data $H_{i}(y_{t},n_{s})$
are observed to change significantly with $p$-Pb centrality. To accommodate
hard-component shape changes two alternative factorizations are possible:

$\displaystyle H_{i}(y_{t},n_{s})$ $\displaystyle\approx$ $\displaystyle
z_{hi}(y_{t},n_{s})\bar{\rho}_{h}\hat{H}_{0i}(y_{t})$ $\displaystyle\approx$
$\displaystyle
z_{hi}(\bar{y}_{t},n_{s})\bar{\rho}_{h}\hat{H}_{0i}(y_{t},n_{s}),$

representing centrality-dependent $y_{t}$ shape variation by
$z_{hi}(y_{t},n_{s})$ (first line) or $\hat{H}_{0i}(y_{t},n_{s})$ (second
line). The first line includes fixed idealized model function
$\hat{H}_{0i}(y_{t})$ defined in terms of a specific $p$-Pb centrality (e.g.
$n=1$ or most central). That strategy is most effective for displaying data-
model deviations as in Sec. VI of Part I but not for adapting the TCM to
accommodate changing data structure. The second line includes variable model
function $\hat{H}_{0i}(y_{t},n_{s})$ that is adjusted to accommodate data,
with $z_{hi}(\bar{y}_{t},n_{s})$ estimated at the data mode which may be
centrality dependent. That strategy is most effective for understanding what
aspects of the hard component change with centrality and how they may be
interpreted physically. The first strategy was employed in Part I as noted in
its Sec. VI. Results from the second strategy are reported below. Spectrum
hard-component variations with $p$-Pb centrality appear to be substantially
different for mesons (pions, kaons) and baryons (protons, Lambdas). The two
cases are considered separately.

###  III.2 Meson hard-component models

The hard components for mesons are observed to have peak modes approximately
independent of $p$-Pb centrality within data uncertainties. However, the peak
widths are observed to vary with centrality above and/or below the mode. The
default TCM model function $\hat{H}_{0}(y_{t})$ consists of a Gaussian on
$y_{t}$ with exponential tail hardspec :

$\displaystyle\hat{H}_{0}(y_{t})$ $\displaystyle\approx$ $\displaystyle
A\exp\left\\{-\frac{(y_{t}-\bar{y}_{t})^{2}}{2\sigma^{2}_{y_{t}}}\right\\}~{}~{}~{}\text{near
mode $\bar{y}_{t}$}$ $\displaystyle\propto$
$\displaystyle\exp(-qy_{t})~{}~{}~{}\text{for larger $y_{t}$ -- the tail}.$

The model can be modified to accommodate meson hard components by defining
separate widths $\sigma_{y_{t}+}\rightarrow\sigma_{+}$ above the mode and
$\sigma_{y_{t}-}\rightarrow\sigma_{-}$ below the mode. Model function
$\hat{H}_{0i}(y_{t},n_{s})$ for a given system is then a composite of
functions defined below and above the mode.

Figure 2 (left) shows pion hard components in the form

$\displaystyle\frac{H_{i}(y_{t},n_{s})}{\bar{\rho}_{h}\hat{H}_{0i}(\bar{y}_{t},n_{s})}$
$\displaystyle\approx$ $\displaystyle
z_{hi}(\bar{y}_{t},n_{s})\frac{\hat{H}_{0i}(y_{t},n_{s})}{\hat{H}_{0i}(\bar{y}_{t},n_{s})},$
(9)

where the left-hand side represents data (points) and the right-hand side
represents the model (curves). The bold dashed curve is based on
$\hat{H}_{0i}(y_{t},n_{s})$ for most-central data ($n=1$) as in Fig. 11 (left)
of Part I to provide a reference. Data for the three most-central event
classes follow the model down to $y_{t}$ = 1.2 ($p_{t}\approx 0.20$ GeV/c).

Figure 2:  Left: Pion data spectrum hard components (points) for 5 TeV $p$-Pb
collisions and centralities denoted by $n\in[1,5]$ (see Table 1) in the form
of Eq. (III.1) (first line). The curves are TCM model function
$\hat{H}_{0i}(y_{t},n_{s})$ plotted in the form appearing on the right-hand
side of Eq. (9). The arrows indicate effective data shifts with increasing
$p$-Pb centrality. Right: Data/model ratio
$H_{i}(y_{t},n_{s})/\bar{\rho}_{h}\hat{H}_{0i}(y_{t},n_{s})$ per Eq. (9). The
dotted line is reference $z_{hi}(\bar{y}_{t},n_{s})=0.72$ for $n=1$.

Figure 2 (right) shows $z_{hi}(y_{t},n_{s})$ derived from data via Eq. (9) as
ratio $H_{i}(y_{t},n_{s})/\bar{\rho}_{h}\hat{H}_{0i}(y_{t},n_{s})$ (curves) vs
the nominal $z_{hi}(\bar{y}_{t},n_{s})$ for the most-central data (dotted
line). The curves in the right panel thus represent deviations via ratio of
data compared to the variable TCM.

Figure 3 shows the same procedure applied to kaons. The $y_{t}$ acceptance for
charged kaons is rather limited. However, the acceptance for neutral kaons
demonstrates that the variable TCM for kaons describes data within statistical
uncertainties from 0.2 GeV/c to 7 GeV/c as is evident from panel (d). The axis
limits in the left panels have been adjusted in each case to provide maximum
sensitivity to data structure in relation to models.

(a)(b)
(c)(d)

Figure 3:  Spectrum data hard components for charged kaons (a,b) and neutral
kaons (c,d) processed the same as for pion data in Fig. 2

Figure 4 (left) shows $\hat{H}_{0i}(y_{t},n_{s})$ model widths above the mode
$\sigma_{+}$ and below the mode $\sigma_{-}$ vs centrality measure $x\nu$. The
inferred parameter values are consistent with linear trends on $x\nu$ (lines):
$\sigma_{+}$ for pions and kaons $\approx 0.57-0.2(x\nu-0.34)$ and
$\sigma_{-}$ for pions $\approx 0.57+(x\nu-0.325)$. The width below the mode
for kaons is held fixed at the value 0.57 obtained for most-central kaon data.

Figure 4:  Left: Gaussian widths for TCM hard-component model
$\hat{H}_{0}(y_{t},n_{s})$ defined in Eq. (III.2) above the mode $\sigma_{+}$
and below the mode $\sigma_{-}$ for pions and kaons. Right: Hard-component
model centroids $\bar{y}_{t}$ for protons and Lambdas shifting with $p$-Pb
centrality according to Eq. (10).

###  III.3 Baryon hard-component models

Peak modes for baryon hard components are observed to shift substantially with
$p$-Pb centrality, and peak amplitudes [described by $z_{hi}(n_{s})$] also
vary substantially with centrality. But shifts and amplitudes are correlated
such that data hard components coincide with a fixed power-law trend at higher
$y_{t}$. TCM model function $\hat{H}_{0i}(y_{t},n_{s};\bar{y}_{t})$ can
accommodate those characteristics if peak mode $\bar{y}_{t}$ shifts with
amplitude $z_{hi}(n_{s})$ according to

$\displaystyle\bar{y}_{t}(n_{s})$ $\displaystyle=$
$\displaystyle\bar{y}_{t0}-(1/q)\ln[z_{hi}(n_{s})/z_{hi0}].$ (10)

following Eq. (III.2) (second line), where $\bar{y}_{t0}$ and $z_{hi0}$
correspond to a reference centrality (e.g. most-central data). Hard-component
data should then be described by

$\displaystyle\frac{H_{i}(y_{t},n_{s})}{\bar{\rho}_{h}\hat{H}_{0i}(\bar{y}_{t},n_{s})}$
$\displaystyle\approx$ $\displaystyle
z_{hi}(\bar{y}_{t},n_{s})\frac{\hat{H}_{0i}[y_{t};\bar{y}_{t}(n_{s})]}{\hat{H}_{0i}(\bar{y}_{t},n_{s})}$
(11)

where the shape of model $\hat{H}_{0i}[y_{t};\bar{y}_{t}(n_{s})]$ remains
fixed.

Figure 5 shows data-model comparisons for protons (a,b) and Lambdas (c,d).
Proton data for more-central collisions follow the model down to $y_{t}$ = 2
($p_{t}\approx 0.5$ GeV/c). The quality of the description is evident from the
right panels.

(a)(b)
(c)(d)

Figure 5:  Spectrum data hard components for protons (a,b) and Lambdas (c,d)
processed the same as pion data in Fig. 2.

Figure 4 (right) shows centroid variation $\bar{y}_{t}(n_{s})$ for protons and
Lambdas (solid points), as determined by $z_{hi}(n_{s})$ values from Table V
of Part I via Eq. (10), vs hard/soft ratio $x\nu$. In Ref. ppbpid  the
apparent centroid shifts for baryon hard components were modeled by an ad hoc
function

$\displaystyle\bar{y}_{t}(n_{s})$ $\displaystyle=$ $\displaystyle
2.96-\delta_{0}(n-4),$ (12)

where $n\in[1,7]$ is the centrality index with $n=1$ most central. The
estimated value $\delta_{0}=0.03$ for uncorrected proton spectra was severely
biased. The value 0.015 for Lambdas corresponds to the open points in the
right panel of Fig. 4 showing reasonable agreement with the present trends
(solid points) derived from precise differential measurements. Those inferred
values for $\bar{y}_{t}(n_{s})$ follow a linear trend on centrality measure
$x\nu$ (line)

$\displaystyle\bar{y}_{t}(n_{s})$ $\displaystyle=$ $\displaystyle
2.875+0.35x\nu.$ (13)

It is notable that the several parameters controlling variable-TCM PID model
functions all vary linearly with hard/soft (jet/nonjet) ratio $x\nu$ within
uncertainties.

###  III.4 Interpreting hard-component evolution

Determination of the $n_{ch}$ evolution of $p$-Pb meson and baryon spectrum
hard-component shapes within their point-to-point uncertainties is reported
above. A physical interpretation may be attempted based on current
understanding of QCD in high-energy nuclear collisions.

The quantitative relation between spectrum hard components and measured jet
properties has been established previously fragevo . A spectrum hard component
(jet fragment distribution) can be expressed as the convolution of a measured
parton/jet energy spectrum with a measured ensemble of fragmentation functions
(FFs) corresponding to the relevant collision system (e.g. $p$-$p$ collisions
mbdijets  as opposed to $e^{+}$-$e^{-}$ collisions eeprd ). A key parameter is
the effective lower bound of the jet spectrum – near 3 GeV according to NSD
$p$-$p$ spectrum data mbdijets .

Hard-component properties are observed to vary with collision charge
multiplicity $n_{ch}$ that may or may not relate to transverse geometry
(centrality) variation. As noted, the $n_{ch}$ (i.e. $\bar{\rho}_{0}$)
variation of 5 TeV $p$-Pb collisions appearing in Table 1 corresponds to
rather weak $p$-Pb transverse geometry variation. Likewise, the quadratic
relation between hard and soft components observed for $p$-$p$ collisions
ppprd  suggests that $p$-$p$ transverse geometry is not relevant for $p$-$p$
dijet production. Yet for $n_{ch}$ variation appearing in Table 1 dijet
production increases 50 fold [$\propto
N_{bin}\bar{\rho}_{sNN}^{2}\rightarrow(3.2/1)\times(16.6/4.2)^{2}$]. Such a
large increase can only be attributed to the depth of the parton splitting
cascade on momentum fraction $x$ corresponding to event-wise parton
distribution functions or PDFs.

Since the combination of PID FF properties and parton energy spectrum
determines PID hard components, and the parton energy spectrum is determined
in part by event-wise PDFs, one may speculate as to the relation of PID
spectrum hard components to event-wise charge density $\bar{\rho}_{0}$. In
particular, the effective lower bound of the parton energy spectrum may depend
directly on $\bar{\rho}_{0}$. What matters is how the parton energy spectrum
manifests as jet fragments of a particular hadron species (i.e. what is the
density of identified-hadron final states).

Evolution of baryon hard components with $n_{ch}$ as in Fig. 5 (left) closely
resembles that expected for variation of the parton spectrum lower bound as
illustrated in Fig. 11 (left) of Ref. fragevo  for 200 GeV $p$-$p$ collisions.
The solid curve corresponds to 3 GeV and the neighboring dotted curves
correspond to $\pm$ 10% energy shifts. The net effect for baryons is a lower-
bound shift upward on $y_{t}$ with increasing $n_{ch}$. In contrast, meson
hard-component widths above and/or below the mode respond to increasing
$n_{ch}$ with the peak centroid shifting downward on $y_{t}$, the net effect
being complementary to baryon evolution.

The relation to a lower bound on the parton energy spectrum appears
contradictory for mesons vs baryons. However, what should matter is what
partons fragment to what hadron species. Demanding lower-mass mesons may
select for lower-energy partons and the converse for higher-mass baryons. It
is notable that hard-component evolution trends exhibit remarkably smooth
variation vs hard/soft ratio $x\nu$ over an $n_{ch}$ interval corresponding to
fifty-fold increase in jet production. Over much of that interval $p$-Pb
collisions are indistinguishable from isolated $p$-N collisions. One may
conclude that $p$-Pb geometric centrality is not relevant to $p$-N jet
modification, a conclusion consistent with an analysis of eventwise-
reconstructed jets from 5 TeV $p$-Pb collisions ppbjets .

###  III.5 Evaluating model accuracy

As noted in Ref. tommodeltests  spectrum data-model comparisons are often
represented by data/model ratios. Ratio values near 1 are interpreted to
indicate acceptable models. However, that procedure can be quite misleading as
discussed for instance in Ref. ppprd . A more meaningful measure of model
validity is the Z-score zscore  defined by

$\displaystyle Z_{i}$ $\displaystyle=$
$\displaystyle\frac{O_{i}-E_{i}}{\sigma_{i}}\rightarrow\frac{\text{data $-$
model}}{\text{statistical error}},$ (14)

where $O_{i}$ is a spectrum datum, $E_{i}$ is the corresponding expectation
(model prediction) and $\sigma_{i}$ is the data r.m.s. statistical uncertainty
(error). For Poisson-distributed samples $\sigma_{i}\approx\sqrt{E_{i}}$ in
Eq. (14), leading to the formula often encountered for the $\chi^{2}$
statistic. Given the Z-score definition in Eq. (14) the relation to the
$\chi^{2}$ statistic can be simply expressed as

$\displaystyle\chi^{2}$ $\displaystyle\equiv$
$\displaystyle\sum_{i=1}^{N}\frac{(O_{i}-E_{i})^{2}}{\sigma_{i}^{2}}=\sum_{i=1}^{N}Z_{i}^{2}$
(15)

for $N$ data points in a spectrum. Given model degrees of freedom
$\nu=N-\text{number of fit parameters}$ one expects $\chi^{2}\sim\nu$, in
which case the r.m.s. value for Z scores for an acceptable fit should be
$\sqrt{\nu/N}$ – somewhat less than 1.

The relation between data/model ratios and corresponding Z-scores is

$\displaystyle\frac{\text{data}}{\text{model}}-1$ $\displaystyle\approx$
$\displaystyle\text{Z-score}\times\frac{\text{error}}{\text{data}},$ (16)

with error/model (exact) $\rightarrow$ error/data (approximate). The
error/data ratio (typically $\ll 1$) can vary by orders of magnitude between
different particle types and collisions systems, and even across $y_{t}$
intervals. Whereas the correct test of model validity is the relation of
Z-scores to 1 the interpretation of data/model ratios relative to 1 is highly
problematic and typically quite misleading.

(a)(b)
(c)(d)

Figure 6:  Left: Data/TCM spectrum ratios for pions (a) and kaons (c). Right:
Z-scores for pions (b) and kaons (b).

The figures in this subsection compare data/model ratios (left panels) to
Z-scores (right panels), for mesons in Fig. 6 and for baryons in Fig. 7. Based
on data/model ratios it would seem that the TCM description for pion data is
much better than that for Lambda data. Yet a comparison of corresponding
Z-scores shows that the data descriptions are actually of comparable quality.

(a)(b)
(c)(d)

Figure 7:  Left: Data/TCM spectrum ratios for protons (a) and Lambdas (c).
Right: Z-scores for protons (b) and Lambdas (d).

Examination of Z-scores for four hadron species shows that the r.m.s. values
are significantly greater than 1. But the excess appears related to
substantial systematic deviations (e.g. for pions near $y_{t}$ = 2) that
cannot be due to some physical phenomenon not accommodated by the TCM. Such
deviations more likely arise from some irregularity in processing the primary
particle data, as for instance was encountered in Sec. VI B of Ref.
tommodeltests . In general, these Z-scores suggest that the PID TCM represents
all physical information carried by $p$-Pb $p_{t}$ spectra.

###  III.6 TCM defined on the continuum

The TCM as utilized above is defined on $y_{t}$ data values for direct
comparisons with data spectrum components as for figures in this subsection.
In following sections the same TCM is defined on a “continuum” of 100 equal-
spaced values on $y_{t}\in[0,5]$ ($p_{t}\in[0,10]$ GeV/c) for investigation of
spectrum ratios and ensemble-mean $\bar{p}_{t}$ trends.

The TCM for PID $y_{t}$ spectra configured to accommodate spectrum data as in
this section is defined by

$\displaystyle\bar{\rho}_{0i}(y_{t})$ $\displaystyle=$ $\displaystyle
S_{i}(y_{t})+H_{i}(y_{t})\approx\frac{d^{2}n_{chi}}{y_{t}dy_{t}d\eta}$
$\displaystyle=$ $\displaystyle
z_{si}(n_{s})\bar{\rho}_{s}\hat{S}_{0i}(y_{t})+z_{hi}(\bar{y}_{t},n_{s})\bar{\rho}_{h}\hat{H}_{0i}(y_{t},n_{s}),$

where the approximation in the first line arises from averaging charge
densities over some limited $\eta$ interval about midrapidity. The spectrum
data reported in Ref. aliceppbpid  and appearing in Fig. 1 (as densities on
$p_{t}$) have the form

$\displaystyle\frac{1}{2\pi}\frac{d^{2}n_{chi}}{p_{t}dp_{t}dy_{z}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{2\pi}\frac{y_{t}}{m_{t}p_{t}}\bar{\rho}_{0i}(y_{t}).$
(18)

That transformation of $\bar{\rho}_{0i}(y_{t})$ is used to produce the TCM
curves in Fig. 1. Consistent with results above there are no visible
deviations between points and curves in Fig. 1 with two exceptions: (a) Meson
data hard components for $n=7$ (most peripheral) fall well below the model
Gaussian below the mode leading to a slight displacement of spectrum data
below the TCM near $y_{t}$ = 2.7. (b) Baryon data hard components for $n=7$
rise above the TCM hard-component model above the mode (apparent in Fig. 5, a
and c) leading to slight elevation of $n=7$ baryon data above the TCM for
large $y_{t}$.

##  IV $\bf p$-$\bf Pb$ PID spectrum and yield Ratios

PID spectrum and yield ratios have often been invoked to argue that hadrons
from high-energy nuclear collisions ($p$-$p$, $p$-A and A-A) emerge from a
partially equilibrated bulk medium via a freeze-out mechanism, thus supporting
inference of QGP formation in those systems. Ratio methods from Ref.
aliceppbpid  are reviewed in the first subsection. The remaining subsections
then consider PID spectrum and yield ratios in light of the present TCM
analysis.

###  IV.1 ALICE ratio methods

Reference aliceppbpid  presents PID spectrum and yield ratios in several
formats as follows (with figure numbers from that paper). Figure 2 shows PID
spectrum ratios vs $p_{t}$ for three hadron combinations and for two $p$-Pb
centralities – 0-5% and 60-80% (centralities as reported by Ref. aliceppbpid ,
see $\sigma^{\prime}/\sigma_{0}$ values in Table 1) corresponding to event
classes $n=1$ and 6 in Table 1. Figure 3 shows integrated-yield $p/\pi$ ratios
vs charge density $\bar{\rho}_{0}$ for $p$-Pb compared to Pb-Pb and inferred
power-law exponents $B(p_{t})$ (referring to $\bar{\rho}_{0}$ dependence) vs
integrated $p_{t}$ interval. The latter are obtained for $p/\pi$ and
$\Lambda/K_{\text{S}}^{0}$. Figure 5 shows integrated-yield ratios for three
hadron species to pions vs $\bar{\rho}_{0}$ comparing $d$-Au and $p$-Pb with
Au-Au and Pb-Pb.

Regarding Fig. 2, baryon/meson ratios “…show a significant enhancement at
intermediate $p_{T}\sim 3$ GeV/c, qualitatively reminiscent of that measured
in Pb-Pb collisions. The latter are generally discussed in terms of collective
flow or quark recombination.” One may ask how “enhancement” is measured,
relative to what?

Regarding Fig. 3, “It is worth noticing that the [yield within $p_{t}$ bins]
ratio $p/\pi$ as a function of $dN_{ch}/d\eta$…follows a power-law
behavior….the same trend is also observed in Pb-Pb collisions.” Regarding Fig.
5, “…a small increase is observed in the $K/\pi$ and $\Lambda/\pi$ ratios…. A
similar rise…in Pb-Pb…collisions…is typically attributed to a reduced
canonical suppression of strangeness production…or to an enhanced strangeness
production in a quark-gluon plasma.”

Regarding suggestions that certain ratio features may imply QGP formation in
small systems one should consider that a simple model describing PID spectrum
data, spectrum ratios and yield ratios within their uncertainties appears to
require no such ad hoc mechanisms. The remainder of this section examines
ratio trends in the context of a PID TCM for 5 TeV $p$-Pb collisions.

###  IV.2 TCM for PID spectrum ratios

A TCM for PID spectrum ratios can be derived as follows. The TCM for
individual PID spectra in Eq. (II.1) depends critically on accurate values for
nonPID $p$-Pb geometry parameters as in Table 1 which can be used to generate
hard/soft ratio $x\nu$. In Sec. IV of Part I coefficients $z_{si}(n_{s})$,
$z_{hi}(n_{s})$ and ratios $\tilde{z}_{i}(n_{s})$ are determined. In Sec. III
of this article model functions $\hat{H}_{0i}(y_{t},n_{s})$ are optimized to
accommodate hard-component variations with $p$-Pb centrality. The TCM for a
PID spectrum ratio including hadron species $i$ and $j$ is then expressed as

$\displaystyle\frac{\bar{\rho}_{0i}(y_{t};n_{s})}{\bar{\rho}_{0j}(y_{t};n_{s})}$
$\displaystyle=$
$\displaystyle\frac{z_{si}(n_{s})}{z_{sj}(n_{s})}\cdot\frac{\hat{S}_{0i}(y_{t})+\tilde{z}_{i}(n_{s})x\nu\hat{H}_{0i}(y_{t},n_{s})}{\hat{S}_{0j}(y_{t})+\tilde{z}_{j}(n_{s})x\nu\hat{H}_{0j}(y_{t},n_{s})}~{}~{}$
(19) $\displaystyle\rightarrow$
$\displaystyle\frac{H_{i}(y_{t},n_{s})}{H_{j}(y_{t},n_{s})}=\frac{z_{hi}(n_{s})}{z_{hj}(n_{s})}\cdot\frac{\hat{H}_{0i}(y_{t},n_{s})}{\hat{H}_{0j}(y_{t},n_{s})},$

where the second line is the limit for large $y_{t}$.

Figure 8 shows spectrum ratios for $p/\pi$ (left) and
$\Lambda/K^{0}_{\text{S}}$ (right). The general features of the ratio trends
are comparable to those in Fig. 2 of Ref. aliceppbpid . However, the data
$p/\pi$ trends presented there are 40% lower than in the left panel of Fig. 8,
consistent with conclusions in Part I. The $p/\pi$ ratios for peripheral
collisions in Ref. aliceppbpid  Fig. 2 are quantitatively similar for $p$-Pb
and Pb-Pb collisions suggesting that the same proton inefficiency noted in
Part I occurs for Pb-Pb as well. The “soft only” trends in Figure 8 are
obtained by omitting the hard components from the TCM and demonstrate in the
left panel that above 1 GeV/c ($y_{t}\approx 2.7$) $p/\pi$ spectrum ratios are
dominated by the hard-component (jet) contribution.

Figure 8:  Left: $p/\pi$ spectrum ratios vs $p_{t}$ for two $p$-Pb
centralities derived from the variable TCM (solid) and from the TCM with soft-
component only (dashed). The reversal of centrality dependence near 1.5 GeV/c
is due to centroid shifts on $y_{t}$ for protons and pions (in opposite
senses) as explained in the text. Right: $\Lambda/K_{\text{S}}^{0}$ spectrum
ratios treated as in the left panel. $K_{\text{S}}^{0}$ spectra are multiplied
by 2 corresponding to $\lambda+\bar{\lambda}$.

An interesting feature of the spectrum ratios is the reversal of sign
(asymmetry) of the variation with centrality: Ratios decrease at lower $p_{t}$
with increasing centrality but increase at higher $p_{t}$. That feature is
emphasized in the top panel of Fig. 3 of Ref. aliceppbpid . The ratio TCM of
Eq. (19) predicts that spectrum ratios should decrease uniformly on $y_{t}$
with increasing centrality assuming fixed model functions
$\hat{S}_{0i}(y_{t})$ and $\hat{H}_{0i}(y_{t})$. Ratio behavior is then
dominated by factors $z_{xi}(n_{s})/z_{xj}(n_{s})$ ($x=s,\,h$). If species $i$
is more massive $z_{xi}(n_{s})$ should decrease more rapidly as noted in Sec.
V of Part I (e.g. its Fig. 9, left).

These results can be compared with Fig. 9 of Ref. ppbpid . The TCM spectrum
ratios generated in that case were based on shifting only the baryon hard
components, via the ad hoc shift function described by Eq. (12). However, Sec.
III of this article establishes that whereas baryon centroids shift to higher
$y_{t}$ with increasing centrality meson centroids shift to lower $y_{t}$. As
discussed below meson centroid shifts should then increase the difference
between central and peripheral spectrum ratio trends.

###  IV.3 Interpreting PID spectrum ratio trends

As demonstrated in Sec. III.5 the TCM provides an excellent data description
for five hadron species including corrected proton spectra. It is therefore
useful to employ the TCM to investigate properties of spectrum ratios and
their physical interpretations. There are two main issues: (a) the origin of
the sign change on $y_{t}$ of the centrality trends and (b) the origin of the
peak near 3 GeV/c.

Figure 9 shows $p/\pi$ spectrum ratios for TCM PID spectra, transforming the
usual linear spectrum ratio vs $p_{t}$ to what is effectively a log-log
representation. The log-log format permits more comprehensive examination of
data trends and isolation of jet and nonjet contributions. The solid and
dashed curves in the left panel are the same as those in Fig. 8 (left).
Reversal in sign of centrality evolution below and above $y_{t}\approx 3.1$ in
Fig. 9 (left) reflects PID spectrum data accurately modeled via the PID TCM to
their statistical limits as described in Sec. III. As noted above, the sign
change results from shifts of hard-component centroids to higher $y_{t}$ for
baryons (numerators) and to lower $y_{t}$ for mesons (denominators).

The effect of hard-component centroid shifts for two hadron species is
illustrated by the dotted curves in Fig. 9 (left). Note that the most dramatic
$n_{ch}$ variation in the $p/\pi$ hard-component ratio occurs for $y_{t}<2$
($p_{t}<0.5$ GeV/c). Above that point the ratio hard component for 60-80%
central is shifted to lower $y_{t}$ and lower amplitude via proton and pion
centroid shifts compared to the 0-5% central curve corresponding to $n=1$
reference model functions. The combination results in the trend reversal.

Figure 9:  Left: Soft+hard spectrum-ratio curves as in Fig. 8 (left) with
proton and pion hard components shifted with centrality (solid) and plotted
here in a log-log format. The dashed curves are the soft-only curves from Fig.
8 (left). The dotted curves are corresponding hard-only ratios. Right: Same as
the left panel except that proton and pion hard components are not shifted as
in Fig. 8. The reversal of centrality dependence near 1.5 GeV/c ($y_{t}\approx
3.1$) then does not occur. Note alteration of dotted curves with and without
shifting.

Figure 9 (right) shows TCM spectrum ratios with no centroid shifts. Note that
the 0-5% curves are the same in the two panels by construction. The ratios
(and individual soft and hard components) then increase with decreasing
centrality consistently on $y_{t}$ per Eq. (19). In Ref. ppbpid  the proton
hard-component model was shifted to illustrate this point in its Fig. 9. The
pion shift to lower $y_{t}$ as in Fig. 2 (left) above, acting in the $p/\pi$
denominator, was expected to significantly increase the asymmetry. Results for
proton and pion peak shifts separately as presented in the following figure
confirm that expectation.

Figure 10 shows $p/\pi$ spectrum ratios (solid) where only the pion hard
component shifts with centrality (left) or only the proton hard component
shifts (right). The bold dotted curves in this figure are solid curves in Fig.
8 (left) representing observed data trends. The curves for 0-5% are unchanged
by construction (corresponding to reference hard-component model functions
with no centroid shifts). Compared to the 60-80% curve (upper dotted) from
Fig. 9 (right) the corresponding 60-80% curves in Fig. 10 (solid) have
translated about half way toward the 60-80% curve (lower dotted) from Fig. 8
(left) representing data. These result demonstrate that proton centroid shifts
to higher $y_{t}$ and pion shifts to lower $y_{t}$ contribute similarly to
sign reversal for the $p/\pi$ spectrum ratio.

Figure 10:  $p/\pi$ spectrum TCM ratios with pion centroid shift only (left)
and proton centroid shift only (right). The dash-dotted curves correspond to a
60-80% model with no centroid shifts as reference. The bold dotted curves are
the solid curves in Fig. 8 (left) that describe data accurately. The lower
solid curves are 60-80% model curves with pion or proton centroid shifts. The
upper solid curves indicate that the 0-5% model (reference) is independent of
centroid shifts by construction.

The general structure of the spectrum ratios, and especially the prominent
peak near 3 GeV/c, are explained as follows: The soft-only trends (dashed
curves above) arise from interplay of slope parameter $T$ and Lévy exponent
$n$ in $\hat{S}_{0}(y_{t})$. At $y_{t}=1$ ($p_{t}\approx 0.15$ GeV/c) the soft
$p/\pi$ ratio is small $O(0.01)$ reflecting statistical-model hadronic
abundances. However, the soft ratio increases strongly because the proton
slope parameter 210 MeV is substantially greater than the pion value 145 MeV.
Above 1 GeV/c ($y_{t}\approx 2.7$) the lower pion Lévy exponent $n=8.5$
(harder tail) then overwhelms the proton value 14 and the $p/\pi$ soft trend
drops rapidly. Above 1 GeV/c ($y_{t}\approx 2.7$) pion and proton hard
components dominate.

The hard-only trends (dotted) in Fig. 9 are similarly explained. The pion
hard-component centroid $y_{t}\approx 2.46$ is substantially lower than the
proton centroid $y_{t}\approx 2.99$ and the pion peak is broader. The pion
hard component is thus much greater than the proton hard component below their
modes. But as $y_{t}$ traverses the peak modes the proton hard component comes
to dominate near and above its 3 GeV/c mode, producing a peak there in the
$p/\pi$ ratio. Above 4 GeV/c ($y_{t}\approx 4$) the difference in peak widths
and exponential (power-law) tail constants $q$ prevails. The pion $q\approx 4$
(harder tail) vs the proton $q\approx 5$ (softer tail) insures that the
$p/\pi$ hard-component ratio then falls off strongly. As is evident from Fig.
8 (left) the structure of the $p/\pi$ ratio on $y_{t}$ is dominated by jet-
related spectrum hard components above 1 GeV/c.

###  IV.4 Comparing $\bf p$-Pb ratios to A-A spectrum ratios

In Fig. 2 of Ref. aliceppbpid  PID spectrum ratios for 5 TeV $p$-Pb collisions
are compared with those from 2.76 TeV Pb-Pb collisions. The ratio structure,
including the asymmetry on $y_{t}$ with increasing centrality, is formally
similar although the effect is much larger for the Pb-Pb data. The ratio peak
near 3 GeV/c in Pb-Pb is said to be “generally discussed in terms of
collective flow or quark recombination.” It is noted that for a Pb-Pb
centrality bin with matching ratio peak amplitude the corresponding Pb-Pb
$\bar{\rho}_{0}$ is 1.7 higher than that for central $p$-Pb.

Figure 11 (left) repeats Fig. 11 (left) of Part I showing the pion hard-
component centroid shifted to lower $y_{t}$ with increasing $p$-Pb centrality.
It is demonstrated in the previous subsection that such displacements are
responsible for the asymmetry on $y_{t}$ of PID spectrum ratios with
increasing centrality, where the same asymmetry effect arises if the ratio
numerator is shifted to higher $y_{t}$ or the denominator is shifted to lower
$y_{t}$. How then should that relate to the Pb-Pb results appearing in Ref.
aliceppbpid ?

Figure 11:  Left: Repeat of Fig. 11 (left) of Part I showing shift of pion
hard-component centroid to lower $y_{t}$ with increasing $p$-Pb centrality.
Right: Pion hard components for 200 GeV $p$-$p$ (open circles) and 0-12%
central Au-Au (solid dots) (Fig. 18, right of Ref. pbpbpid ). The TCM
reference for central Au-Au is the dash-dotted curve – the TCM for minimum-
bias $p$-$p$ (dashed) multiplied by $\nu\approx 6$. Comparison of Au-Au
central pion data with the TCM reference reveals a centroid shift to lower
$y_{t}$ much larger than that for $p$-Pb at left.

Figure 11 (right) shows PID pion spectrum hard components for 200 GeV minimum-
bias (MB) $p$-$p$ collisions (open circles) and 0-12% central 200 GeV Au-Au
collisions (solid dots). The dashed curve is the TCM for 200 GeV MB $p$-$p$
collisions while the dash-dotted curve shows the TCM prediction for central
Au-Au collisions assuming Glauber linear superposition (GLS) – the dashed
curve scaled up by $\nu=$ 6 as the ratio of N-N binary collisions to
participant pairs for central Au-Au. The important result: the pion data hard
component is shifted to lower $y_{t}$ with increasing Pb-Pb centrality
compared to linear superposition, formally equivalent to the shift in the left
panel. It is also significant that the effective peak shift is correlated with
suppression at higher $y_{t}$ that is usually interpreted to demonstrate “jet
quenching.”

Given this Au-Au example it is reasonable to interpret the Pb-Pb data in Fig.
2 (middle) of Ref. aliceppbpid  as arising from some combination of proton
hard component shifting to higher $y_{t}$ and pion hard component shifting to
lower $y_{t}$ (as in Fig. 11, right). The correspondence with a spectrum
feature associated with jet quenching buttresses the conclusion that the most
prominent features of spectrum ratios and their centrality evolution are
mainly jet related. It should also be noted that $p/\pi$ spectrum ratios for
peripheral collisions in Fig. 2 of Ref. aliceppbpid  are the same within
uncertainties for $p$-Pb and Pb-Pb collisions, suggesting that the proton
detection inefficiency evaluated for $p$-Pb collisions in Part I is also
present in the Pb-Pb data presented in Fig. 2 of Ref. aliceppbpid . See Ref.
alicepbpbpidspec  for PID spectrum TCM analysis methods applied to 2.76 TeV
Pb-Pb spectrum data. Reference aliceppbpid  also states that “The separation
power achieved in p-Pb collisions is identical to that for pp collisions”
suggesting a common property of any ALICE PID spectrum data.

###  IV.5 Power-law behavior of PID spectrum ratios

In its Fig. 3 Ref. aliceppbpid  represents the asymmetry on $y_{t}$ of
spectrum ratios as in Fig. 8 in terms of power-law exponent
$B(p_{t};\bar{\rho}_{0})$, i.e.
$p/\pi\propto\bar{\rho}_{0}^{\,B(p_{t};\bar{\rho}_{0})}$. In Ref. aliceppbpid
values of $B(p_{t};\bar{\rho}_{0})$ for successive values of $p_{t}$ are
apparently derived via power-law fits to ratio trends on $n_{ch}$. The same
result can also be achieved via logarithmic derivatives:

$\displaystyle B(y_{t};\bar{\rho}_{0})$ $\displaystyle\approx$
$\displaystyle\frac{d}{d\ln\bar{\rho}_{0}}\ln\left[\frac{\bar{\rho}_{0p}(y_{t};\bar{\rho}_{0})}{\bar{\rho}_{0\pi}(y_{t};\bar{\rho}_{0})}\right]$
$\displaystyle\approx$ $\displaystyle
B_{p}(y_{t};\bar{\rho}_{0})-B_{\pi}(y_{t};\bar{\rho}_{0}),$

where the switch to transverse rapidity $y_{t}$ facilitates the discussion
below. The logarithm of a ratio results in a difference between two hadron
species represented by the second line. Results in Fig. 8 (left) make clear
that near the centrality-trend reversal (asymmetry) of the $p/\pi$ ratio hard
components $H_{i}(y_{t})$ dominate. Also, in the neighborhood of the $p/\pi$
ratio peak the hard-component peaks are approximately Gaussian with mean
$\bar{y}_{t}$ and width $\sigma_{y_{t}}$ leading to the following progression
for species $i$:

$\displaystyle B_{i}(y_{t};\bar{\rho}_{0})$ $\displaystyle\approx$
$\displaystyle\bar{\rho}_{0}\frac{d}{d\bar{\rho}_{0}}\ln\left[{H_{i}(y_{t};\bar{y}_{t},\bar{\rho}_{0})}\right]$
$\displaystyle=$
$\displaystyle\bar{\rho}_{0}\frac{d\bar{y}_{t}}{d\bar{\rho}_{0}}\cdot\frac{d}{d\bar{y}_{t}}\ln\left[{H_{i}(y_{t};\bar{y}_{t},\bar{\rho}_{0})}\right]$
$\displaystyle\approx$
$\displaystyle\bar{\rho}_{0}\frac{d\bar{y}_{t}}{dn}\frac{dn}{d\bar{\rho}_{0}}\cdot(y_{t}-\bar{y}_{tn})/\sigma^{2}_{y_{t}}.$

For five of seven $p$-Pb centrality classes $n\in[2,6]$ the expression in the
first line is evaluated in terms of finite differences between data hard
components for four nearest-neighbor centrality ($\bar{\rho}_{0}$) pairs, and
the results are averaged. Given the last line above, based on a Gaussian
model, the averaged data trend is approximated as

$\displaystyle\bar{B}_{i}(y_{t})$ $\displaystyle\approx$
$\displaystyle\hat{B}_{i}(y_{t}-\bar{y}_{t}).$ (22)

Figure 12 shows inferred data power-law exponents (points) for protons (left)
and Lambdas (right). The solid curves correspond to Eq. (IV.5) applied to TCM
model function $\hat{H}_{0}(y_{t};n_{s})$ shifted by $\Delta\bar{y}_{t}=0.015$
for each increment in centrality index $n$ (Table 1) as per Ref. ppbpid . The
horizontal line segment corresponds to the exponential (power-law) tail on
$\hat{H}_{0}(y_{t},n_{s})$. Note that the Lambda data trend extends high
enough on $y_{t}$ to reveal that the data exponential tail for baryons does
not shift significantly with centrality, as is evident from Fig. 5.

Figure 12:  Power-law exponent $B(y_{t})$ (points) obtained from data hard
components in Fig. 5 (a) via Eq. (IV.5) (first line). Solid curves are derived
from the same procedure applied to TCM hard components
$\hat{H}_{0}(y_{t},n_{s})$ shifted on $y_{t}$ with $n_{s}$ as described in
Ref. ppbpid  Sec. 6.2. Note that $y_{t}\approx 3.1$ above is equivalent to
$p_{t}\approx 1.55$ GeV/c in Fig. 3 (middle) of Ref. aliceppbpid .

The slope of the $\bar{B}_{i}(y_{t})$ trend can be predicted as follows (see
Table 1): Assume centrality class $n=4$ with $d\bar{y}_{t}/dn\approx 0.015$ as
noted above, $\bar{\rho}_{0}\approx 23$, $d\bar{\rho}_{0}/dn\approx 7$ and
$\sigma_{y_{t}}\approx 0.47$. Based on Eq. (IV.5) (third line) baryon
$\hat{B}_{i}\approx 0.22$ in Eq. (22) corresponding to the trends in Fig. 12.
The dashed curve at left is $\bar{B}_{p}(y_{t})=0.22(y_{t}-3.2)$.

For the full $p/\pi$ ratio the pion contribution $B_{\pi}(y_{t})$ in Eq.
(IV.5) must contribute. The denominator should nominally contribute a negative
term as in Eq. (IV.5). However, as shown in Fig. 2 the factor
$d\bar{y}_{t}/dn$ for pions is itself negative (the hard component shifts to
lower $y_{t}$) so the result should be further increase in the slope of total
$B(y_{t})$. Those relations are visualized in Fig. 10.

As demonstrated above, various features of Figs. 2 and 3 in Ref. aliceppbpid
are actually manifestations of jet-related PID spectrum hard components
shifting to lower or higher $y_{t}$ with increasing $n_{ch}$. The chosen
plotting formats in Ref. aliceppbpid  in effect obscure the consequences of
jet production in $p$-Pb collisions as reported in Sec. III above.

###  IV.6 PID yield ratios

In its Fig. 5 Ref. aliceppbpid  presents centrality evolution for three hadron
yield ratios from 5 TeV $p$-Pb and various A-A collision systems. According to
Eq. (19) (first line) integration of numerator and denominator leads to

$\displaystyle\frac{\bar{\rho}_{0i}(n_{s})}{\bar{\rho}_{0j}(n_{s})}$
$\displaystyle=$
$\displaystyle\frac{z_{si}(n_{s})}{z_{sj}(n_{s})}\cdot\frac{1+\tilde{z}_{i}(n_{s})x\nu}{1+\tilde{z}_{j}(n_{s})x\nu}$
$\displaystyle\rightarrow$ $\displaystyle\frac{z_{0i}}{z_{0j}}$

as the TCM for yield ratios. The second line of Eq. (IV.6) is a trivial result
given the definition of $z_{si}(n_{s})$. The TCM that describes PID $p$-Pb
spectra within statistical uncertainties (see Sec. III.5) thus predicts yield
ratios independent of $n_{ch}$. The approximate centrality independence of
parameters $z_{0i}$ is confirmed in Sec. V B of Part I.

The constant values predicted by the TCM (see Table 2 for $z_{0i}$ estimates)
are $K/\pi\approx 0.128/0.82=0.156$, $p/\pi\approx 0.065/0.82=0.079$ and
$\Lambda/\pi\approx 0.034/0.82=0.041$. Those values can be compared with the
ratio trends in Fig. 5 of Ref. aliceppbpid . For $K/\pi$ the reported ratios
increase from 0.12 to 0.145. For $p/\pi$ the reported ratios are approximately
constant at 0.057. For $\Lambda/\pi$ the reported ratios increase from 0.032
to 0.040. Given the large overall systematic uncertainties the $K/\pi$ and
$\Lambda/\pi$ increases might not be significant. However, the uncertainties
“uncorrelated across multiplicity bins” (shaded boxes) are small enough to
suggest that those variations are quite significant. Certainly significant is
the discrepancy between the $p/\pi\approx 0.079$ TCM prediction and the
reported 0.057 ratio in Fig. 5 of Ref. aliceppbpid  that would suggest a
proton $z_{0i}\approx 0.057\cdot 0.82=0.047\rightarrow 0.72\times 0.065$.

The relation between $p/\pi$ and $\Lambda/\pi$ ratios is illuminated by a
comparison of proton and Lambda spectra in Fig. 7 of Part I. Uncorrected (i.e.
published) proton data below 0.6 GeV/c ($y_{t}\approx 2.15$) divided by 2
agree with corresponding Lambda data within a few percent. That result
strongly argues that the $p/\pi$ ratio should be approximately twice the
$\Lambda/\pi$ ratio $\approx 0.04$. The reported ratio data $p/\pi\approx
0.057$ ($\approx 70$% of the expected 0.079) buttresses the correction
procedure reported in Part I.

Additional information comes from consideration of sum rules applied to
charged hadrons, particularly the condition $\sum_{i}z_{si}(n_{s})\approx 1$,
with the limiting case (for $n_{s}\rightarrow 0$) $\sum_{i}z_{0i}\approx 1$.
Note that the $z_{si}(n_{s})$ are inferred from published data spectra at low
$y_{t}$ where no data require correction and this sum rule is accurately
satisfied. The sum rule can then be used to describe a relation among ratios
of charged hadrons:

$\displaystyle K/\pi+p/\pi$ $\displaystyle\approx$ $\displaystyle 1/\pi-1$
(24)

where the respective $z_{0i}$ are represented by particle symbols. Assuming
that $z_{0i}$ for pions is $\approx 0.8$ within a few percent then one should
expect $K/\pi+p/\pi\approx 0.25$, and the sum for TCM ratios is $\approx$
0.24. But the sum of corresponding ratio data in Fig. 5 of Ref. aliceppbpid
varies with centrality from 0.18 to 0.20. The shortfall of $\approx 0.04$ \-
0.06 is highly significant compared to the uncertainties “uncorrelated across
multiplicity bins” of $\approx 0.002$ reported in Ref. aliceppbpid . Part of
that shortfall (0.022) appears to be a $\approx 30$% reduction in the total
integrated proton yield, or a larger fraction above 0.6 GeV/c as noted in Part
I. Another substantial part (0.021) is an apparent $\approx 15$% reduction in
the K± yield (averaged over centrality), possibly due to extrapolation from a
limited $p_{t}$ acceptance (see large error bars on charged-kaon $\bar{p}_{t}$
data in Fig. 14).

##  V Ensemble-mean $\bf\bar{p}_{t}$ trends

The updated TCM for PID spectra presented in the previous sections may be
tested by comparison with measured PID $\bar{p}_{t}$ values. In this section
the $\bar{p}_{t}$ TCM is generalized to include $\bar{p}_{ti}$ hard-components
$\bar{p}_{thi}(n_{s})$ and ratios
$\tilde{z}_{i}(n_{s})=z_{hi}(n_{s})/z_{si}(n_{s})$ varying with centrality
assuming that all hadron species share common $p$-Pb centrality parameters
$x(n_{s})$ and $\nu(n_{s})$ presented in Table 1.

###  V.1 Ensemble-mean $\bf\bar{p}_{t}$ TCM

The ensemble-mean total $p_{t}$ $\bar{P}_{ti}$ for identified hadrons of
species $i$ integrated over some angular acceptance $\Delta\eta$ as derived
from Eq. (II.1) (second line) is

$\displaystyle\bar{P}_{ti}$ $\displaystyle=$
$\displaystyle\Delta\eta\int_{0}^{\infty}dp_{t}\,p_{t}^{2}\,\bar{\rho}_{0i}(p_{t})=\bar{P}_{tsi}+\bar{P}_{thi}~{}~{}$
$\displaystyle=$ $\displaystyle
z_{si}(n_{s})n_{s}\bar{p}_{tsi}+z_{hi}(n_{s})n_{h}\bar{p}_{thi}(n_{s}).$

Based on spectrum data, and just as for unidentified hadrons, it is concluded
that soft-component $\bar{p}_{tsi}$ is a universal quantity for each hadron
species $i$. Hard-component $\bar{p}_{thi}(n_{s})$ trends are inferred from
spectrum hard components as they are described in Sec. III. The corresponding
TCM for conventional ratio $\bar{p}_{ti}$ is

$\displaystyle\frac{\bar{P}_{ti}}{\bar{n}_{chi}}$ $\displaystyle\equiv$
$\displaystyle\bar{p}_{ti}(n_{s})\approx\frac{\bar{p}_{tsi}+\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})\,\bar{p}_{thi}(n_{s})}{1+\tilde{z}_{i}(n_{s})x(n_{s})\,\nu(n_{s})},~{}~{}~{}~{}$
(26)

assuming that $p_{t}$ integrals extend down to $p_{t}=0$ (i.e. by
extrapolation of $\bar{\rho}_{0i}$ data). The lower limit on centrality for
$\bar{p}_{ti}$ is then $\bar{p}_{tsi}$. The limit for very large multiplicity
is the limiting value for $\bar{p}_{thi}(n_{s})$. Data trends and
corresponding TCM are illustrated in Fig. 8 of Ref. ppbpid .

In Ref. ppbpid  ratio $\tilde{z}_{i}=z_{hi}/z_{si}$ for each hadron species
was held fixed independent of centrality with values given by its Table 4.
That table also includes fixed values for $\bar{p}_{tsi}$ and $\bar{p}_{th0i}$
derived from fixed model functions $\hat{S}_{0i}(y_{t})$ and
$\hat{H}_{0i}(y_{t})$ respectively as defined by parameters in its Table 2.
Centrality parameters $x(n_{s})$ and $\nu(n_{s})$, independent of hadron
species, correspond to Table 1 of the present article. In this study ratios
$\tilde{z}_{i}(n_{s})$ are taken from Sec. V of Part I, and as noted
$\bar{p}_{thi}(n_{s})$ are derived from varying data hard components as
derived in Sec. III.

Soft-components $\bar{p}_{tsi}$ are obtained from model functions
$\hat{S}_{0}(y_{t})$ for pions, kaons, protons and Lambdas as $0.40\pm 0.02$,
$0.60\pm 0.02$, $0.74\pm 0.02$ and $0.78\pm 0.02$ GeV/c.

###  V.2 Hard components $\bf\bar{p}_{thi}(n_{s})$

Table 3 presents hard-component $\bar{p}_{thi}(n_{s})$ values for four hadron
species obtained by evaluating sums over hard-component data on data points
(no extrapolation). Proton and Lambda TCM model functions are identical. What
sets the proton data apart is an apparent excess on the low-$y_{t}$ side of
the hard-component peak as seen in Fig. 13 (a) of Part I. Results for
$K^{\pm}$ are unreliable due to the limited $p_{t}$ coverage.
$K_{\text{S}}^{0}$ values are used instead.

Table 3: Spectrum hard-component means $\bar{p}_{thi}(n_{s})$ for four hadron species from six centralities of 5 GeV $p$-Pb collisions with units GeV/c. $K^{\pm}$ values are copied from $K_{\text{S}}^{0}$ values because of the latter’s much larger $y_{t}$ coverage. These value are obtained by integrating over data hard components within their $p_{t}$ acceptances.  $n$ | $\pi^{\pm}$ | $K^{\pm}$ | $p$ | $K_{\text{S}}^{0}$ | $\Lambda$
---|---|---|---|---|---
1 | $1.05$ | – | $1.60$ | $1.34$ | $1.72$
2 | $1.07$ | – | $1.58$ | $1.35$ | $1.71$
3 | $1.07$ | – | $1.59$ | $1.38$ | $1.70$
4 | $1.09$ | – | $1.56$ | $1.39$ | $1.67$
5 | $1.14$ | – | $1.51$ | $1.40$ | $1.62$
6 | $1.12$ | – | $1.50$ | $1.48$ | $1.58$

Table 4 presents hard-component $\bar{p}_{thi}(n_{s})$ values derived from
variable model functions $\hat{H}_{0i}(y_{t},n_{s})$, inferred from PID hard-
component data in Sec. III, integrated over 100 points distributed uniformly
on $y_{t}\in[0,5]$ (i.e. within $p_{t}\in[0,10]$ GeV/c). Those values are used
below in discussion of a PID $\bar{p}_{t}$ TCM.

Table 4: Spectrum hard-component means $\bar{p}_{thi}(n_{s})$ for four hadron species from seven centralities of 5 GeV $p$-Pb collisions with units GeV/c. $K^{\pm}$ values are copied from $K_{\text{S}}^{0}$ values because of the latter’s much larger $y_{t}$ coverage. These values are obtained by integrating over hard-component models from the variable TCM derived in this section.  $n$ | $\pi^{\pm}$ | $K^{\pm}$ | $p$ | $K_{\text{S}}^{0}$ | $\Lambda$
---|---|---|---|---|---
1 | $1.10$ | – | $1.67$ | $1.32$ | $1.66$
2 | $1.13$ | – | $1.65$ | $1.33$ | $1.65$
3 | $1.15$ | – | $1.63$ | $1.34$ | $1.63$
4 | $1.18$ | – | $1.61$ | $1.35$ | $1.61$
5 | $1.21$ | – | $1.59$ | $1.36$ | $1.59$
6 | $1.29$ | – | $1.55$ | $1.39$ | $1.56$
7 | $1.30$ | $-$ | $1.51$ | $1.42$ | $1.55$

Figure 13:  Left: Mean-$p_{t}$ hard components $\bar{p}_{thi}(n_{s})$ for
pions, kaons, protons and Lambdas from Tables 3 (open circles) and 4 (solid
dots) vs hard/soft ratio $x\nu$. The lines are described in the text. Right:
Mean values $\bar{P}_{ti}/n_{si}$ as defined by Eq. (V.3) for four hadron
species vs hard/soft ratio $x\nu$. The lines are described in the text. The
charged-kaon points are shifted to the right for visibility.

Figure 13 (left) shows hard-component mean values $\bar{p}_{thi}(n_{s})$ for
four hadron species inferred from data presented in Table 3 and from model
functions derived in Sec. III presented in Table 4. Open points are obtained
by simple integration of data hard components over their $p_{t}$ acceptances
and are therefore substantially biased. Solid points are obtained by
integration“on a continuum” of variable TCM hard components derived in Sec.
III. Lambda open points are high because the data lower bound is 0.65 GeV/c,
and proton open points are low because of a systematic excess below the mode,
both evident in Fig. 13 of Part I. Mean values generally increase
substantially with hadron mass, and that trend is consistent with measured
fragmentation functions for jets from $e^{+}$-$e^{-}$ collisions at the large
electron-positron collider LEP (see Fig. 7 of Ref. ppbpid ).

Meson mean values decrease with centrality whereas baryon mean values
increase. Those trends are a manifestation of hard-component centroid shifts
reported in Sec. III and are interpreted in Sec. IV.3 to cause certain PID
spectrum-ratio features. The trends (straight lines) for pions, kaons and
protons/Lambdas respectively are $1.345-0.8x\nu$, $1.425-0.35x\nu$ and
$1.50+0.5x\nu$ GeV/c.

###  V.3 Total $\bf P_{t}$ vs soft hadron yield $\bf n_{s}$

Especially in the context of small collision systems the relevance of an
overarching ensemble mean $\bar{p}_{ti}(n_{s})$ relating total $\bar{P}_{ti}$
to total charge $n_{chi}$ for hadron species $i$ can be questioned. The TCM
for $p$-Pb collisions describes a final state in which two components are
closely related but clearly distinct. The structure of Eq. (26) already relies
on two distinct mean values – $\bar{p}_{tsi}$ and $\bar{p}_{thi}$. A more-
instructive mean-value definition that illustrates the simplicity of TCM
structure has the form

$\displaystyle\frac{\bar{P}_{ti}}{n_{si}}$ $\displaystyle=$
$\displaystyle[1+\tilde{z}_{i}(n_{s})x(n_{s})\,\nu(n_{s})]\,\bar{p}_{ti}(n_{s}).$
$\displaystyle=$
$\displaystyle\bar{p}_{tsi}+\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})\,\bar{p}_{thi}(n_{s}),$

where $n_{si}=z_{si}n_{s}$. In what follows that relation is used to reexamine
$p_{t}$ mean-value trends for 5 TeV $p$-Pb collisions.

Figure 13 (right) shows transformed data (points) and fixed TCM references
(lines) evaluated according to Eq. (V.3). The solid points are
$\bar{p}_{ti}(n_{s})$ data from Fig. 4 of Ref. aliceppbpid  converted to the
format of Eq. (V.3) via its first line. The open points are derived from
$\tilde{z}_{i}(n_{s})$ values reported in Part I and from
$\bar{p}_{thi}(n_{s})$ values (solid points) from the left panel using
$x(n_{s})\nu(n_{s})$ values from Table 1. The lines are Eq. (V.3) (second
line) evaluated with fixed parameter values $\tilde{z}_{i}(n_{s})$ and
$\bar{p}_{thi}(n_{s})$ corresponding to centrality class $n=4$. The baryon
points fall below the fixed TCM reference for peripheral collisions and above
the reference for central collisions. That trend is due to a combination of
$\tilde{z}_{i}(n_{s})$ and $\bar{p}_{thi}(n_{s})$ (for baryons) increasing
with centrality. In contrast, the meson points show no significant deviation
from the fixed TCM reference.

Figure 14 (left) provides the reason for the last observation. What determines
the slope of the Eq. (V.3) $\bar{P}_{ti}/n_{si}$ trend on $x\nu$ is the
product $\tilde{z}_{i}(n_{s})\bar{p}_{thi}(n_{s})$. Those trends are shown in
Fig. 14 (left). Parameter $\tilde{z}_{i}(n_{s})$ always increases in magnitude
with centrality. When combined with baryon $\bar{p}_{thi}(n_{s})$ increasing
trends the baryon-product rate of increase is greater whereas combined with
the meson $\bar{p}_{thi}(n_{s})$ decreasing trends the meson-product variation
is negligible. The straight-line trends for pions through Lambdas respectively
are $0.93$, $3.6$, $7.6(1+x\nu)$ and $8.4(1+x\nu)$ GeV/c. The various trends
emphasize two issues: (a) jets (spectrum hard components) dominate $p_{t}$
production in high-energy nuclear collisions, and (b) the nonPID hard/soft
ratio $x\nu$ accurately describes the variation of jet production for four
hadron species.

Figure 14:  Left: The product $\tilde{z}_{i}(n_{s})\bar{p}_{thi}(n_{s})$ for
pions, kaons, protons and Lambdas with $\tilde{z}_{i}(n_{s})$ from Fig. 8 of
Part I and $\bar{p}_{thi}(n_{s})$ from Table 4 vs hard/soft ratio $x\nu$. The
lines are described in the text. Right: Published $\bar{p}_{ti}$ data from
Ref. aliceppbpid  Fig. 4 (solid points) and values inferred in the present
study (open circles) from the same published PID spectra from Fig. 1 vs
corrected charge density $\bar{\rho}_{0}=n_{ch}/\Delta\eta$ for five hadron
species. The curves are fixed TCM model Eq. (26) with parameters corresponding
to $n=4$ and $\alpha=0.0127$. The charged-kaon points are shifted to the right
for visibility.

###  V.4 Ensemble-mean $\bf\bar{p}_{t}$ TCM data description

Figure 14 (right) shows published PID $\bar{p}_{t}$ data from Ref. aliceppbpid
(solid points) and TCM results (open points and curves). The published error
bars from Ref. aliceppbpid  are included in this format. The TCM points and
lines in Figure 13 (right) are transformed to conventional ensemble mean
$\bar{p}_{ti}(n_{s})$ via Eq. (V.3) (first line). Comparison of the present
study with results of Ref. aliceppbpid  is as follows: The kaon result from
the present study (open circles) is statistically indistinguishable from the
published $K_{\text{S}}^{0}$ data (inverted solid triangles). Both trends are
slightly above the TCM reference for peripheral and slightly below the
reference for central collisions. While the product
$\tilde{z}_{i}(n_{s})\bar{p}_{thi}(n_{s})$ for mesons has negligible variation
(left panel) conventional mean $\bar{p}_{ti}(n_{s})$ includes the expression
$1+\tilde{z}_{i}(n_{s})x(n_{s})\,\nu(n_{s})$ in its denominator, the
increasing $\tilde{z}_{i}(n_{s})$ leading to an overall decrease for kaons and
pions.

The published Lambda data agree with the present study within their systematic
uncertainties as do the charged-kaon data. However, uncorrected proton data
fall well below the present results, presumably due to the detection
inefficiency identified in Part I. The baryon data rise substantially faster
than the TCM reference as expected from the left panel. The published pion
data agree well with the present study given the effective soft-component
value $\bar{p}_{tsi}\approx 0.40$ GeV/c. In Part I Sec. III it was
demonstrated that pion spectra require an explicit model for a resonance
contribution. Without that contribution $\bar{p}_{tsi}\approx 0.44$ would
clearly disagree with $\bar{p}_{t}$ data.

##  VI Systematic uncertainties

The present study (Part II) considers systematic uncertainties throughout the
text. Three issues are considered in this section: (a) The TCM parameter
inventory, its origins and uncertainties; (b) TCM prediction accuracy and
determination of data “fine structure” and (c) TCM physical interpretations vs
other spectrum models.

###  VI.1 TCM parameter inventory

Different elements of the TCM may arise from different collision systems that
are subsequently unified within an overall physical context. The full spectrum
TCM employed in the current study is defined in Eq. (II.1) (second line). Its
several elements are discussed below.

NonPID charge densities $\bar{\rho}_{s}$ and $\bar{\rho}_{h}$ (and hence
$p$-Pb “geometry” in terms of jet production) are based on 5 TeV $p$-Pb
$\bar{p}_{t}$ data analyzed in Ref. tommpt . The $p$-Pb $\bar{p}_{t}$ data are
described there by the TCM within data uncertainties out to
$\bar{\rho}_{0}\approx 115$ (in contrast to data extending only to
$\bar{\rho}_{0}\approx 45$ for Ref. aliceppbpid ). Indicators of the quality
of the $p$-Pb geometry description are the low-$y_{t}$ tails of the
$K^{0}_{\text{S}}$ hard components in Fig. 3 (left) and the high-$y_{t}$ tails
of the Lambda hard components in Fig. 5 (left). The hard components are only
rescaled by factors $\bar{\rho}_{h}$ and $\hat{H}_{0i}(\bar{y}_{t})$. The
latter are peak amplitudes all near 0.29 that vary by a few percent and only
with variation of the widths of the $\hat{H}_{0i}(y_{t})$ model functions.
Thus, the stability of the PID hard-component tail structure in the two cases
is indicative of the accuracy of the nonPID $\bar{\rho}_{h}$ estimates.

Coefficients $z_{si}(n_{s})$ and $z_{hi}(n_{s})$ are $n_{ch}$-dependent PID
fractions of soft and hard charge densities $\bar{\rho}_{s}$ and
$\bar{\rho}_{h}$. The analysis in Part I demonstrates that those coefficients
are accurately generated by simple functions of hard/soft ratio $x\nu$ with
linear dependence on hadron mass $m_{i}$, determined by only two parameters,
and on hadron species fractions $z_{0i}$ comparable to statistical-model
predictions. Those coefficients were demonstrated to be self-consistent at the
percent level in Part I (e.g. its Fig. 10, right).

Fixed TCM model functions $\hat{S}_{0i}(y_{t})$ and $\hat{H}_{0i}(y_{t})$ are
determined by comparisons with data. However, they are not fitted to
individual spectra or collision systems. In Table II of Part I the model
parameters for nonPID hadrons ($h$) are obtained from Ref. alicetomspec , a
systematic survey of $p_{t}$ spectra from $p$-$p$ collisions over a range of
collision energies. The interpolated nonPID values for 5 TeV $p$-$p$
collisions (not directly measured) are compatible with the PID values for 5
TeV $p$-Pb collisions reported in Table 2 of the present study. The TCM,
serving as a fixed reference, is thus compatible with an array of collision
systems and collision energies within data uncertainties.

In the present study PID TCM hard components are varied systematically to
accommodate data as reported in Sec. III. The resulting parameter variations
are simply described in terms of linear variation with hard/soft ratio $x\nu$
as in Fig. 4. Once again, the TCM does not rely on fits to individual spectra
for its definition. The accuracy of the description is made clear by Z-score
distributions reported in Sec. III.5.

###  VI.2 TCM prediction accuracy

The variable TCM adjusted to accommodate PID spectrum data can be used to
predict secondary data features, one example being ensemble $\bar{p}_{t}$
trends as in Sec. V. Hard-component means $\bar{p}_{thi}(n_{s})$ derived from
variable TCM model functions in Sec. III are combined in Eq. (V.3) with nonPID
$x\nu$ presented in Table 1 and $\tilde{z}_{i}(n_{s})$ trends from Part I to
produce PID $\bar{p}_{t}$ trends corresponding to spectra extrapolated over
$y_{t}\in[0,5]$. In Fig. 13 (right) the resulting entries (open points) are
compared with a fixed TCM reference (dotted lines) to obtain “fine structure”
of the $\bar{p}_{t}$ trends. Systematic deviations from the fixed TCM are
explained by the linear (on $x\nu$) trends in Fig. 14 (left) to reveal
$\bar{p}_{t}$ features at the percent level.

Another example is spectrum ratio trends as in Sec. IV. The TCM ratio model
defined in Eq. (19) is used to reproduce the centrality asymmetry that appears
in Fig. 8 and to trace its origins to translation of hard-component peak
centroids to lower or higher $y_{t}$ with increasing $n_{ch}$ that is revealed
in Sec. III. As with $\bar{p}_{t}$ data, sensitivity at the statistical limits
of spectrum data is possible via differential studies relative to the fixed
TCM as reference.

###  VI.3 TCM physical interpretation

An important aspect of systematic uncertainties is the interpretability of
model results. One may ask whether the model has a simple structure and well-
defined implementation so that different groups may obtain equivalent results
with the same model applied to equivalent data. An example of substantial
disagreement is blast-wave (BW) model analysis of PID spectra from 13 TeV
$p$-$p$ collisions in Refs. cleymans  and alicepppid . Equation (14) of Ref.
cleymans  defining the BW model is markedly different in form from Eq. (1) of
Ref. aliceppbpid  repeated from Ref. blastwave . As noted in Fig. 12 of Ref.
tommodeltests  BW model parameters $\langle\beta_{t}\rangle$ and $T_{kin}$
inferred in two cases differ by a factor 2 or more, providing one reason to
question the model’s validity.

In contrast, the TCM as defined in this and previous studies has a simple and
clear algebraic structure that makes possible differential extraction of all
information contained in particle data down to statistical uncertainties, as
demonstrated in Sec. III.5. The TCM soft component is arguably related to
longitudinal dissociation of projectile nucleons, a manifestation of an
eventwise PDF in terms of a soft hadron distribution. Its $p_{t}$ spectrum
(i.e. Lévy distribution) is consistent with thermal emission from an
incompletely thermalized source or a source structured due to a longitudinal
parton splitting cascade. Note that for exponent $1/n\rightarrow 0$ the Lévy
distribution goes to a Maxwell-Boltzmann distribution on $m_{t}$. The TCM hard
component is quantitatively comparable with a convolution of measured
fragmentation functions with a measured parton/jet energy spectrum fragevo .
The direct quantitative connection between spectrum structure at low $p_{t}$
(e.g. $\approx 0.2$ GeV/c) and the jet contribution at higher $p_{t}$ (e.g.
above 4 GeV/c), as described in Part I, Secs. II D, III B and IV B, seems to
preclude any role for hydrodynamic expansion in $p$-Pb collisions.

##  VII Discussion

This section compares text in Ref. aliceppbpid  relating to interpretation of
PID spectrum trends with some alternative viewpoints. Relevant issues include
full utilization of information carried by PID particle data and
acknowledgement of substantial jet contributions to nuclear collisions as
determined by earlier high-energy physics (HEP) research and as described by
conventional QCD principles.

###  VII.1 $\bf p$-Pb $\bf p_{t}$ spectra and the QGP narrative

Reference aliceppbpid  presents a certain narrative relating to QGP formation
in high-energy nuclear collisions in discussing its $p$-Pb PID spectrum
measurements. Quoted text below is labeled A, I, R, D, C for paper Abstract,
Introduction, Results, Discussion and Conclusions.

The importance of $p_{t}$ spectra for interpreting high-energy collision data
is acknowledged: (I) “The $p_{T}$ distributions and yields of particles of
different mass at low and intermediate momenta of less than a few GeV/c (where
the vast majority of particles [e.g. jet fragments] is produced), can provide
important information about the system created in high-energy hadron
reactions.” Also acknowledged is the importance of smaller systems for
interpreting results from heavy-ion collisions: (I) “The interpretation of
heavy-ion results depends crucially on the comparison with results from
smaller collision systems such as proton-proton (pp) or proton-nucleus (pA).”
However, the effectiveness of small systems for testing claims associated with
A-A collisions is effectively compromised: (I) [because the charge density in
$p$-A is high] “Therefore the assumption that final state dense matter effects
can be neglected in pA may no longer be valid.”

A broader context is established with presentation of the QGP scenario
believed by many to describe at least heavy-ion collisions: (I) “The bulk
matter created in high-energy nuclear reactions can be quantitatively
described in terms of hydrodynamic and statistical models. The initial hot and
dense partonic matter rapidly expands and cools down, ultimately undergoing a
transition to a hadron gas phase.” Presumed rapid expansion of hot and dense
partonic matter is then related to spectrum structure: (I) “This [collective
hydrodynamic flow] results in a characteristic dependence of the shape of the
transverse momentum ($p_{T}$) distribution on the particle mass….” (A) “The
transverse momentum distributions exhibit a hardening as a function of event
multiplicity, which is stronger for heavier particles. This behavior [in
$p$-Pb collisions] is similar to what has been observed in pp and Pb-Pb
collisions at the LHC.” (R) The [$p$-Pb] $p_{T}$ distributions show a clear
evolution, becoming harder as the multiplicity increases. The change is most
pronounced for protons and lambdas. They show an increase of the slope
[actually decrease of the slope] at low $p_{T}$, similar to the one observed
in heavy-ion collisions.” (R) “The latter [in Pb-Pb collisions] are generally
discussed in terms of collective flow or quark recombination.”

The possibility of hydrodynamic expansion affecting spectrum features then
leads to introduction of the blast-wave spectrum model: (D) “In heavy-ion
collisions, the flattening [i.e. decreased slope] of transverse momentum
distribution and its mass ordering find their natural explanation in the
collective radial expansion of the system. This picture can be tested in a
blast-wave framework with a simultaneous fit to all particles for each
multiplicity bin. This parametrization assumes a locally thermalized medium,
expanding collectively with a common velocity field and undergoing an
instantaneous common freeze-out.” (R) “Several parametrizations [of spectra]
have been tested, among which the blast-wave function…gives the best
description of the data over the full $p_{T}$ range” [emphasis added, fits are
independently applied to individual spectra]. (D) “In contrast with the
individual [blast-wave] fits discussed above, the simultaneous fit to all
particle species under consideration can provide insight on the (common)
kinetic freeze-out properties of the system.” “…the actual values of the fit
parameters depend substantially on the fit range [on $p_{t}$].” (D) “As can be
seen in Fig. 6, the parameters show a similar trend as the ones obtained in
Pb-Pb. Within the limitations of the blast-wave model, this observation is
consistent with the presence of radial flow in p-Pb collisions.” However, as
noted in Ref. hydro , in the presence of radial flow “…the particle yield is
thus depleted at low $p_{T}$. The heavier the particle and the larger the flow
velocity $v_{T}$, the larger the depletion.” Thus, precise measurements of PID
spectrum structure down to very low $p_{t}$ compared to a reference would be
essential to claim radial flow in $p$-Pb collisions. However, the fact that
for example proton $z_{si(n_{s})}$ values obtained near 0.15 GeV/c lead to
accurate predictions of jet fragment yields near 4 GeV/c (Part I, Sec. III B)
makes radial flow in $p$-Pb collisions quite unlikely.

Within quoted text above one encounters a tendency to emphasize similarities
between data trends in small and large systems with the implication that QGP
formation may then occur as well in the former. But when such comparisons lead
to counterintuitive results ad hoc explanations are introduced: (D) Comparing
$p$-Pb and Pb-Pb data “…at similar $dN_{ch}/d\eta$ the [blast-wave]
$\langle\beta_{T}\rangle$ values are significantly higher in $p$-Pb
collisions. While in Pb-Pb collisions high multiplicity events are obtained
through multiple soft interactions, in $p$-Pb collisions the high multiplicity
selection biases the sample towards harder [$p$-N] collisions. This could lead
to the larger $\langle\beta_{T}\rangle$ parameter obtained from the blast-wave
fits. Under the assumptions of a collective hydrodynamic expansion, a larger
radial velocity in $p$-Pb collisions has been suggested as a consequence of
stronger radial gradients.”

The conclusions of Ref. aliceppbpid  include (C) “These data represent a
crucial set of constraints for the modeling of proton-lead collisions at the
LHC. The transverse momentum distributions show a clear evolution with
multiplicity, similar to the pattern observed in high-energy pp and heavy-ion
collisions, where in the latter case the effect is usually attributed to
collective radial expansion.”

###  VII.2 Full utilization of spectrum information

Reference aliceppbpid  provides little significant quantitative information
actually derived from spectrum data. Spectra are described qualitatively, and
in multiple instances, as “hardening” with increasing $n_{ch}$, increasingly
so with increasing hadron mass. Spectrum data are said to provide a “crucial
set of constraints” but no specifics are provided. The word “constraints”
could be associated with information extracted from spectrum data, but
corresponding information is not made apparent. The article concludes with the
statement “The transverse momentum distributions show a clear evolution with
multiplicity…,” but nothing clear emerges from the paper’s presentation about
spectrum evolution or its causes.

In contrast, material presented in Part I (Part I) and Secs. III, IV and V of
the present study (Part II) provides exhaustive detail on spectrum structure
and its evolution with $n_{ch}$. Spectrum hard components, previously related
quantitatively to measured jet properties eeprd ; hardspec ; fragevo , are
isolated accurately at the level of spectrum-data statistical uncertainties.
That $p$-Pb $p_{t}$ spectra “harden” with increasing $n_{ch}$ is a simple
consequence of the quadratic relation between the nonjet (soft) hadron yield
and jet production ppprd , which in turn is reflected in the measured
systematics of jet cross sections for high-energy $p$-$p$ collisions jetspec2
. That the “hardening” trend increases with hadron mass again corresponds to
hard-component (i.e. jet fragmentation) systematics as reflected in the
$\tilde{z}_{i}(n_{s})$ trends presented in Part I Sec. V (e.g. its Fig. 8).

Substantial effort is extended in Ref. aliceppbpid  to extract information of
a sort in the form of spectrum and yield ratios (its Figs. 2, 3 and 5).
However, the physical mechanisms that determine such ratio evolution remain
obscure within that presentation. Referring to its Fig. 2, Ref. aliceppbpid
asserts that “The ratios $p/\pi$ and $\Lambda/K^{0}_{\text{S}}$ show a
significant enhancement [relative to what?] at intermediate $p_{T}\sim 3$
GeV/c, qualitatively reminiscent of that measured in Pb-Pb collisions
[emphasis added]. The latter are generally discussed in terms of collective
flow or quark recombination.” While the features are indeed qualitatively
similar for two collision systems the underlying mechanism(s) are not
substantively identified. Similarly, a “power-law” trend is inferred from
$n_{ch}$ dependence of the $p$-Pb $p/\pi$ spectrum ratio as in Fig. 3 of Ref.
aliceppbpid  and compared with a similar Pb-Pb trend. But the origin of the
trend in either system is not determined.

In contrast, Sec. IV provides a detailed explanation of spectrum ratio shapes
and evolution with $n_{ch}$ in terms of PID spectrum hard components. The
well-known $p/\pi$ peak near 3 GeV/c is a simple consequence of the separation
of corresponding pion and proton hard-component modes at $y_{t}\approx 2.46$
and 3.0 ($p_{t}\approx 0.80$ and 1.4 GeV/c) respectively combined with the
difference in their power-law tails, as demonstrated in Sec. IV.3. Centrality
dependence of the $p/\pi$ spectrum ratio reflects shifts to higher and lower
$y_{t}$ respectively of the proton and pion hard components that are clearly
demonstrated in Sec. III. The power-law trend in Fig. 3 of Ref. aliceppbpid
is a consequence of the Gaussian shape of spectrum hard components as
demonstrated in Fig. 12 of the present study. That there are similarities
between trends in $p$-Pb data and in Pb-Pb data suggests that Pb-Pb spectra be
analyzed in similar ways to extract jet-related hard components for direct
comparison. But that has already been done in Ref. pbpbpid .

Various ratio formats invoked by Ref. aliceppbpid  may respond to interesting
spectrum features relating to $p$-Pb centrality dependence but also tend to
discard essential information carried by spectrum data. Spectrum ratios alone
may not be interpretable within the conventional context. In contrast,
differential TCM analysis makes clear (at the percent level) what aspects of
collision dynamics produce those manifestations and quantitatively how those
features are related to fundamental QCD processes.

###  VII.3 Jet contributions to nuclear collisions

The narrative introduced in Ref. aliceppbpid  and summarized in Sec. VII.1
invokes argument from analogy to interpret $p$-Pb spectrum data in terms of a
popular interpretation of A-A data as demonstrating QGP formation (hot and
dense partonic matter) and transverse flows. Such arguments reverse a
convention adopted upon commencement of the relativistic heavy ion collider
(RHIC) program that inference of QGP formation in A-A collisions (e.g. Au-Au
collisions) based on certain data features should depend on absence of those
features in smaller collision systems (e.g. $d$-Au collisions daufinalstate )
serving as controls. With the appearance of certain features in $p$-Pb and
even $p$-$p$ collisions at the LHC the control concept has been abandoned and
interpretations based on QGP formation are favored. In retrospect, there was
lacking a full understanding of the jet physics of small collision systems, at
least among those pursuing research within the RHIC program.

The larger context is a conflict between two narratives – the popular flow/QGP
narrative and a QCD-based jet-production narrative – that compete to explain
certain critical data features. Arguments such as those summarized in Sec.
VII.1 tend to favor the flow/QGP narrative as a desired outcome and provide no
significant role for minimum-bias jet formation. For instance, it is argued
that [because the “charge density” in $p$-A is high] “Therefore the assumption
that final state dense matter effects [i.e. QGP formation] can be neglected in
pA may no longer be valid.” But as noted above the increasing $\bar{\rho}_{0}$
in $p$-Pb collisions is due primarily to the soft component, which shows no
sign of modification in any system, and secondarily to the jet-related hard
component whose minor evolution with $n_{ch}$ (see Sec. III) can be
interpreted in terms of well-known jet properties. The presence of the Pb
nucleus per se is apparently irrelevant. What jet (hard-component) evolution
is evident varies smoothly from charge densities below NSD to a ten-fold
increase.

The signature manifestation of QCD in high-energy hadron-hadron collisions is
jet production. The physics of jet production in hadronic collisions has been
established quantitatively over several decades by the HEP community. If small
systems are to serve as meaningful controls jet production within such systems
must be fully understood by those embarking on a search for QGP formation in
heavy-ion collisions.

Transport of longitudinal projectile momentum to transverse degrees of freedom
is a fundamental issue. The QGP scenario invokes particle (hadron or parton?)
energy loss (“stopping”) during the collision followed by partial
equilibration to a high-density medium and consequent generation of transverse
matter/energy gradients that drive radial expansion prior to final-state
hadron formation. That scenario, if valid, might result in certain
modifications of hadron spectra invoked as signals of QGP formation, for
instance spectrum “hardening” and mass dependence as described in Ref.
aliceppbpid .

In contrast, minimum-bias dijet production provides a mechanism for energy
transport from longitudinal to transverse degrees of freedom in elementary
collisions that is predicted by conventional QCD theory and has been
thoroughly investigated via HEP jet measurements. MB dijet production produces
PID spectrum manifestations somewhat similar to what is expected for QGP
formation: hardening of spectra with increasing $n_{ch}$ and with increasing
hadron mass. But those trends are quantitatively predicted by the quadratic
relation between jet production in elementary collisions and nonjet soft
hadron production (projectile dissociation – longitudinal hadronization of
low-$x$ gluons) – as in the TCM.

The discussion in Ref. aliceppbpid  includes the assertion “In heavy-ion
collisions, the flattening of transverse momentum distribution and its mass
ordering find their natural explanation in the collective radial expansion of
the system [emphasis added].” But in elementary collisions (at least) similar
spectrum trends “find their natural explanation” in QCD-predicted dijet
production as demonstrated by the present study and others fragevo ; jetspec2
; mbdijets . As noted, the comparison between small and large collision
systems represents, in effect, a competition between MB diject production and
QGP formation to explain pivotal data manifestations. If $p$-Pb data look like
Pb-Pb data in some sense does that mean $p$-Pb collisions exhibit physics
assumed for Pb-Pb collisions (what is implied in Ref. aliceppbpid ) or do Pb-
Pb collisions actually manifest a variant of simple QCD physics predicted for
elementary collisions?

To buttress claimed QGP formation in A-A collisions one would have to master
minimum-bias jet contributions to small collision systems and then demonstrate
that something truly novel occurs in larger collision systems. The ambiguity
between transport mechanisms and their manifestations described above may be
resolved by sufficiently differential analysis, as in the present study,
applied consistently to small and large collision systems.

###  VII.4 Baryons, strangeness and jets

The centrality trends of coefficients $z_{si}(n_{s})$, $z_{hi}(n_{s})$ and
hard/soft ratio $\tilde{z}_{i}(n_{s})$ in Part I Sec. V A reveal the dominant
role of jet production in the abundance of baryons and strangeness in high-
energy nuclear collisions. PID hard/soft ratio
$\bar{\rho}_{hi}/\bar{\rho}_{si}=\tilde{z}_{i}(n_{s})x\nu$ is the key
parameter. $p$-Pb nonPID ratio $x\nu=\bar{\rho}_{h}/\bar{\rho}_{s}$ ranges
from about 0.06 to 0.32 for all hadron species. $\tilde{z}_{i}(n_{s})$ however
increases with hadron mass from about 0.8 for pions to 6 for baryons. There is
apparently no sensitivity to strangeness per se. The $z_{xi}(n_{s})$ trends
are determined solely by hadron mass $m_{i}$ and overall abundances $z_{0i}$.

For most-central $p$-Pb collisions (at least for the spectrum data reported in
Ref. aliceppbpid ) and $x\nu\approx 0.32$ the hard/soft (jet/nonjet) ratio for
Lambdas (for instance) is $6.5\times 0.32\approx 2.1$ for integrated yields.
Above 1 GeV/c Lambda spectra are completely dominated by jet fragments. For
kaons and central collisions the yield ratio is $2.5\times 0.32\approx 0.8$,
and again spectra above about 1 GeV/c are dominated by jet fragments. Even for
pions the jet contribution plays a dominant role above 2 GeV/c.

Minimum-bias jets appearing near midrapidity in high-energy N-N collisions
have their origin in low-$x$ gluons as part of eventwise PDFs. The results in
Part I Sec. V A (among others) indicate that large-angle scattered gluons in
the form of MB dijets produce large amounts of $p_{t}$, strangeness and
baryons near midrapidity. Within the flow/QGP narrative those same indicators
may be associated not with jets but with spectrum “hardening” (due to radial
flow), parton recombination and strangeness enhancement in a dense QCD medium
or QGP. To resolve that conflict requires detailed differential spectrum
analysis as reported in the present study and Part I.

##  VIII Summary

This article presents Part II of a two-part study of identified-hadron (PID)
$p_{t}$ spectra from 5 TeV $p$-Pb collisions. In Part I a previously-reported
preliminary PID spectrum two-component (soft + hard) model (TCM) was extended
in several ways, including incorporation of a resonance model for pions,
correction of proton spectra for detection inefficiency and determination of
TCM coefficients $z_{si}(n_{s})$ and $z_{hi}(n_{s})$ directly from spectra.
Minimum-bias data hard components $H_{i}(y_{t},n_{s})$ were extracted from
spectrum data with an improved method, and their evolution with $p$-Pb
centrality was determined relative to a fixed PID TCM serving as a reference.

Whereas the analysis described in Part I is based on the TCM as a fixed model,
with data deviations suggesting novel physics, the analysis described in this
article (Part II) includes adjustment of the TCM to describe all data
accurately within their statistical uncertainties as determined by standard
statistical measures (Z-scores). No physically-significant data-model
deviations remain.

Based on the differential study in Part I it is evident that different model
parametrizations are required for meson and baryon hard components. However,
the models for strange and nonstrange baryons are identical. The small-but-
significant model differences between strange and nonstrange mesons may be
entirely due to the substantial mass difference between pions and kaons. The
parameter variations can be expressed as simple linear functions of TCM
hard/soft (jet/nonjet) ratio $x\nu$.

Given its precise representation of PID spectrum data for $p$-Pb collisions,
including not only actual data acceptances but accurate extrapolations over
the interval $y_{t}\in[0,5]$ ($p_{t}\in[0,10]$ GeV/c), the variable TCM is
then applied to a study of PID spectrum and yield ratios and ensemble-mean
$\bar{p}_{t}$ trends. Variation of $p_{t}$ spectrum ratios with charge
multiplicity $n_{ch}$ is predicted by the TCM. Certain notable features of PID
spectrum ratios are thereby explained, including the asymmetry in centrality
evolution centered (for proton/pion) near $y_{t}$ = 3.1 ($p_{t}\approx 1.55$
GeV/c) and the prominent peak near 3 GeV/c. The former arises from shifts of
jet-related hard components with increasing $n_{ch}$ to higher $y_{t}$ for
baryons in the numerator vs shifts to lower $y_{t}$ for mesons in the
denominator. The approximate power-law dependence on $n_{ch}$ of spectrum
ratios for specific $p_{t}$ values is simply due to the approximate Gaussian
shape of spectrum hard components. A prominent peak near 3 GeV/c arises
because of two factors: pion and proton hard-component peak modes are well
separated on $y_{t}$, and the pion peak width is substantially greater than
the proton width. The Lambda-to-kaon ratio features are similarly explained.

PID yield ratios are likewise predicted by the TCM in terms of fractions
$z_{0i}$ that should be directly comparable with the statistical model of
hadron production. A notable consequence of the TCM description is that total-
yield ratios should be independent of $n_{ch}$ or $p$-Pb centrality.
Significant deviations from that prediction suggest possible biases in the
processing of particle data.

PID ensemble-mean $\bar{p}_{t}$ trends are similarly predicted by the TCM.
Comparisons of PID $\bar{p}_{t}$ data to a fixed TCM reference reveal subtle
details that are nevertheless fully explained by variation with $n_{ch}$ of
hard/soft ratios $\tilde{z}_{i}(n_{s})\equiv z_{hi}(n_{s})/z_{si}(n_{s})$ as
well as hard-component peak centroids and shapes as established in the present
study.

Several conclusions can be drawn from the combined study reported in Parts I
and II:

(a) Variations of strangeness production (enhancement and suppression) and
baryon abundance are conventionally associated with possible quark-gluon
plasma (QGP) formation. The present study reveals that strong variation of
strangeness and baryon number with $p$-Pb collision $n_{ch}$ is dominated by
jet production trends, and jet-related strangeness and baryon production in
turn depends only on hadron mass. Thus, strangeness per se and baryon identity
(not net baryon number) per se do not significantly determine hadron abundance
in smaller collision systems. Hadron mass is the determining factor.

(b) The only “centrality” (i.e. $n_{ch}$) dependent aspect of spectrum
structure relative to a fixed PID TCM reference is evolution of hard-component
shapes, differently for meson and baryon hard components. The basics of jet
formation in the context of a QCD description of scattered-parton production
and fragmentation suggest that the shape evolution arises mainly from control
of event-wise parton distribution functions (PDF) by the choice of average
event charge multiplicity $n_{ch}$.

(c) Evolution of certain $p_{t}$ spectrum features with A-B centrality (e.g.
increased spectrum “hardening” – to a greater extent for more-massive hadrons)
has been attributed to radial flow. The present study reports a TCM based only
on longitudinal projectile-nucleon dissociation and transverse jet production
that provides exhaustive descriptions of four hadron species for seven
centrality classes of 5 TeV $p$-Pb collisions. The TCM is thus sufficient as a
spectrum model. An earlier study of $p$-$p$ spectrum data not based on a
priori assumptions demonstrated that the TCM emerges as a necessary model. No
other model provides equivalent accuracy and universality. Since all
information carried by spectrum data is fully represented by the TCM one may
conclude that the PID $p$-Pb spectrum data addressed in the present study
provide no significant evidence for radial flow.

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