#  A two-component model of hadron production applied to $\bf p_{t}$ spectra
from 5 TeV and 13 TeV $\bf p$-$\bf p$ collisions at the large hadron collider

Thomas A. Trainor  CENPA 354290, University of Washington, Seattle, Washington
98195

###### Abstract

The ALICE collaboration at the large hadron collider (LHC) recently reported
high-statistics $p_{t}$ spectrum data from 5 TeV and 13 TeV $p$-$p$
collisions. Particle data for each energy were partitioned into event classes
based on the total yields within two disjoint pseudorapidity $\eta$ intervals
denoted by acronyms V0M and SPD. For each energy the spectra resulting from
the two selection methods were then compared to a minimum-bias INEL $>0$
average over the entire event population. The nominal goal was determination
of the role of jets in high-multiplicity $p$-$p$ collisions and especially the
jet contribution to the low-$p_{t}$ parts of spectra. A related motivation was
response to recent claims of “collective” behavior and other nominal
indicators of quark-gluon plasma (QGP) formation in small collision systems.
In the present study a two-component (soft + hard) model (TCM) of hadron
production in $p$-$p$ collisions is applied to the ALICE spectrum data. As in
previous TCM studies of a variety of A-B collision systems the jet and nonjet
contributions to $p$-$p$ spectra are accurately separated over the entire
$p_{t}$ acceptance. Distinction is maintained among spectrum normalizations,
jet contributions to spectra and systematic biases resulting from V0M and SPD
event selection. The statistical significance of data-model differences is
established. The effect of spherocity (azimuthal asymmetry measure nominally
sensitive to jet production) on ensemble-mean $p_{t}$ vs event multiplicity
$n_{ch}$ is investigated and found to have little relation to jet production.
The general results of the TCM analysis are as expected from a conventional
QCD description of jet production in $p$-$p$ collisions.

###### pacs:

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

††preprint: Version 2.0

##  I Introduction

Claims of quark-gluon plasma (QGP) formation in Au-Au collisions at the
relativistic heavy ion collider (RHIC) and subsequently in Pb-Pb collisions at
the large hadron collider (LHC) were based on certain data features initially
seen as unique to more-central A-A collisions and not appearing in small
asymmetric A-B control systems such as $d$-Au or $p$-Pb where QCD theory
suggested that QGP formation should be unlikely. However, in recent years
similar features have been observed in LHC data for $p$-Pb collisions and
high-charge-multiplicity $p$-$p$ collisions and have been interpreted as
evidence for QGP formation in small collision systems ppcms ; dusling . But
the recent LHC results could also be interpreted to indicate that data
features conventionally associated with QGP formation may result from
unexceptional QCD processes.

The ALICE collaboration recently published a comprehensive high-statistics
study of $p_{t}$ spectra from 5 TeV and 13 TeV $p$-$p$ collisions alicenewspec
. The analysis employs two methods to sort collision events into ten
multiplicity classes each and application of spherocity $S_{0}$, a measure of
the azimuthal nonuniformity of distributed $\vec{p}_{t}(\phi)$, to estimate
the “jettiness” of events. Several methods are applied to determine variation
of spectrum shape with charge multiplicity, event-selection method and
spherocity.

The study reported in Ref. alicenewspec  is motivated in part by claimed
observation in $p$-$p$ and $p$-Pb collisions of evidence for radial and
elliptic flow (“collectivity”) aliflows1 ; aliflows2  as well as strangeness
enhancement alistrange  similar to that observed in more-central $A$-$A$
collisions and attributed there to QGP formation. Observation of such effects
in low-density systems occupying small space-time volumes runs counter to
initial theoretical expectations concerning QGP formation. The study seeks to
understand hadron production associated with jets in relation to soft particle
production: “The aim of this study is to investigate the importance of jets in
high-multiplicity pp collisions and their contribution to charged-particle
production at low $p_{T}$.”

The phenomenology of high-energy $p$-$p$ collision data serves as an essential
reference for high-energy $p$-$A$ and $A$-$A$ collisions, specifically
regarding claims of novel physical mechanisms such as QGP formation perfect
or possible manifestations of hydrodynamic flows even in small collision
systems ppflow ; moreppflow . One can formulate a set of critical questions
addressed to available $p$-$p$ data: (a) What is the evidence for or against
azimuthally symmetric radial flow, and for or against elliptic flow and
“higher harmonics” as manifestations of azimuthal asymmetry? (b) Are nominally
flow-related azimuthal features certainly disjoint from jet production? (c)
Are jet contributions to spectra significantly modified by a dense medium
(i.e. QGP)? (d) Does the charge-multiplicity ($n_{ch}$) dependence of certain
data features reflect the onset of such a medium with increasing particle
density? If the QGP scenario is valid then systematic behavior of various data
features should be synchronized with emergence of a common underlying dense
medium. Are comprehensive data trends consistent with such expected
synchronization?

A two-component (soft + hard) model (TCM) of hadron production near mid-
rapidity in A-B collisions was initially derived from the charge-multiplicity
$n_{ch}$ dependence of $p_{t}$ spectra from 200 GeV $p$-$p$ collisions ppprd .
The $n_{ch}$ dependence of yields, spectra and two-particle correlations has
played a key role in establishing (a) the nature of hadron production
mechanisms in $p$-$p$ collisions and (b) that the TCM hard component of
$p_{t}$ spectra is quantitatively consistent with predictions based on event-
wise reconstructed jets fragevo ; jetspec ; jetspec2 . The question of
recently-claimed collectivity or flows in small ($p$-$p$ and $p$-$A$) systems
has been addressed in terms of the resolved TCM soft and hard components and
evidence (or not) for radial flow in differential studies of $p_{t}$ spectra
hardspec ; ppquad ; ppbpid .

Reference alicetomspec  reported previous TCM analysis of 13 TeV $p$-$p$
spectra that serves as a precursor to the present study (see App. A). Then-
available spectrum data alicespec  were presented only as ratios to minimum-
bias spectra and only for a limited $n_{ch}$ range. The present study extends
those results utilizing both 5 TeV and 13 TeV data. Spectra for the two
energies and for two methods of event selection, V0M and SPD, are decomposed
into soft and hard components for ten event multiplicity classes each.
Spectrum biases resulting from event selection methods are evaluated and
compared. The quality of the TCM description is determined via data-model
differences (not ratios) compared to statistical uncertainties (Z-scores) as a
significance test. A TCM for ensemble-mean $p_{t}$ or $\bar{p}_{t}$ is defined
and applied to $\bar{p}_{t}$ vs $n_{ch}$ data from Ref. alicenewspec .
Evolution of the $\bar{p}_{t}$($n_{ch}$) trend with spherocity $S_{0}$ is
examined in detail – especially the relation of $S_{0}$ to dijet production.
An ironic result emerges.

The TCM serves as an accurate reference for A-B collision systems that is not
derived from fits to individual spectra. The TCM is required to describe
diverse data formats applied to a broad array of collision systems self-
consistently. It precisely separates jet and nonjet data contributions,
greatly facilitating and simplifying data interpretation. Data-TCM deviations
may reveal systematic data biases as in the present study or identify new
physics beyond conventional models as in Ref. ppquad .

This article is arranged as follows: Section II summarizes $p$-$p$ spectrum
data, methods and conclusions reported in Ref. alicenewspec . Section III
describes the TCM for $p$-$p$ $p_{t}$ spectra. Section IV summarizes selection
biases resulting from two event-selection criteria and their evolution with
event multiplicity $n_{ch}$. Section V reviews some $p_{t}$ spectrum shape
measures. Section VI describes the TCM for ensemble $\bar{p}_{t}$ and reviews
results from Ref. alicenewspec  for evolution of $\bar{p}_{t}$($n_{ch}$)
trends with varying spherocity $S_{0}$. Section VII discusses systematic
uncertainties. Sections VIII and IX present discussion and summary.

##  II 5 and 13 T$\bf e$V $\bf p$-$\bf p$ $\bf p_{t}$ spectrum data

Reference alicenewspec  reports $p_{t}$ spectra from 5 and 13 TeV $p$-$p$
collisions for two event selection methods (V0M and SPD) and for ten charge-
multiplicity classes each. For each energy the same minimum-bias (INEL $>0$)
event ensemble is effectively sorted into multiplicity classes in two ways.
The $p_{t}$ acceptance is $p_{t}\in[0.15,20]$ GeV/c. The angular acceptance is
$2\pi$ on azimuth and $|\eta|\leq 0.8$ on pseudorapidity. The total event
numbers are 105 and 60 million for 5 and 13 TeV respectively. The basic
$p_{t}$ spectra are further analyzed via several methods. As noted, the stated
main goal of the study is to understand the role of jets in high-multiplicity
$p$-$p$ collisions.

###  II.1 Motivation and strategy

The context presented for the spectrum analysis reported in Ref. alicenewspec
is claimed observation at the LHC of collectivity – i.e. radial aliflows1 ;
aliceppbpid  and anisotropic (e.g. elliptic, higher harmonic) aliflows2  flows
– and strangeness enhancement alistrange  in $p$-$p$ and $p$-Pb collisions
whereas those phenomena had been designated as indicators of QGP formation
only in high-density $A$-$A$ collision systems. A variety of models based on
hydrodynamics, string percolation, multiparton interactions or fragmentation
of saturated gluon states are “…able to describe…qualitatively well…some
features of data.” However, concerns have been expressed about interpretations
of data from small collision systems in terms of QGP formation without a more-
rigorous examination of data and models thoughts .

It is asserted that a $p_{t}$ spectrum “carries information of the dynamics of
soft and hard interactions.” Reference is made to three $p_{t}$ intervals:
$p_{t}>10$ GeV/c is said to be “quantitatively well described by perturbative
QCD (pQCD) calculations.”111A pQCD (i.e. jet) description is quantitatively
consistent with the $p$-$p$ $p_{t}$ spectrum hard component down to 0.5 GeV/c
fragevo . Below that limit one must “resort to phenomenological QCD inspired
models [i.e. Monte Carlo models].” Novel effects claimed for $p$-$p$ and
$p$-Pb collisions are said to appear in $p_{t}<2$ GeV/c and $p_{t}\in[2,10]$
GeV/c. Reference alicenewspec  asserts that “The present paper reports a novel
multi-differential analysis aimed at understanding charged-particle production
associated to partonic scatterings with large momentum transfer and their
possible correlations with soft particle production.” In essence, Ref.
alicenewspec  poses the question: what is the jet contribution to hadron
spectra at low $p_{t}$?

###  II.2 $\bf p$-$\bf p$ $\bf p_{t}$ spectrum data

Figures 1 and 2 show spectra (points) for 5 TeV and 13 TeV respectively.
Panels (a) and (c) show spectra for sorting criteria V0M and SPD respectively
multiplied by successive powers of ten from lowest to highest multiplicity in
a conventional log-log plot format. The solid curves represent the TCM
described in Sec. III. The TCM is not derived from fits to individual spectra.
Panels (b) and (d) show data/model ratios based on the TCM. Line types for the
four highest-$n_{ch}$ classes vary as solid, dashed, dotted and dash-dotted.
That convention is applied consistently in what follows unless explicitly
noted. In those ratios certain “noise” components appear as common to multiple
spectra. It is possible that those features arise from efficiency corrections
generated by a Monte Carlo simulation with a more-limited number of events.
Power-law fits to spectra above 6 GeV/c are used to infer a power-law exponent
$n$ which is observed to decrease in magnitude with increasing $n_{ch}$ (but
see Sec. V.1). Note that spectra as plotted in Ref. alicenewspec  and Figs. 1
and 2 (a,c) below are in the form $d^{2}n_{ch}/dp_{t}d\eta$ whereas
$\bar{\rho}_{0}(p_{t};n_{ch})$ corresponding to $\bar{\rho}_{0}(y_{t};n_{ch})$
as defined in Eq. (III.1) includes an additional factor $p_{t}$ in its
denominator.

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

Figure 1:  Left: $p_{t}$ spectra from ten (V0M) or nine (SPD) multiplicity
classes of 5 TeV $p$-$p$ collisions for V0M (a) and SPD (c) event selection.
Right: data/TCM spectrum ratios for data in the left panels for V0M (b) and
SPD (d) event selection.

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

Figure 2:  Left: $p_{t}$ spectra from ten multiplicity classes of 13 TeV
$p$-$p$ collisions for V0M (a) and SPD (c) event selection. Right: data/TCM
spectrum ratios for data in the left panels for V0M (b) and SPD (d) event
selection.

###  II.3 Ensemble-mean $\bf\bar{p}_{t}$ vs $\bf n_{ch}$ and spherocity

Spectra are also sorted according to spherocity $S_{0}\in[0,1]$ or
$S_{0}(\%)\in[0\%,100\%]$ [see Eq. (14)], said to be a measure of the
“jettiness” of the event-wise $\vec{p}_{t}(\phi)$ azimuth distribution, where
$S_{0}\approx 1$ or 100% is associated with near-isotropic events. For ten
classes of spherocity $S_{0}$ ensemble-mean $\bar{p}_{t}$ vs multiplicity
$n_{ch}$ trends for 13 TeV data are evaluated. Event multiplicity classes are
defined within $|\eta|<0.8$ (i.e. SPD selection). Reference alicenewspec
states: “Since the goal of the present study is to separate jet events from
isotropic ones, we study different spherocity classes for a given multiplicity
value.” “…the most jet-like and [most] isotropic events will be referred to as
0-10% and 90-100% spherocity classes, respectively.”

###  II.4 Summary and conclusions

The abstract of Ref. alicenewspec  asserts that $p_{t}$ spectra exhibit little
energy dependence (between 5 and 13 TeV), and the high-$p_{t}$ tails of
spectra increase faster than linearly with event multiplicity $n_{ch}$.
Regarding the spherocity study “For low- (high-) spherocity events,
corresponding to jet-like (isotropic) events, the average $p_{T}$ is higher
(smaller) than that measured in INEL $>0$ pp collisions.”

Although it is reported that ensemble-mean $\bar{p}_{t}$ increases with
decreasing spherocity (expected to indicate increased jettiness) the observed
shape of $\bar{p}_{t}$ vs $n_{ch}$ doesn’t change significantly with
spherocity (contradicting the expectation, see Sec. VI.3). Spherocity results
are said to “illustrate the difficulties for the [Monte Carlo] models to
describe different observables once they are differentially analyzed as a
function of several variables.”

The summary states that “For a fixed center-of-mass energy, particle
production above $p_{T}=0.5$ GeV/c exhibits a remarkable multiplicity
dependence. Namely, for transverse momenta below 0.5 GeV/c, the ratio of the
multiplicity dependent spectra to those for INEL$>0$ pp collisions is rather
constant, and for higher momenta, it shows a significant $p_{t}$ dependence.
The behavior observed for each of the two multiplicity estimators are
consistent within the $\langle dN_{ch}/d\eta\rangle$ interval defined by the
V0M multiplicity estimator, which gives a $\langle dN_{ch}/d\eta\rangle$ reach
of $\sim 25$. For the highest V0M multiplicity class, the ratio increases
going from $p_{T}=0.5$ GeV/c up to $p_{T}\approx 4$ GeV/c, then for higher
$p_{T}$, it shows a smaller increase.” Those qualitative observations contrast
with highly differential and quantitative TCM results from the present study
as presented below in Secs. III \- VI.

##  III $\bf p$-$\bf p$ $\bf p_{t}$ spectrum TCM

The $p_{t}$ spectrum TCM, first reported for 200 GeV $p$-$p$ collisions in
Ref. ppprd , is basically consistent with the TCM first reported in Ref.
pancheri  in response to UA1 “minijets” from the CERN S$p\bar{p}$S. The TCM
provides an accurate description of yields, spectra and two-particle
correlations for A-B collision systems based on linear superposition of
$N$-$N$ or parton-parton collisions. In general, the TCM serves as a
predictive reference for any collision system. Deviations from the TCM then
provide systematic and quantitative information on details of collision
mechanisms. In this section the spectrum TCM is reviewed and then applied to 5
TeV and 13 TeV $p_{t}$ spectra for two event selection methods from Ref.
alicenewspec .

###  III.1 $\bf p_{t}$ spectrum TCM for unidentified hadrons

The $p_{t}$ or $y_{t}$ spectrum TCM is by definition the sum of soft and hard
components with details inferred from data (e.g. Ref. ppprd ). For $p$-$p$
collisions

$\displaystyle\bar{\rho}_{0}(y_{t};n_{ch})$ $\displaystyle\approx$
$\displaystyle\frac{d^{2}n_{ch}}{y_{t}dy_{t}d\eta}$ $\displaystyle\approx$
$\displaystyle\bar{\rho}_{s}(n_{ch})\hat{S}_{0}(y_{t})+\bar{\rho}_{h}(n_{ch})\hat{H}_{0}(y_{t}),$

where $n_{ch}$ is an event-class index and factorization of the dependences on
$y_{t}$ and $n_{ch}$ is a central feature of the spectrum TCM inferred from
200 GeV $p$-$p$ spectrum data in Ref. ppprd . The motivation for transverse
rapidity $y_{ti}\equiv\ln[(p_{t}+m_{ti})/m_{i}]$ (applied to hadron species
$i$) is explained in Sec. III.2. The $y_{t}$ integral of Eq. (III.1) is
$\bar{\rho}_{0}\equiv n_{ch}/\Delta\eta=\bar{\rho}_{s}+\bar{\rho}_{h}$, a sum
of soft and hard charge densities. $\hat{S}_{0}(y_{t})$ and
$\hat{H}_{0}(y_{t})$ are unit-normal model functions independent of $n_{ch}$.
The centrally-important relation
$\bar{\rho}_{h}\approx\alpha\bar{\rho}_{s}^{2}$ with $\alpha=O(0.01)$ is
inferred from $p$-$p$ spectrum data ppprd ; ppquad ; alicetomspec .
$\bar{\rho}_{s}$ is then obtained from measured $\bar{\rho}_{0}$ as the root
of the quadratic equation
$\bar{\rho}_{0}=\bar{\rho}_{s}+\alpha\bar{\rho}_{s}^{2}$ with $\alpha$
determined by an energy trend derived from $p$-$p$ spectrum data covering a
large energy interval alicetomspec . It is important to distinguish TCM model
elements from spectrum data soft and hard components. It is useful to recall
that $y_{t}$ values 1, 2, 3, 4 and 5 are approximately equivalent to $p_{t}$
values 0.15, 0.5, 1.4, 3.8 and 10 GeV/c.

###  III.2 $\bf p_{t}$ spectrum TCM model functions

The $p$-$p$ $p_{t}$ spectrum soft component is most efficiently described on
transverse mass $m_{t}$ whereas the spectrum hard component is most
efficiently described on transverse rapidity $y_{t}$. The spectrum TCM thus
requires a heterogeneous set of variables for its simplest definition. The
components can be easily transformed from one variable to the other by
Jacobian factors defined below.

Given spectrum data in the form of Eq. (III.1) the unit-normal spectrum soft-
component model $\hat{S}_{0}(y_{t})$ is defined as the asymptotic limit of
data spectra normalized in the form
$X(y_{t})\equiv\bar{\rho}_{0}(y_{t};n_{ch})/\bar{\rho}_{s}$ as $n_{ch}$ goes
to zero. Hard components of data spectra are then defined as complementary to
soft components, with the explicit form

$\displaystyle
Y(y_{t})\equiv\frac{1}{\alpha\bar{\rho}_{s}}\left[X(y_{t})-\hat{S}_{0}(y_{t})\right],$
(2)

directly comparable with TCM model function $\hat{H}_{0}(y_{t})$.

The data soft component for a specific hadron species $i$ is typically well
described by a Lévy distribution on $m_{ti}=\sqrt{p_{t}^{2}+m_{i}^{2}}$. The
unit-integral soft-component model is

$\displaystyle\hat{S}_{0i}(m_{ti})$ $\displaystyle=$
$\displaystyle\frac{A_{i}}{[1+(m_{ti}-m_{i})/n_{i}T_{i}]^{n_{i}}},$ (3)

where $m_{ti}$ is the transverse mass-energy for hadron species $i$ with mass
$m_{i}$, $n_{i}$ is the Lévy exponent, $T_{i}$ is the slope parameter and
coefficient $A_{i}$ is determined by the unit-normal condition. Model
parameters $(T_{i},n_{i})$ for several species of identified hadrons have been
inferred from 5 TeV $p$-Pb spectrum data as described in Ref. ppbpid . A soft-
component model function for unidentified hadrons can be defined as the
weighted sum

$\displaystyle\hat{S}_{0}(m_{t})$ $\displaystyle=$
$\displaystyle\sum_{i}z_{0i}\hat{S}_{0i}(m_{ti}),$ (4)

where the weights for charged hadrons follow $\sum_{i}z_{0i}=1$. In the
present context model function $\hat{S}_{0}(m_{t})$ should not be confused
with spherocity $S_{0}$ introduced in Ref. alicenewspec .

The unit-normal hard-component model is a Gaussian on
$y_{t\pi}\equiv\ln[(p_{t}+m_{t\pi})/m_{\pi}]$ (as explained below) with
exponential (on $y_{t}$) or power-law (on $p_{t}$) tail for larger $y_{t}$

$\displaystyle\hat{H}_{0}(y_{t})$ $\displaystyle\approx$ $\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},$

where the transition from Gaussian to exponential on $y_{t}$ is determined by
slope matching fragevo . The $\hat{H}_{0}$ tail density on $p_{t}$ varies
approximately as power law $1/p_{t}^{q+1.8}\approx 1/p_{t}^{n}$. Coefficient
$A~{}(\approx 0.3)$ is determined by the unit-normal condition. Model
parameters $(\bar{y}_{t},\sigma_{y_{t}},q)$ are derived as described in App. A
except as noted in the main text.

All spectra are plotted vs pion rapidity $y_{t\pi}$ with pion mass assumed.
The motivation is comparison of spectrum hard components demonstrated to arise
from a common underlying jet spectrum on $p_{t}$ fragevo , in which case
$y_{t\pi}$ serves simply as a logarithmic measure of hadron $p_{t}$ with well-
defined zero. $\hat{S}_{0}(m_{t})$ in Eq. (4) is transformed to $y_{t\pi}$ via
the Jacobian factor $m_{t\pi}p_{t}/y_{t\pi}$ to form $\hat{S}_{0}(y_{t\pi})$
for unidentified hadrons. $\hat{H}_{0}(y_{t})$ in Eq. (III.2) is always
defined on $y_{t\pi}$ as noted. In general, plotting spectra on a logarithmic
rapidity variable provides improved access to important spectrum structure in
the low-$p_{t}$ interval where the majority of jet fragments appear.

###  III.3 $\bf p$-$\bf p$ $\bf p_{t}$ spectrum data

Figures 1 and 2 (a,c) show the general relation between the TCM (solid) and
ALICE data (points). The TCM is not the result of fits to individual spectra.
The curves actually represent predictions derived from a self-consistent
description of $p$-$p$ spectra covering the energy interval 17 GeV to 13 TeV
(Ref. alicetomspec  and App. A). The data/TCM ratios in (b,d) provide
important information on biases resulting from V0M and SPD event sorting
methods.

The TCM format of Figs. 3 and 4 then provides a more-differential
decomposition of spectrum data into soft and hard components. Panels (a,c)
show full data spectra (thin solid) in the normalized form $X(y_{t})$ defined
above that are directly comparable with soft-component model
$\hat{S}_{0}(y_{t})$ (bold dashed). Below 0.5 GeV/c ($y_{t}\approx 2$) the
data curves closely follow the model. The same $\hat{S}_{0}(y_{t})$ model is
used for both event-selection methods.

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

Figure 3:  Left: Normalized $y_{t}$ spectra in the form $X(y_{t})$ from ten
(V0M) or nine (SPD) multiplicity classes of 5 TeV $p$-$p$ collisions for V0M
(a) and SPD (c) event selection. Right: Normalized spectrum hard components in
the form $Y(y_{t})$ for data in the left panels for V0M (b) and SPD (d) event
selection. The bold dashed curves are TCM model functions.

Panels (b,d) show inferred data hard components $Y(y_{t})$ defined by Eq. (2)
(thin, several line styles) compared to TCM hard-component model
$\hat{H}_{0}(y_{t})$ (bold dashed). Deviations from $\hat{H}_{0}(y_{t})$ below
$y_{t}=2$ appear in every $p$-$p$ collision system (e.g. 200 GeV as reported
in Ref. ppprd ). The horizontal dotted lines provide a check on proper
normalization of hard-component model $\hat{H}_{0}(y_{t})$ The data hard
component for the lowest multiplicity class is not shown because there is in
effect very little jet contribution to those events due to strong selection
bias. Note that full spectra in panels (a,c) for the lowest $n_{ch}$ class are
approximately consistent with $\hat{S}_{0}(y_{t})$. The same
$\hat{H}_{0}(y_{t})$ model is used for both event-selection methods.

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

Figure 4:  Left: Normalized $y_{t}$ spectra in the form $X(y_{t})$ from ten
multiplicity classes of 13 TeV $p$-$p$ collisions for V0M (a) and SPD (c)
event selection. Right: Normalized spectrum hard components in the form
$Y(y_{t})$ for data in the left panels for V0M (b) and SPD (d) event
selection. The bold dashed curves are TCM model functions.

Although Ref. alicenewspec  states that $p$-$p$ $p_{t}$ spectra show “little
energy dependence” both soft and hard components exhibit significant energy
dependence as previously reported in Ref. alicetomspec . The TCM parameters
$n$ (soft-component exponent) and $q$ (hard-component exponent) vary
systematically with $\log(\sqrt{s})$. The variation is apparent as shifts of
$\hat{S}_{0}(y_{t})$ and $\hat{H}_{0}(y_{t})$ intercepts on $y_{t}$ (at plot
lower bounds) to larger values with increasing energy. The former may be
related to increasing depth of the longitudinal splitting cascade on proton
momentum fraction $x$ with increasing energy alicetomspec , and the latter is
certainly related to expected evolution of the underlying jet spectrum with
increasing $p$-$p$ collision energy jetspec2 .

###  III.4 Spectrum TCM parameter summary

Table 1 presents TCM parameters for 5 TeV and 13 TeV $p$-$p$ collisions.
Entries are grouped as soft-component parameters $(T,n)$, hard-component
parameters $(\bar{y}_{t},\sigma_{y_{t}},q)$, hard/soft ratio parameter
$\alpha$ and NSD (non-single-diffractive) soft density $\bar{\rho}_{sNSD}$.
For unidentified hadrons soft component $\hat{S}_{0}(m_{t})$ may be
approximated by Eq.(3) for pions only with parameters as in Table 1. Slope
parameter $T=145$ MeV is held fixed for all cases consistent with data. Its
value is determined within a low-$y_{t}$ interval where the hard component is
negligible. For the present analysis Eq. (3) was evaluated separately for
pions, kaons and protons with $T_{i}=140$, 200 and 210 MeV respectively. Those
expressions were then combined to form $\hat{S}_{0}(m_{t})$ via Eq. (4) with
$z_{0i}=0.82$, 0.12 and 0.06 respectively. For each energy the same exponent
$n$ was applied to three hadron species. Lévy exponent $n$ values and hard-
component $q$ values are as reported in Table 1.

$\bar{\rho}_{sNSD}$ values are derived from the universal NSD trend
$\bar{\rho}_{sNSD}\approx 0.81\ln(\sqrt{s}/\text{10 GeV})$ inferred from
spectrum and yield data. $\bar{\rho}_{0NSD}$ values are derived from the TCM
relation $\bar{\rho}_{0}\approx\bar{\rho}_{s}+\alpha\bar{\rho}_{s}^{2}$. The
NSD $\bar{\rho}_{0}$ values can be compared with $\bar{\rho}_{0}=5.91$ and
7.60 for 5 and 13 TeV INEL $>0$ events from Ref. alicenewspec  [above its Eq.
(1)]. The 13 TeV number contrasts with $6.46\pm 0.19$ in its Fig. 5 (center
panel). Appendix A describes previous TCM analysis of 13 TeV $p$-$p$ $p_{t}$
spectrum data from Ref. alicespec .

Table 1: $p_{t}$ spectrum TCM parameters for 5 TeV and 13 TeV NSD $p$-$p$ collisions within $\Delta\eta\approx 2$.  $\sqrt{s}$ (TeV) | T(̇MeV) | $n$ | $\bar{y}_{t}$ | $\sigma_{y_{t}}$ | $q$ | $100\alpha$ | $\bar{\rho}_{s\text{NSD}}$ | $\bar{\rho}_{0\text{NSD}}$
---|---|---|---|---|---|---|---|---
5.0 | 145 | 8.5 | 2.63 | 0.58 | 4.0 | 1.45 | 5.0 | 5.3
13.0 | 145 | 7.8 | 2.66 | 0.60 | 3.8 | 1.70 | 5.8 | 6.3

It should be noted that the $\alpha$ values for 5 TeV and 13 TeV in Table 1
are 0.0145 and 0.0170 whereas Table 2 includes values reported in Ref.
alicetomspec  for alpha as 0.013 and 0.015. The earlier values were based on
limited 13 TeV $p$-$p$ spectrum data from Ref. alicespec . The updated values
better accommodate the much more complete spectrum data reported in Ref.
alicenewspec . The $\approx 12$% increase in $\alpha$ values is compatible
with estimated uncertainties in Fig. 16 (left) of Ref. alicetomspec  but also
favors a prediction of the $\alpha(\sqrt{s})$ trend based on measured $p$-$p$
jet cross sections and fragmentation functions (dashed curve in that panel).

##  IV $\bf p_{t}$ spectrum biases and evolution

V0M and SPD event-selection methods bias event structure (e.g. $p_{t}$
spectra) in different ways. Biases are examined here relative to a TCM
reference. The TCM for $p$-$p$ collisions assumes linear superposition of
parton-parton interactions consistent with basic QCD (e.g. published jet
measurements). The TCM is not fitted to individual spectra; it serves as a
fixed reference for comparison of biases from different event-selection
methods.

###  IV.1 Spectrum ratios vs $\bf n_{ch}$

In its Figs. 2 and 3 Ref. alicenewspec  presents ratios of $p_{t}$ spectra for
ten $n_{ch}$ classes to those for minimum-bias INEL $>0$ ensemble averages for
each of two event selection criteria. It is noted that “While at low $p_{T}$
($<0.5$ GeV/c) the ratios exhibit a modest $p_{T}$ dependence, for $p_{T}>0.5$
GeV/c they strongly depend on multiplicity and $p_{T}$.” It is possible to
arrive at more-detailed quantitative conclusions using TCM-based techniques
first reported in Ref. ppprd .

Figure 5 shows spectra from 13 TeV $p$-$p$ collisions for V0M (left) and SPD
(right) event classes and for data from Ref. alicenewspec  (solid) and
corresponding TCM (dashed). The various spectra are in ratio to the TCM
spectra for $n_{ch}$ class 5 where 1-10 goes from higher to lower $n_{ch}$.
The spectra have first been normalized by the corresponding soft-component
density $\bar{\rho}_{s}$ as they appear in Fig. 4 (a,c).

Figure 5:  Spectrum ratios for data (solid) and TCM (dashed) from 13 TeV
$p$-$p$ collisions in ratio to the TCM spectrum for event class V for V0M
(left) and SPD (right) event selection.

The corresponding Figs. 2 and 3 in Ref. alicenewspec  show ratios of $p_{t}$
spectra to a single minimum-bias INEL $>0$ reference spectrum. As such the
spectrum ratios include three sources of systematic variation: (a) mean event
multiplicity varying from class to class, (b) varying jet contribution
relative to charge multiplicity and (c) bias effects that are of major
interest. It is then essentially impossible to sort out what cause produces
which effect.

Figure 5 removes variation due to mean event multiplicity, but strong
variation of jet contributions relative to total yield is still confused with
bias effects. As a consequence of that plotting format, below $y_{t}$ = 2 (0.5
GeV/c) spectra nearly coincide as a result of the chosen normalization and are
nearly constant on $y_{t}$, qualitatively consistent with the observation in
Ref. alicenewspec  (modulo distortions from selection bias discussed in the
next subsection). Above that point the ratios vary strongly with $n_{ch}$ and
$p_{t}$, also qualitatively consistent with Ref. alicenewspec . However, such
variation is to be expected given that jet production varies approximately
quadratically with $n_{ch}$.

Close examination of the left panel reveals that the data spectrum for the
highest V0M $n_{ch}$ class corresponds to the TCM (dashed) within statistics.
Then with decreasing $n_{ch}$ spectra are suppressed relative to the TCM at
higher $p_{t}$ but are enhanced at lower $p_{t}$, the enhancement mode moving
to lower $p_{t}$. Correlated suppression and enhancement lead to trends in
Fig. 12 where spectrum integrals adhere to the TCM trend
$\bar{\rho}_{h}\propto\bar{\rho}_{s}^{2}$ for all event classes even as the
spectrum hard components are significantly modified in shape with decreasing
$n_{ch}$.

###  IV.2 V0M and SPD biases relative to the TCM

Event sorting or selection in Ref. alicenewspec  is based on different angular
acceptances denoted by acronym. SPD denotes tracklets (two hits plus vertex)
within $|\eta|<0.8$, the same angular acceptance as for the $p_{t}$ spectra.
V0M denotes a “forward estimator” with combined acceptances $\eta\in[2.8,5.1]$
and $\eta\in[-3.7,-1.7]$ that is said to “minimize the possible
autocorrelations introduced by the use of the mid-pseudorapidity estimator.”
The term “autocorrelations” is here misused. The autocorrelation function
(special case of cross-correlation function) is an established statistical
method for analyzing time series autocorr . A better term is selection bias
wherein event selection is based on the same particle sample (e.g. mid-
rapidity hadrons) as the object of study (mid-rapidity $p_{t}$ spectra).

Figure 6 shows data/TCM ratios for 13 TeV $p$-$p$ collisions and for V0M
(left) and SPD (right) event classes. In contrast to Fig. 5 the systematic
variation of jet yield relative to total yield is largely canceled in the
data/TCM ratio. The same TCM reference is used for both selection methods.
What remains is the bias effects of interest. The 5 TeV results are similar.
The two methods bias spectra substantially but in apparently different ways.
V0M for smaller multiplicities suppresses spectra at higher $y_{t}$ but
produces complementary enhancement at lower $y_{t}$. SPD for smaller
multiplicities also suppresses higher $y_{t}$ and enhances lower $y_{t}$, but
for greater $n_{ch}$ there is apparently strong enhancement for higher $y_{t}$
whereas V0M produces no significant corresponding effect. However, spectrum
ratios exaggerate structure at higher $y_{t}$ while concealing important
structure at lower $y_{t}$ alicetomspec . For example, compare these results
with data-model differences in ratio to statistical errors (Z-scores) in Fig.
8.

Figure 6:  Data/TCM spectrum ratios from 13 TeV $p$-$p$ collisions for V0M
(left) and SPD (right) event selection.

As can be seen in Figs. 3 and 4 (b,d) the physical process biased by event
selection is jet production represented by the TCM spectrum hard component.
The TCM hard-component reference has fixed exponent $q$ corresponding to
$p_{t}$ power-law exponent $n$. The constant trend for the data/TCM ratio
above $y_{t}$ = 4 in Fig. 6 (left) is consistent with $n$ or $q$ independent
of $n_{ch}$, whereas the result in Fig. 6 (right) is consistent with strong
decrease of those parameters with increasing $n_{ch}$. See Figs. 9 (right) and
17 (left). The different forms of spectrum bias relating to V0M and SPD as
indicated by Fig. 6 can thus be understood in terms of jet production.

Minimum-bias jet production near midrapidity arises from three processes: (a)
separate parton splitting cascades within projectile protons resulting from
inelastic scattering, (b) occasional large-angle scattering of cascade
(participant) partons and (c) fragmentation of scattered participant partons
to dijets. A proton splitting cascade (event-wise parton distribution function
or PDF) is sensitive to initial conditions and fluctuates strongly from event
to event gosta . Likewise, the fragment distribution within a jet (also a
splitting cascade) fluctuates strongly from jet to jet. The biases indicated
in Fig. 6 result from sorting the same minimum-bias INEL $>0$ event population
according to two different criteria.

Since V0M selection is derived from particle yields at higher $\eta$ outside
the acceptance for spectra it cannot significantly bias the jet formation
process itself since most jets, near the lower bound of the jet spectrum, are
derived from low-$x$ partons appearing near midrapidity within a longitudinal
cascade. However, V0M selection for low $n_{ch}$ may influence the underlying
jet spectrum resulting from the event-wise PDF, i.e. softening the jet
spectrum. The result is a shift of the data hard component to lower $y_{t}$
without changing its shape, consistent with Fig. 6 (left).

Since SPD selection is based on particle yields within the same acceptance as
for $p_{t}$ spectra it relates primarily to low-$x$ partons and low-energy
jets. It cannot significantly bias the event-wise PDF at larger $x$ (or
$\eta$) but can strongly bias the parton scattering and fragmentation process
near midrapidity. Figure 6 (right) suggests that whereas lower SPD values
produce a bias similar to V0M (softened jet spectrum), for higher SPD values
the effective jet spectrum and mean fragmentation function are biased to
harder distributions leading to evolution of the hard-component tail. The
manifestation in the data/TCM ratio seems dramatic but a very small fraction
of all particles is actually involved. Figures 3, 4 and 8 provide a more
transparent picture of hard-component biases arising from V0M and SPD event
selection.

The role of fluctuations warrants further consideration. It would be
informative to have a 2D plot of event density on SPD vs V0M. Given the jet
production scenario described above one may conjecture qualitatively that for
large V0M yields (and hence event-wise PDF) the SPD event multiplicity and
especially jet contribution is free to fluctuate strongly over a large range
whereas the multiplicity mean value for fixed V0M remains modest as observed.
In contrast, for large SPD yields the largest jet fluctuations are singled
out, V0M is pinned to its highest value (V0M fluctuations are thus limited)
and the multiplicity mean value for fixed SPD is large as observed. In Fig. 8
the most significant bias structure (the bipolar excursion on $y_{t}$) has
maximum amplitude for the lowest values of V0M and SPD. The trend is
consistent with a biased underlying jet spectrum (via the event-wise PDF).

###  IV.3 Significance of data-model differences

Data/model spectrum ratios may exaggerate deviations at higher $p_{t}$
compared to deviations at lower $p_{t}$. A more transparent representation is
based on statistical-significance measures. The Z-score zscore  compares data-
model differences to their statistical uncertainties

$\displaystyle Z_{i}$ $\displaystyle=$
$\displaystyle\frac{O_{i}-E_{i}}{\sigma_{i}},$ (6)

where $O_{i}$ is an observation (e.g. spectrum data), $E_{i}$ is an
expectation (e.g. predictions derived from a model) and $\sigma_{i}$ is the
statistical uncertainty of the observation.

Figure 7 shows statistical errors $\sigma_{i}$ (solid) accompanying published
VOM and SPD spectrum data for 5 and 13 TeV $p$-$p$ collisions which exhibit
step-like structures. If used to process data within ratios those structures
are injected into the result. Smooth approximations (dashed) are introduced to
represent statistical errors without step-like structures. The approximated
error curves are used for data-model comparisons below.

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

Figure 7:  Statistical errors (solid) as reported in Ref. alicenewspec  for 5
TeV (left) and 13 TeV (right) $p$-$p$ collisions. To avoid the step-like
structures in the data errors simple models (dashed, Gaussians on $y_{t}$) are
used to approximate the error trends.

Figure 8 shows data-TCM differences in ratio to statistical uncertainties for
5 TeV (left) and 13 TeV (right) $p$-$p$ collisions and for V0M (upper) and SPD
(lower) event selection. Such Z-score results can be contrasted with data/TCM
ratios as in Fig. 6. Whereas in the ratio format biases for V0M and SPD appear
quite different, the Z-scores in Fig. 8 reveal the statistical significance of
data-model deviations. Despite noticeable differences at higher $p_{t}$ the
most significant bias effects are similar. The bias amplitude in terms of
Z-scores seems to track with fraction of total cross section rather than with
$n_{ch}$.

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

Figure 8:  Data-TCM differences in ratio to statistical errors (Z-scores) for
5 TeV (left) and 13 TeV (right) and for V0M event selection (upper) and SPD
selection (lower). The highest multiplicity classes in each case are
represented by solid, dashed, dotted and dash-dotted curves in descending
order.

##  V $\bf p_{t}$ spectrum shape measures

Reference alicenewspec  analyzed the evolution of $p_{t}$ spectrum shapes with
varying charge multiplicity $n_{ch}$ in two ways: (a) power-law model fits to
spectra to infer trends for power-law exponent $n$ and (b) variation of
integrated yields within three $p_{t}$ intervals compared to a minimum-bias
reference as a function of $n_{ch}$. This section considers such shape-measure
results in the context of the TCM.

###  V.1 Power-law fits to high-$\bf p_{t}$ intervals

Reference alicenewspec  fitted a power-law function to 13 TeV $p$-$p$ spectra
above 6 GeV/c ($y_{t}\approx 4.5$) to estimate exponents $n$ vs $n_{ch}$ for
V0M and SPD events. A related result can be obtained without curve fitting via
a logarithmic derivative applied directly to data hard components $Y(y_{t})$
as in Figs. 3 and 4 (b,d) or $Y(p_{t})=y_{t}Y(y_{t})/m_{t}p_{t}$:

$\displaystyle-\frac{d\ln[Y(y_{t})]}{dy_{t}}\rightarrow
q~{}~{}\text{for}~{}~{}y_{t}>4.1$ (7)
$\displaystyle-\frac{d\ln[Y(p_{t})]}{d\ln(p_{t})}\rightarrow
n~{}~{}\text{for}~{}~{}y_{t}>4.1.$

If $dn_{ch}/y_{t}dy_{t}\propto\exp(-qy_{t})$ [high-$y_{t}$ tail of
$\hat{H}_{0}(y_{t})$] then
$dn_{ch}/p_{t}dp_{t}=(y_{t}/m_{t}p_{t})dn_{ch}/y_{t}dy_{t}\propto
1/p_{t}^{q+2}\approx 1/p_{t}^{n}$. $n\approx q+2$ is a rough estimate, but
$n\approx q+1.8$ is established by direct comparison of results from Eq. (7)
(upper and lower) for the same spectra. Power-law exponent $n$ invoked by Ref.
alicenewspec  should not be confused with TCM Lévy exponent $n$ for soft
component $\hat{S}_{0}(m_{t})$.

Figure 9 (left) shows logarithmic derivatives vs $y_{t}$ (thin solid) for
eight $n_{ch}$ classes of 13 TeV V0M $p$-$p$ collisions. The bold dashed curve
results from applying the same technique to TCM hard-component model
$\hat{H}_{0}(p_{t})$. Horizontal dotted lines represent values $n\approx
q+1.8$ expected for 5 (upper) and 13 (lower) TeV $p$-$p$ collisions from TCM
$q$ energy trends related to Ref. alicetomspec  (see Table 1). The crossed
solid lines for $y_{t}=2.65$ and $n=1.8$ remind that while the mode of the
hard component on $y_{t}$ is near 2.65 (where the upper line of Eq. (7) would
pass through zero) the mode on $p_{t}$ is near $p_{t}=0.5$ GeV/c
($y_{t}\approx 2$ where the lower line of Eq. (7) would pass through zero).
See Fig. 20 (right) and associated text.

In this data format the spike artifacts common to all spectrum classes are
most evident since differential measures are sensitive to short-wavelength
structure. As noted, it may be that those artifacts arise from efficiency
corrections derived from Monte Carlo data with more-limited statistics than
the spectrum data themselves.

Figure 9:  Left: Logarithmic derivatives defined by Eq. (7) (second line, thin
solid) for ten $n_{ch}$ classes of 13 TeV $p$-$p$ collisions and V0M events.
The dashed curve is the same operation applied to TCM hard component
$\hat{H}_{0}(p_{t})$. The crossed solid lines are explained in the text.
Right: Power-law exponents $n$ obtained by averaging the last ten points in
the left panel (i.e. $y_{t}>4.1$, $p_{t}>4.2$ GeV/c). Points corresponding to
$n=q+1.8$ with $q$ from Fig. 17 (left) are raised by 0.02 for visibility. The
curve is derived from Eq. (17). Power-law exponent $n$ should not be confused
with TCM $\hat{S}_{0}$ exponent $n$.

Figure 9 (right) shows exponents $n$ vs $n_{ch}$ for three spectrum data
types. The $n$ values are in each case averages over the ten highest-$y_{t}$
data points ($y_{t}>4.1$ or $p_{t}>4.2$ GeV/c) in the left panel. That limit
is lower than the fit interval $p_{t}>6$ GeV/c applied in Ref. alicenewspec .
However, the V0M results in the left panel indicate that the exponent trend is
approximately constant down to $y_{t}=4.1$, and the added points provide more-
stable $n$ values. The values for highest and lowest multiplicity classes are
not plotted because of excessive noise in the log-derivative results. The V0M
data trends on $n_{ch}$ are approximately constant, with values close to 5.60
(13 TeV) and 5.80 (5 TeV). The dotted lines correspond to TCM
$\hat{H}_{0}(y_{t})$ $q=$ 3.8 for 13 TeV and 4.0 for 5 TeV related to Ref.
alicetomspec  (see Table 1). The SPD data trend varies strongly, decreasing
(i.e. to harder spectra) with increasing $n_{ch}$. Those trends are consistent
with spectrum data trends in Figs. 1 and 2 (b,d) showing deviations from fixed
TCM references. The solid curve for 13 TeV SPD data is derived from Eq. (17)
with $n=q+1.8$. The 5 TeV SPD data are more scattered than other data trends
and are thus omitted to improve visual access to the 13 TeV SPD trend relevant
to Fig. 5 of Ref. alicenewspec .

In its summary Ref. alicenewspec  concludes “…within uncertainties, the
functional form of $n$ as a function of $\langle dN_{ch}/d\eta\rangle$ is the
same for the two multiplicity estimators [V0M and SPD] used in this analysis.
Moreover, $n$ is found to decrease with $\langle dN_{ch}/d\eta\rangle$.” The
results in Fig. 9 contradict that conclusion: The power-law trend is
dramatically different for V0M and SPD, and $n$ does not decrease with charge
density for V0M event selection. The contrast between V0M and SPD event
selection is most evident in Fig. 6. In the left panel the effective exponent
at high $p_{t}$ relative to the fixed TCM value is itself fixed (the ratio is
approximately constant above $y_{t}$ = 4). The right panel shows dramatic
variation of the effective SPD exponent from high (soft) to low (hard) with
increasing $n_{ch}$.

###  V.2 Spectrum response to selection bias

Figures 4 and 6 of Ref. alicenewspec  deal with other manifestations of
selection bias in $p_{t}$ spectrum structure. Figure 6 compares yields
integrated within three specific $p_{t}$ intervals $\Delta p_{t}$ for ten
multiplicity classes in ratio to yields in the same intervals from the INEL
$>0$ minimum-bias class. The resulting data trends are compared to a linear
$y=x$ trend corresponding to no change in spectrum shape with $n_{ch}$. Figure
4 compares V0M and SPD spectra for nearly-equal charge densities
$\bar{\rho}_{0}\approx 20$. Ratio SPD/V0M drops to 0.85 near 4 GeV/c for both
5 and 13 TeV spectra. What follows is an effort to understand systematics
details and provide a physical interpretation.

Figure 10 (left) shows integrated yields $n_{ch}(\Delta p_{t})$ for three
$p_{t}$ intervals from spectra for ten $n_{ch}$ classes of SPD and V0M events
in ratio to yields $n_{ch,\text{INEL}}(\Delta p_{t})$ in the same intervals
(points) from a 13 TeV TCM $p_{t}$ spectrum defined on data $p_{t}$ values
with $\bar{\rho}_{0}=7.60$ as for INEL $>0$ events in Ref. alicenewspec . The
solid lines are TCM references resulting from the same method applied to TCM
spectra defined “on a continuum” (i.e. on 100 equal-spaced $y_{t}$ points
extending down to $y_{t}$ = 0). Soft and hard TCM model functions do not vary
with $n_{ch}$. The different slopes of the TCM lines result only from the
different fractions of soft and hard components in each $p_{t}$ interval. The
log-log plot format ensures that suppression at smaller $n_{ch}$ is as visible
as enhancement at larger $n_{ch}$. The effects of selection bias are indicated
by deviations of data from the TCM trends, not from the $y=x$ dotted line
(that assumes no change in jet production). As for Fig. 5 above or Figs. 2 and
3 in Ref. alicenewspec  distinction should be maintained between variations
due to generic jet trends and bias effects relative to those variations.

Figure 10:  Left: Ratios of yields (points) from $p_{t}$ spectra within three
$\Delta p_{t}$ intervals for ten $p$-$p$ V0M or SPD event classes to yields
from the same intervals and from INEL $>0$ events. The independent variable is
mean event multiplicity $n_{ch}$ in ratio to INEL $>0$ mean multiplicity
$n_{ch,\text{INEL}}$. The solid reference curves result from the same method
applied to TCM spectra defined on a $y_{t}$ continuum. Right: Ratios of data
spectra to corresponding TCM references for class II of V0M data and class VII
of SPD data (dashed), which both approximately correspond to
$\bar{\rho}_{0}=20$. The solid curve is SPD/V0M data spectra compared directly
in ratio, reportedly demonstrating that SPD is softer than V0M for the same
charge multiplicity. However, the corresponding TCM ratio (dash-dotted)
demonstrates that most of the deviation from unity is simply due to the V0M vs
SPD multiplicity difference. The two panels relate to Figs. 6 and 4
respectively of Ref. alicenewspec .

Several features are apparent. For V0M events points for different $\Delta
p_{t}$ at higher $n_{ch}$ approximately coincide, consistent with Fig. 6
(left) where the ratios for higher $n_{ch}$ are nearly flat on $p_{t}$. In
contrast, higher-$n_{ch}$ points for SPD vary strongly with $p_{t}$ interval
$\Delta p_{t}$, consistent with Fig. 6 (right). For lowest $n_{ch}$ classes
there is strong suppression below the TCM references for both V0M and SPD,
also consistent with spectrum ratios in Fig. 6, but suppression for V0M is
significantly greater than for SPD.

Figure 10 (right) shows 13 TeV data/TCM spectrum ratios (dashed) for class II
V0M/TCM and class VII SPD/TCM, data/data ratio SPD/V0M (solid) that appears in
Fig. 4 of Ref. alicenewspec  and the corresponding ratio of TCM spectra (dash-
dotted). Commenting on the SPD/V0M spectrum ratio in its Fig. 4 (identical to
the solid curve here) Ref. alicenewspec  states that “For transverse momenta
within 0.5-3 GeV/c the spectra [sic] for the [V0M] multiplicity class II is
harder than that for the [SPD] multiplicity class VII′.” However, the 5%
difference in $\bar{\rho}_{0}$ for V0M (20.5) and SPD (19.5) plays a
significant role as indicated by the TCM ratio (dash-dotted). Since TCM model
functions $\hat{S}_{0}$ and $\hat{H}_{0}$ are fixed, charge density
$\bar{\rho}_{0}$ determines the TCM spectrum shape. The TCM ratio demonstrates
the effect of the V0M vs SPD multiplicity difference: at least half of the
SPD/V0M ratio deviation from unity arises from the difference in
$\bar{\rho}_{0}$. Absent detailed comparisons with a model the term “harder”
may mean a modified jet fragment distribution or simply more or less jet
production according to $\bar{\rho}_{h}\approx\alpha\bar{\rho}_{s}^{2}$.

One should also note that class VII is comparatively low $n_{ch}$ for SPD
whereas class II is relatively high $n_{ch}$ for V0M. Given the trends in Fig.
8 (b,d) one should then expect a substantial difference in bias for the two
cases, as observed. Concerning the short-wavelength structure, the peaks near
5 GeV/c in the data/TCM ratios certainly correspond to the bipolar structure
near $y_{t}=4.4$ in Fig. 9 (left) that is common to all $n_{ch}$ classes and
therefore most probably results from the inefficiency correction. That
structure then cancels in the SPD/V0M data ratio even though the spectra are
from quite different event classes.

###  V.3 Spectrum running integrals

In Sec. IV of Ref. ppprd  running integrals of $y_{t}$ spectra were the basis
for discovery of the two-component structure of 200 GeV $p$-$p$ $p_{t}$
spectra without a priori assumptions. It was observed that spectra normalized
not with total charge density $\bar{\rho}_{0}$ but with a “soft component”
$\bar{\rho}_{s}$ defined as the root of
$\bar{\rho}_{0}=\bar{\rho}_{s}+\alpha\bar{\rho}_{s}^{2}$ with $\alpha\approx
O(0.01)$ coincided below $y_{t}=2$ ($p_{t}\approx 0.5$ GeV/c) within data
uncertainties and that the endpoints of spectrum running integrals also
followed a trend consistent with the above quadratic equation. The same
approach is applied here to 13 TeV $p$-$p$ spectrum data. One purpose is
demonstration that the $p_{t}$ spectrum TCM is required by $p$-$p$ data for
any energy, is not imposed a priori.

Figure 11 shows running integrals of $y_{t}$ spectra for ten $n_{ch}$ classes
of 13 TeV $p$-$p$ collisions each for V0M (left) and SPD (right) event
selection criteria derived from data (solid) and TCM (dashed) spectra. Spectra
are normalized by $\bar{\rho}_{s}$ inferred from $\bar{\rho}_{0}$ reported in
Ref. alicenewspec  using the quadratic equation defined above with
$\alpha=0.017$ for 13 TeV, 12% higher than reported in Ref. alicetomspec . The
uncorrected spectrum running integrals are then defined as

$\displaystyle\Sigma^{\prime}(p_{t};n_{ch},p_{t,cut})$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{\rho}_{s}}\int_{p_{t,cut}}^{p_{t}}dp_{t}^{\prime}p_{t}^{\prime}\bar{\rho}_{0}(p_{t}^{\prime};n_{ch}).$
(8)

Note that a factor $p_{t}^{\prime}$ in the integrand is required in order to
be consistent with the definition of $\bar{\rho}_{0}(y_{t};n_{ch})$ in Eq.
(III.1). Running integrals on data $p_{t}$ values within the ALICE $p_{t}$
acceptance can be simply corrected for incomplete $p_{t}$ acceptance. The
correction is addition of estimated $1-\xi$ corresponding to $p_{t,cut}\approx
0.15$ GeV/c (see Fig. 14, left and related text).

Figure 11:  Running integrals for data (solid) and TCM (dashed) $p_{t}$
spectra from 13 TeV $p$-$p$ collisions and for V0M (left) and SPD (right)
event selection. Running integrals have been corrected for incomplete $p_{t}$
acceptance as described in the text. The bold dotted curve corresponds to
model $\hat{S}_{0}(y_{t})$.

The corrected data running integrals can be expressed in TCM form on $y_{t}$
as

$\displaystyle\Sigma(y_{t})$ $\displaystyle=$
$\displaystyle\Sigma_{s}(y_{t})+\Sigma_{h}(y_{t}).$ (9)

As in Ref. ppprd  the corrected data running integrals coincide below
$y_{t}=2$ ($p_{t}\approx 0.5$ GeV/c) within data uncertainties then separate
and achieve saturation above $y_{t}=4.0$ ($p_{t}\approx 3.8$ GeV/c). Running
integral $\Sigma_{0}(y_{t})$ (bold dotted) of TCM soft component
$\hat{S}_{0}(y_{t})$ is defined as the limit of data running integrals as
$n_{ch}\rightarrow 0$ where $\Sigma_{h}(y_{t})\rightarrow 0$. The functional
form of $\hat{S}_{0}(y_{t})$ given by Eqs. (3) and (4) is then observed to
generate the required limiting form of $\Sigma_{0}(y_{t})$ within data
uncertainties. $\Sigma_{0}(y_{t})$ saturates at 1 by definition. The same form
is used for V0M and SPD data.

The data trends in Fig. 11 demonstrate the following: (a) The shape of data
soft component $\Sigma_{s}(y_{t})$ does not vary significantly with $n_{ch}$,
is consistent with $\Sigma_{0}(p_{t})$. (b) The complementary data hard
components $\Sigma_{h}(y_{t};n_{ch})$ are consistent with an erf$(y_{t})$
function as running integral, demonstrating that data spectrum hard components
are similarly-shaped peaked distributions with mode near $y_{t}$ = 2.7
($p_{t}\approx 1$ GeV/c) as demonstrated in Ref. ppprd . It is then of
interest to examine the $n_{ch}$ trend of the data running-integral endpoints
for TCM consistency.

Figure 12 shows running-integral endpoints vs $\bar{\rho}_{s}$ for 5 TeV
(left) and 13 TeV (right) derived from data spectra for V0M (solid dots) and
SPD (open circles) event selection. The result for uncorrected spectra is
$\xi+x(n_{s})$. The inefficiency correction is then $1-\xi\approx 0.16$ for
the TCM defined on SPD $p_{t}$ values (dashed lines) as noted.

Figure 12:  Endpoints of data running integrals in Fig. 11 for V0M (solid
dots) and SPD (open circles) and for 5 TeV (left) and 13 TeV (right) $p$-$p$
collisions. The dashed lines are TCM endpoints on SPD $p_{t}$ values. The
dotted lines are TCM trends $1+x(n_{s})\rightarrow 1+\alpha\bar{\rho}_{s}$
with $\alpha=0.017$ (0.0145) for 13 (5) TeV.

Those results demonstrate experimentally the precise quadratic relation
between hard and soft data components: (a) All spectra coincide precisely for
$p_{t}<0.5$ GeV/c given normalization via $\bar{\rho}_{s}$ as computed from
measured $\bar{\rho}_{0}$ and (b) corrected-running-integral endpoints fall
along straight-line loci $1+\alpha\bar{\rho}_{s}$ (dotted lines) with
$\alpha\bar{\rho}_{s}\approx x(n_{s})$. The endpoint trends in turn accurately
indicate the fractional contribution $x/(1+x)$ of jet fragments to total
spectra. For the highest multiplicity classes $x\approx 0.5$ and 33% of
hadrons are jet fragments.

In terms of TCM model functions, spectrum data demonstrate that one spectrum
component scales linearly with $\bar{\rho}_{s}$ at lower $p_{t}$ and that a
second component scales quadratically with $\bar{\rho}_{s}$ at higher $p_{t}$.
The $p_{t}$ structures of the data components vary little with $n_{ch}$.
Running integrals provide a less-sensitive way to probe spectrum details
compared to Figs. 3 and 4. However, there are no assumptions about spectrum
structure, per Ref. ppprd . It is notable that the straight-line trends in
Fig. 12 are followed down to the lowest event multiplicities although the data
hard components are substantially biased there. The present study demonstrates
that the $p_{t}$ spectrum TCM, with its quadratic relation between soft and
hard components, is necessary to describe 13 TeV $p_{t}$ spectra (modulo
biases described and interpreted in Sec. IV).

##  VI $\bf p$-$\bf p$ ensemble $\bf\bar{p}_{t}$ systematics

Ensemble-mean $\bar{p}_{t}$ data are inferred from hadron spectra via
integration. The $\bar{p}_{t}$ values in Ref. alicenewspec  are obtained by
integrating over $p_{t}\in[0.15,10]$ GeV/c. Accurate values corresponding to
ideal spectrum data could be obtained by extrapolating data spectra with a
reliable spectrum model. Values obtained with unextrapolated spectra may be
strongly biased, and correct interpretation of biased experimental
$\bar{p}_{t}$ values may be difficult. This section demonstrates how to relate
a TCM reference to biased $\bar{p}_{t}$ values obtained from spectrum data.
The resulting bias is here estimated and corrected.

Reference alicempt  reported a comprehensive analysis of $\bar{p}_{t}$ data vs
event multiplicity for $p$-$p$, $p$-Pb and Pb-Pb collision systems. The strong
increase of $\bar{p}_{t}$ with $n_{ch}$ for $p$-$p$ collisions was there
interpreted in terms of color reconnection as modeled within the PYTHIA Monte
Carlo. The data trends were said to “pose a challenge to most of the existing
models.” A TCM analysis of the same data was presented in Ref. alicetommpt .
The observed $\bar{p}_{t}$ vs $n_{ch}$ trends were found to be consistent with
the jet (hard component) contribution to hadron production in all cases.

Reference alicenewspec  introduces spherocity measure $S_{0}$ intended to
select more- or less-“jetty” events according to its value. Variation of jet
production in a given event sample is expected to bias the $\bar{p}_{t}$ vs
$n_{ch}$ trend. It is noted that as $S_{0}$ decreases $\bar{p}_{t}$ increases,
possibly due to an increased jet contribution to spectra as anticipated. In
addition to biases resulting from incomplete $p_{t}$ acceptance and from V0M
and SPD event selection methods, biases from event selection via spherocity
$S_{0}$ are considered.

###  VI.1 Ensemble $\bf\bar{p}_{t}$ TCM for $\bf p$-$\bf p$ collisions

The TCM for ensemble-mean $\bar{p}_{t}$ data from $p$-$p$ collisions is
summarized. It is assumed that due to incomplete $p_{t}$ acceptance (lower
bound $p_{t,cut}\approx 0.15$ GeV/c) only a fraction $\xi<1$ of the $p_{t}$
spectrum soft component is accepted. It is also evident that for such a low
cutoff value the entire spectrum hard component is accepted. The $\bar{p}_{t}$
values obtained from TCM model functions as defined by Table 1 are as follows:
For models defined on the continuum ($\xi\approx 1$) the mean values are
$\bar{p}_{ts}\approx 0.48$ GeV/c and $\bar{p}_{th}\approx 1.39$ GeV/c. For
models defined on spectrum data $p_{t}$ values the mean values are
$\bar{p}_{ts}\approx 0.58$ GeV/c (corresponding to $\xi\approx 0.84$) and
$\bar{p}_{th}\approx 1.38$ GeV/c.

The TCM for charge densities averaged over some angular acceptance
$\Delta\eta$ (1.6 for $|\eta|<0.8$ as in Ref. alicenewspec ) is

$\displaystyle\bar{\rho}_{0}$ $\displaystyle=$
$\displaystyle\bar{\rho}_{s}+\bar{\rho}_{h}$ $\displaystyle=$
$\displaystyle\bar{\rho}_{s}[1+x(n_{s})],$
$\displaystyle\frac{\bar{\rho}_{0}^{\prime}}{\bar{\rho}_{s}}$ $\displaystyle=$
$\displaystyle\frac{n_{ch}^{\prime}}{n_{s}}~{}=~{}\xi+x(n_{s}),$

where $x(n_{s})\equiv\bar{\rho}_{h}/\bar{\rho}_{s}\approx\alpha\bar{\rho}_{s}$
is the $p$-$p$ ratio of hard-component to soft-component yields ppprd  and
$\alpha(\sqrt{s})$ is defined in Ref. alicetomspec . $\bar{\rho}_{0}^{\prime}$
is an uncorrected charge density corresponding to incomplete $p_{t}$
acceptance with $\xi<1$.

The TCM for extensive ensemble-mean total $p_{t}$ integrated within some
angular acceptance $2\pi$ and $\Delta\eta$ from $p$-$p$ collisions for given
$(n_{ch},\sqrt{s})$ can be expressed as

$\displaystyle\bar{P}_{t}$ $\displaystyle=$
$\displaystyle\bar{P}_{ts}+\bar{P}_{th}$ $\displaystyle=$ $\displaystyle
n_{s}\bar{p}_{ts}+n_{h}\bar{p}_{th}.$

The conventional intensive ratio of extensive quantities

$\displaystyle\frac{\bar{P}_{t}^{\prime}}{n_{ch}^{\prime}}\equiv\bar{p}_{t}^{\prime}$
$\displaystyle\approx$
$\displaystyle\frac{\bar{p}_{ts}+x(n_{s})\bar{p}_{th}(n_{s})}{\xi+x(n_{s})}$
(12)

(assuming $\bar{P}_{ts}^{\prime}\approx\bar{P}_{ts}$ tommpt )222Because of the
additional factor $p_{t}^{\prime}$ in
$\bar{P}_{ts}^{\prime}=\int_{p_{t.cut}}^{\infty}dp^{\prime}_{t}{p^{\prime}_{t}}^{2}\bar{\rho}_{s}(p_{t}^{\prime})$
the effect of the low-$p_{t}$ cutoff is minimal and
$\bar{P}_{ts}^{\prime}\approx\bar{P}_{ts}$. in effect partially cancels dijet
manifestations represented by ratio $x(n_{s})$ that may be of considerable
interest. The alternative ratio

$\displaystyle\frac{n_{ch}^{\prime}}{n_{s}}\bar{p}_{t}^{\prime}\approx\frac{\bar{P}_{t}}{n_{s}}$
$\displaystyle=$ $\displaystyle\bar{p}_{ts}+x(n_{s})\bar{p}_{th}(n_{s})$
$\displaystyle\approx$
$\displaystyle\bar{p}_{ts}+\alpha(\sqrt{s})\,\bar{\rho}_{s}\,\bar{p}_{th}(n_{s},\sqrt{s})$

preserves the simplicity of Eq. (VI.1) and provides a convenient basis for
testing the TCM hypothesis precisely.

###  VI.2 Spherocity event selection

Reference alicenewspec  studies ensemble $\bar{p}_{t}$ trends vs spherocity

$\displaystyle
S_{0}(\%)=100\times\frac{\pi^{2}}{4}\min_{\bf\hat{n}_{s}}\left(\frac{\sum_{i}|\vec{p}_{t,i}\times{\bf\hat{n}_{s}}|}{\sum_{i}\vec{p}_{t,i}}\right)^{2}\in[0,100],$
(14)

where $\bf\hat{n}_{s}$ is a unit vector varied to minimize $S_{0}$. Limit 0%
would result for a single $p_{t}$ vector. Limit 100% would result for an
infinite number of $p_{t}$ vectors uniformly distributed on azimuth with equal
magnitudes, in which case the quantity in parenthesis becomes
$(1/\pi)\int_{0}^{\pi}d\phi\sin(\phi)$ = $2/\pi$. Events are sorted into ten
spherocity classes with the intent to identify events based on the particle
fraction arising from jets, i.e. the event “jettiness.”

###  VI.3 ALICE ensemble $\bf\bar{p}_{t}$ results

Figure 13 (left) shows uncorrected (biased) ensemble $\bar{p}_{t}^{\prime}$
values vs $n_{ch}$ (thin solid) from 13 TeV $p$-$p$ collisions for ten classes
of spherocity $S_{0}$. According to Ref. alicenewspec  uncorrected
$\bar{p}_{t}$ is determined within acceptance $p_{t}\in[0.15,10]$ GeV/c but
$n_{ch}$ is corrected by extrapolation to zero. The bold dash-dotted curve is
a simple unweighted average of those spectra. As a result of the incomplete
$p_{t}$ acceptance the $\bar{p}_{t}$ values are biased upward as for
$\bar{p}_{t}^{\prime}$ in Eq. (12) with $\xi\approx 0.84$. The method of event
selection on $n_{ch}$ is based on $n_{ch}$ appearing in angular acceptance
$|\eta|<0.8$ (i.e. SPD). The solid dots and open circles are $\bar{p}_{t}$
values obtained from TCM spectra defined on a “continuum” (100 points equally
spaced on $y_{t}\in[0,6]$) for V0M and SPD $n_{ch}$ which are then unbiased
($\xi\approx 1$). The dashed curves are TCM trends for 5 and 13 TeV described
by Eq. (12) with $\xi=1$ and hard component $\bar{p}_{th}(n_{s})=1.39$ GeV/c
fixed. The open squares are the TCM defined on SPD data $p_{t}$ values and
integrated without extrapolation (i.e. are biased). The open triangles are
from 13 TeV SPD $p_{t}$ spectra reported in Ref. alicenewspec  also integrated
without extrapolation (biased). The $\bar{p}_{t}$ bias for uncorrected data is
about 0.1 GeV/c (open squares vs upper dashed curve), consistent with Fig. 14
(left).

Figure 13:  Left: Ensemble-mean $\bar{p}_{t}$ vs $n_{ch}$ trends for 13 TeV
events (thin solid) vs spherocity $S_{0}$. The dash-dotted curve is an
unweighted mean of those curves. The open triangles are $\bar{p}_{t}$ from 13
TeV SPD data. The open squares are from uncorrected TCM spectra on data
$p_{t}$ values. The open and solid circles are the TCM “on a continuum” (see
text). The dashed curves are TCM Eq. (12). Right: $\bar{p}_{t}$ vs $n_{ch}$
trends in the left panel converted per Eq. (VI.1) with $\xi=1$ for dots and
circles and 0.84 for open squares and open triangles (corrected).

Figure 13 (right) shows the same data and curves (with the exception of
spherocity-related curves, see Fig. 15) in the form of Eq. (VI.1). The TCM
trends follow straight-line loci whose slopes are determined by the spectrum
jet contribution, controlled in this case by parameter $\alpha$. The TCM
defined on data points (open squares) and 13 TeV data SPD $\bar{p}_{t}$ values
(open triangles) have been corrected according to Eq. (VI.1) with $\xi\approx
0.84$ in Eq. (VI.1) (see below). The dashed lines are Eq. (VI.1) with
$\bar{p}_{ts}=0.48$ GeV/c and $\bar{p}_{th}=1.39$ GeV/c as described in Sec.
VI.1.

Figure 14 (left) shows the consequence of an incomplete $p_{t}$ acceptance for
calculation of $\bar{p}_{t}$ per Ref. tommpt . The solid curve is the running
integral of $1-\hat{S}_{0}(p_{t})$ that determines the inefficiency parameter
$\xi$ as a function of spectrum lower bound $p_{t,cut}$, the fraction of
$\hat{S}_{0}(p_{t})$ that survives the cut. The lowest hatched band indicates
$\bar{p}_{t}$ soft component $\bar{p}_{ts}\approx 0.48$ GeV/c determined from
TCM fixed spectrum soft-component $\hat{S}_{0}(y_{t})$ with no cutoff.
$p_{ts}^{\prime}=\bar{p}_{ts}/\xi$ (dashed) is the biased $\bar{p}_{t}$ soft
component resulting from the cutoff. For cutoff $p_{t,cut}\approx 0.15$ GeV/c
in Ref. alicenewspec  $\xi\approx 0.84$ (as in the previous paragraph) and
$\bar{p}_{ts}^{\prime}\approx 0.58$ GeV/c, i.e. about 0.1 GeV/c higher than
the unbiased value. The right panel is explained below.

Figure 14:  Left: Fraction $\xi$ (solid) of TCM soft component
$\hat{S}_{0}(y_{t})$ integrated above cutoff $p_{t,cut}$ and biased
$\bar{p}_{ts}^{\prime}=\bar{p}_{ts}/\xi$ (dashed). For nominal acceptance cut
$p_{t,cut}\approx 0.15$ GeV/c $\xi\approx 0.84$ and
$\bar{p}_{ts}^{\prime}\approx 0.58$ GeV/c (hatched bands). Right: Biased
$\bar{p}_{ts}^{\prime}$ vs spherocity $S_{0}$ (in percent) for corrected
$\bar{p}_{t}$ vs $n_{ch}$ trends compared to $\bar{p}_{ts}\approx 0.48$ GeV/c
(hatched band). Note that symbol $\bar{p}_{ts}^{\prime}$ represents different
bias sources in the two panels – low-$p_{t}$ acceptance cutoff vs spherocity
$S_{0}$.

Figure 15 (left) shows $\bar{p}_{t}$ vs $n_{ch}$ data trends in Fig. 13 (left)
as they vary with spherocity $S_{0}$ (thin solid), the unweighted ensemble
average (bold dash-dotted) and the TCM on 13 TeV SPD data $p_{t}$ values (open
squares) transformed according to Eq. (VI.1) with $\xi\approx 0.84$ that can
be compared with Fig. 13 (right). The open triangles are 13 TeV SPD data
$\bar{p}_{t}$ values treated the same. In this plotting format it is evident
that for most data spectra the primary variation with spherocity $S_{0}$ is
soft-component mean $\bar{p}_{ts}$ ($y$-axis intercepts in this plot) since
the jet contribution (i.e. $\bar{p}_{th}$), determining slopes in this format,
shows little variation with $S_{0}$. The upper dotted line is
$0.48+1.39\alpha\bar{\rho}_{s}$ consistent with TCM $\bar{p}_{ts}=0.48\pm
0.01$ GeV/c and $\bar{p}_{th}=1.39\pm 0.015$ GeV/c. The lower dotted line with
$\bar{p}_{ts}=0.42$ GeV/c and $\bar{p}_{th}=1.30$ is consistent with the
lowest $S_{0}$ curve for $\bar{\rho}_{s}>15$.

Figure 15:  Left: 13 TeV $\bar{p}_{t}$ vs $n_{ch}$ data vs spherocity $S_{0}$
(thin solid, dash-dotted) from Fig. 13 (left) transformed and corrected via
Eq. (VI.1) with $\xi\approx 0.84$. The open squares are the corrected 13 TeV
TCM on data $p_{t}$ values from Fig. 13 (right). $\bar{p}_{ts}$ is derived
from the TCM $\hat{S}_{0}(y_{t})$ model function. Right: $\bar{p}_{t}$ hard
component $\bar{p}_{th}(n_{s})$ derived from the contents of the left panel
via Eq. (VI.1). For the $S_{0}$ data
$\bar{p}_{ts}\rightarrow\bar{p}_{ts}^{\prime}(S_{0})$ values from Fig. 14
(right) are used in Eq. (VI.1). The dotted lines in left and right panels
indicate the TCM trend (upper) and the lowest of the $S_{0}$ curves above
$\bar{\rho}_{s}=15$.

Figure 14 (right) shows empirically-determined biased soft-component values
$\bar{p}_{ts}^{\prime}$ for the individual $\bar{p}_{t}(n_{ch})$ trends in
Fig. 15 (left). The estimated values are simply described by the power-law
expression $\bar{p}_{ts}^{\prime}(S_{0})\approx[0.048/S_{0}(\%)]^{0.088}$
GeV/c (straight line) over the entire $S_{0}$ range. The hatched band denotes
the unbiased soft-component mean $\bar{p}_{ts}\approx 0.48$ GeV/c. The bias in
this instance is not due to a $p_{t}$ cutoff (which has been corrected), is
instead the result of the imposed $S_{0}$ condition. The two points above the
hatched band thus do not “cause” the two elevated trends in Fig. 15 (right).

Figure 15 (right) shows $\bar{p}_{th}(n_{s})$ inferred from data in the left
panel via Eq. (VI.1) (second line) with $\bar{p}_{ts}^{\prime}(S_{0})$ taken
from Fig. 14 (right) for each spherocity class. The value 1.39 GeV/c (hatched
band) is just the $\bar{p}_{th}$ corresponding to the 13 TeV TCM hard-
component $\hat{H}_{0}(y_{t})$ in Sec. III.2. The adopted criterion for
determining $\bar{p}_{ts}$ values in Fig. 14 (right) is a requirement that
$\bar{p}_{th}(n_{s})$ trends be as level as possible above
$\bar{\rho}_{s}=15$, consistent with a constant $\bar{p}_{th}(n_{s})$ trend in
Eq. (VI.1) reflecting the TCM. The horizontal dotted lines are the dotted
lines at left transformed to obtain $\bar{p}_{th}(n_{s})$ according to Eq.
(VI.1).

Deviations from the TCM reference in Fig. 15 (right) could in general carry
significant new information about collision dynamics. However, the data mainly
indicate that while jet production increases about 30-fold over the interval
$\bar{\rho}_{s}\in[6,33]$ it continues to follow the observed
$\bar{\rho}_{h}\propto\bar{\rho}_{s}^{2}$ trend precisely no matter what the
value of $S_{0}$.

The comparison between V0M and SPD $\bar{p}_{th}$ values indicates that the
SPD hard component mean $p_{t}$ is biased down by almost 0.1 GeV/c independent
of $S_{0}$. The effect is evident in Fig. 6 in that the data trend on $y_{t}$
for higher $n_{ch}$ and for V0M is flat (i.e. agreeing with the TCM) whereas
for SPD the hard component is shifted down on $y_{t}$ (hence the tilt). Thus,
for large SPD $n_{ch}$ the high-$p_{t}$ tail is hardened while the main part
of the hard component is softened. Trends for higher $S_{0}$ (thin solid)
agree with the basic SPD data. Only the lowest two $S_{0}$ values show
significant hardening of the hard component.

Information carried by $p_{t}$ spectra sorted by $S_{0}$ thus has three
contributions: (a) bias of $\bar{p}_{ts}$ as in Fig. 14 (right) indicates bias
of the spectrum soft component in response to imposition of an azimuthal
(a)symmetry condition via $S_{0}$, (b) bias of the spectrum hard component in
the form of strong modification of the hard-component shape with low $n_{ch}$
condition as in Fig. 8 resulting in lower $\bar{p}_{t}$ independent of
$S_{0}$, and differently for V0M and SPD, and (c) minor modification of the
jet contribution to $\bar{p}_{t}$ varying with $S_{0}$ as in Fig. 15 (right).
Note that for the SPD data used in the $S_{0}$ study the absolute jet
production increases (relative to NSD $p$-$p$ collisions) by $\approx$ 30-fold
because of the $n_{ch}$ variation (see Fig. 7 of Ref. alicenewspec ) while the
largest effect of $S_{0}$ variation on the jet-related $\bar{p}_{t}$ hard
component is less than 10% (see Fig. 15, right).

###  VI.4 ALICE ensemble $\bf\bar{p}_{t}$ summary

Results in this section suggest several conclusions: (a) The basic
$S_{0}$-related $\bar{p}_{t}$ data in Fig. 13 (left) (thin solid) are strongly
biased by the incomplete $p_{t}$ acceptance (i.e. lower bound $p_{t,cut}$).
The TCM results in the same panel illustrate the consequence of determining
$\bar{p}_{t}$ by integration down to zero $p_{t}$. In contrast, integrating
TCM spectra on data $p_{t}$ values (open squares, with cutoff) is close to
(but not equal to) integrating spectra from Ref. alicenewspec  without
extrapolation (open triangles).

(b) The $p_{t,cut}$ bias can be removed via Eq. (VI.1) as shown in Fig. 13
(right). The biased TCM data in the left panel (open squares) are then in good
agreement with unbiased TCM data in the right panel. The bias correction
relies on estimating efficiency $\xi\approx 0.84$ based on a TCM soft-
component model that provides good data descriptions down to 0.15 GeV/c as
demonstrated in Sec. III.3.

(c) The conventional $\bar{p}_{t}$ ratio format of Eq. (12) and Fig. 13 (left)
mixes two distinct physical mechanisms and is therefore difficult to
interpret. Aside from bias corrections the data format corresponding to Eq.
(VI.1) and Fig. 13 (right) provides a clear distinction between $\bar{p}_{t}$
data soft and hard components, especially some differential features of the
isolated jet contribution $\bar{p}_{th}(n_{s})$.

(d) Even with complete $p_{t}$ acceptance the data spectra from Ref.
alicenewspec  are still strongly biased by the event selection methods, both
SPD and V0M, as illustrated by comparison with fixed (not fitted) TCM spectra
in Sec. II.2 (ratios in right panels). With demand for lower $n_{ch}$, spectra
for any $S_{0}$ are increasingly softened leading to decreased $\bar{p}_{t}$
relative to the TMC trend in Fig. 13 (left) (compare open squares and open
triangles) and Fig. 15 (left, note the strong drop-off near and below the NSD
$\bar{\rho}_{s}$ value). The effect is most apparent in Fig. 15 (right)
because $\bar{p}_{th}(n_{s})$ corresponds to slopes in the left panel.

(e) Event selection via spherocity $S_{0}$ was intended primarily to bias jet
production alicenewspec . Data indicate that $S_{0}$ instead biases the
spectrum soft component as evident from the nearly-linear trends in Fig. 15
(left) where the slopes (jet contribution) vary little while the $y$-axis
intercepts (soft component) vary strongly according to a simple power-law
trend shown in Fig. 14 (right). The origin of the bias is simple to identify.
Large values of $S_{0}$ favor uniform azimuth distributions of nearly-equal
$p_{t}$ values. That condition then biases against higher-$p_{t}$
contributions to the soft component which reduces $\bar{p}_{ts}$. Strong
$S_{0}$ bias of $\bar{p}_{ts}$ would occur even in the absence of jets. The
inferred soft-component power-law $\bar{p}_{ts}^{\prime}(S_{0})$ trend in Fig.
14 (right) is consistent for all values of $S_{0}$. Figure 15 (right) shows
that if $S_{0}$ bias of $\bar{p}_{ts}$ is corrected there is significant bias
of $\bar{p}_{th}(n_{s},S_{0})$ beyond what is evident in Figs. 1 and 2 (b,d),
but the bias magnitude is small compared to variation of jet production with
$n_{ch}$.

(f) Just as for $p_{t}$ spectra, the TCM provides a basic reference for
understanding ensemble-mean $\bar{p}_{t}$ trends. The TCM is not fitted to
individual data sets or collision systems. It is, in effect, a global
representation of a large data volume including yields, spectra and two-
particle correlations from multiple A-B collision systems over a broad range
of collision energies. Deviations of properly-corrected particle data from the
TCM then reveal differential details of the information carried by those data.

These conclusions can be compared with those reported in Ref. alicenewspec .
In the introduction appears “The aim of this study is to investigate the
importance of jets in high-multiplicity pp collisions and their contribution
to charged-particle production at low $p_{T}$.” The contribution of jets to
hadron spectra at lower $p_{t}$ in $p$-$p$ collisions has been established in
a series of papers over fifteen years ppprd ; eeprd ; fragevo ; hardspec ;
anomalous ; jetspec ; jetspec2 , none of which is cited in Ref. alicenewspec .
The Ref. alicenewspec  abstract states “Within uncertainties, the functional
form of $\langle p_{t}\rangle(n_{ch})$ is not affected by the spherocity
selection,” but that statement conflicts with actual data properties as
revealed in Fig. 15 (right). $S_{0}$ bias relating to the jet contribution to
spectra is small compared to that for soft component $\bar{p}_{ts}$.
Imposition of an $S_{0}$ condition on spectra does result in systematic bias
of the jet-related spectrum hard component which is detected only by means of
a highly differential analysis technique based on the TCM. As spectra and
ensemble $\bar{p}_{t}$ data are presented in Ref. alicenewspec  that
information is inaccessible.

##  VII Systematic uncertainties

Estimation of data systematic uncertainties typically addresses the
reliability of numerical values resulting from physical measurements. In
addition to inevitable statistical fluctuations the reliability of instrument
calibrations and resulting data corrections is estimated. That approach is
adequate if the interpretation of numerical values is unambiguous. In the
present case where data volumes of 60M or 105M events are the basis for
analysis systematic errors may dwarf statistical errors, making their correct
estimation all that more important.

In this analysis the two-component model, a formal procedure with elements
inferred from lower-energy $p$-$p$ spectrum data, is used to separate two
disjoint contributions to measured 13 TeV $p$-$p$ $p_{t}$ spectra. For the
analysis of systematic uncertainties in this study both the accuracy of the
separation and physical interpretation of the results are in question. This
section addresses the following questions: (a) What is the overall accuracy of
the TCM for $p$-$p$ $p_{t}$ spectrum data descriptions? (b) What is the
significance and physical interpretation of selection-bias trends relative to
the TCM? (c) What is the accuracy and interpretation of spectrum
parametrizations related to jets? (d) What is the effect of spherocity as a
basis for jet preference and the physical interpretation of results?

###  VII.1 Overall accuracy of the TCM per Sec. III

Based on Figs. 6 and 8 it might be argued that the TCM shows highly
significant deviations from $p$-$p$ spectrum data and is therefore a poor
model. But the TCM is a predictive reference based on a broad survey of yield,
spectrum and correlation data ppprd ; ppquad ; alicetomspec ; anomalous ;
ppbpid ; jetspec2 ; fragevo  suggesting that the concept of systematic
uncertainty should be reconsidered. How should the TCM be required to describe
data and how well does it do that? Do significant data-model deviations reveal
a faulty model or unexpected information carried by particle data?

In the present study the TCM is applied as a fixed reference to a minimum-bias
ensemble of $p$-$p$ $p_{t}$ spectra sorted into multiplicity classes via two
methods. The fixed TCM then provides highly differential information on
resulting spectrum bias. If the TCM were fitted to individual spectra the bias
trends would then be represented by varying fit parameters whose physical
interpretation might be difficult. By examining resulting data deviations from
a fixed model in relation to statistical errors as in Fig. 8 a likely physical
interpretation may be possible that was not intuitively obvious beforehand.

The fixed TCM thus provides a valuable reference that is highly constrained by
the requirement to describe $p$-$p$ $p_{t}$ spectra for all energies from 17
GeV to 13 TeV via simple systematic variation of model parameters and
agreement with predictions based on jet data as illustrated in App. A. The
model parameters that appear in Table 1 are those appearing in Table 2 with
the following exceptions: The values for $q$ used in the present study are 4.0
and 3.8 respectively for 5 and 13 TeV whereas the values in Table 2 are 3.85
and 3.65. The values for $\sigma_{y_{t}}$ used in the present study are 0.58
and 0.60 respectively for 5 and 13 TeV whereas the values in Table 2 are 0.58
and 0.615. Those changes arise because the 13 TeV spectrum data employed in
Ref. alicetomspec  were in effect SPD data. In the present study it was
decided to favor V0M data with the TCM. The motivation is evident in Fig. 9
(right). The values for $\alpha$ estimated in the present study are 0.0145 and
0.0170 whereas the values in Table 2 are 0.013 and 0.015 based on the earlier
13 TeV data with its limited $n_{ch}$ reach (maximum $\bar{\rho}_{0}\approx
12$ vs 54 for SPD in the later study) and 1M total events vs 60M for the later
study. With those changes the TCM provides a good description of V0M data for
the six highest $n_{ch}$ classes as demonstrated in Fig. 8. Deviations for
lower $n_{ch}$ classes carry important new information about biases as noted.

The TCM may be further parametrized to describe $n_{ch}$ trends of spectrum
data in detail as illustrated in App. A. The $n_{ch}$ dependence of 13 TeV SPD
spectra can then be described within data uncertainties mainly by
accommodating hard-component trends. The soft component exhibits no
significant $n_{ch}$ dependence in its shape. Within the TCM context variation
of hard-component parameters with $n_{ch}$ may then be physically interpreted
in terms of mean jet characteristics altered due to selection bias.

###  VII.2 Bias trends and “fit” quality per Sec. IV

For the Ref. alicenewspec  analysis three issues are important for data
interpretation: (a) spectrum normalization for each $n_{ch}$ class, (b) the
jet contribution to spectra for each $n_{ch}$ class and (c) selection bias for
each $n_{ch}$ class and each event selection method. Without a well-defined
reference model it is difficult to distinguish among those issues.

Figure 16 (left) shows Fig. 10 (right) repeated for further consideration. The
dashed curves for V0M/TCM and SPD/TCM spectrum ratios are those in Fig. 2 (b)
and (d) for event classes II (V0M) and VII (SPD), each with
$\bar{\rho}_{0}\approx 20$. The data spectrum ratio SPD/V0M (solid) is
consistent with the lower panel of Fig. 4 in Ref. alicenewspec . The
statistical noise artifact common to V0M and SPD spectra (dashed curves)
cancels in ratio as an example of common-mode noise rejection. As noted in
Sec. V.2 the SPD/V0M trend is interpreted by Ref. alicenewspec  to indicate
that the V0M spectrum is “harder” than the SPD spectrum. However, comparison
of the corresponding TCM reference spectra (dash-dotted) indicates that most
of the deviation from unity is simply due to the difference in charge density
$\bar{\rho}_{0}$ for the two event classes.

Figure 16:  Left: Repeat of Fig. 10 (right) for further discussion. Right:
$\chi_{\nu}^{2}$ values [second line of Eq. (VII.2)] for each curve in Fig. 8.
Data are observed to approximate power-law trends.

The data-model comparisons of Fig. 8 illustrate the statistical significance
of the data-model deviations. It is useful to compare the “fit” quality of the
TCM compared to V0M and SPD data spectra, although the TCM is not fitted to
individual spectra. The quality of the data description for a given model can
be estimated by the reduced $\chi^{2}$ statistic $\chi^{2}_{\nu}$

$\displaystyle\chi^{2}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{\nu}\sum_{i=1}^{N}\frac{(O_{i}-E_{i})^{2}}{\sigma_{i}^{2}}$
$\displaystyle\approx$ $\displaystyle\frac{1}{N}\sum_{i=1}^{N}Z_{i}^{2},$

where $O_{i}$ are observations, $E_{i}$ are model predictions and $\nu$ is the
number of degrees of freedom – the number of observations $N$ less the number
of model parameters. In the present context a comparable model quality measure
is approximated by the second line based on Z-scores as defined in Eq. (6).
Since the TCM is not fitted to individual spectra and the resulting statistic
is not compared to a $\chi^{2}$ probability distribution the second form is
applied.

Figure 16 (right) shows $\chi_{\nu}^{2}$ values [second line of Eq. (VII.2)]
for each curve in Fig. 8. Note that $\chi^{2}$ is ordinarily employed as a
measure of goodness of fit but in the present case is a measure of the amount
of new information carried by data relative to the TCM reference. The V0M and
SPD trends are separately equivalent when comparing 5 TeV and 13 TeV data.
Both event selection types happen to approximate power-law trends on
$\bar{\rho}_{0}$.

The vertical dotted line at $\bar{\rho}_{0}=20$ locates class II of V0M (20.5)
and class VII of SPD (19.5) events for 13 TeV. There is a factor 5 difference
between $\chi_{\nu}^{2}$ values. However, the difference would be much greater
if not for the structures in Fig. 16 (left) that are common to all spectra as
noted in connection with Fig. 9 (left). Without that noise the higher $n_{ch}$
classes of V0M events are approximately statistically consistent with the TCM.

###  VII.3 Power-law fits: data vs TCM per Sec. V

This refers to Fig. 5 of Ref. alicenewspec  vs Fig. 9 of the present study and
the substantial differences in the two results. One result arises from a
power-law fit to a limited $p_{t}$ interval; the other result is obtained with
a log-derivative applied to spectrum data over a larger interval. The latter
data are averaged over an interval chosen based on differential data in Fig. 9
(left) that directly indicates what $p_{t}$ interval can be approximated by a
power law.

Numerically, the power-law model fits result in variation for V0M from 6.0
down to 5.7 and for SPD from 6.2 down to 4.9 whereas the log-derivative
produces a fixed value 5.6 for V0M and variation from 5.7 down to 5.2 for SPD.
The fixed log-derivative results for V0M are consistent with data/TCM ratios
in Fig. 2 (d): i.e. no significant deviation from the TCM hard-component shape
above $y_{t}$ = 4. The same conclusion can be inferred from Fig. 4 (b). The
varying $n$ results for SPD are consistent with Eq. (17) for $2/q$ in App. A.
Note that plotting estimates of $2/q$ rather than $q$ leads to a simple linear
trend as in Fig. 17 (left) indicating that the effective high-$p_{t}$ “width”
of the spectrum hard component (i.e. $2/q$) increases linearly with SPD yield,
an informative result.

It is unfortunate that relatively large artifacts appear in the data as
sources of systematic uncertainty, as in Fig. 9 (left). The statistical power
of 60 million 13 TeV events is thereby degraded, and in the case of the log-
derivative requires rejecting the lowest and highest $n_{ch}$ classes.
Nevertheless, the consistency of both V0M and SPD exponent data with their
respective smooth trends is indicative of the power of the log-derivative
method.

###  VII.4 Ensemble $\bf\bar{p}_{t}$ per Sec. VI

Although Ref. alicenewspec  estimates systematic uncertainties for ensemble
$\bar{p}_{t}$ (quoted as 1-2%) the substantial bias from incomplete $p_{t}$
acceptance is not discussed. It is stated that “The efficiency correction [to
$\bar{p}_{t}$]…is found to be $\sim 1$%”, and “The effect of track cuts on
$\langle p_{T}\rangle$ was found to be…of the order of 1%.” The only mention
of the low-$p_{t}$ cutoff is “This [primary particle composition] uncertainty
takes into account the extrapolation of the spectra to low $p_{T}$….” It is
further stated that “The transverse momentum spectra…are fully
corrected…[emphasis added].” As demonstrated in Sec. VI.3 the $p_{t}$ cutoff
bias to $\bar{p}_{ts}^{\prime}$ is about 0.1 GeV/c corresponding to 10-20% of
the $\bar{p}_{t}$ data values, compared to ALICE estimated systematic
uncertainties $O(1\%)$ as noted above.

Given that the total excursion of uncorrected $\bar{p}_{t}^{\prime}$ in Fig.
13 (left) is about 65% of the soft-component value 0.48 GeV/c, interpretation
of uncorrected data is problematic. The nominal goal of the ALICE spectrum
study is determination of jet contributions to spectra, and $\bar{p}_{t}$ data
are essential for achieving that goal. But one can contrast Fig. 13 (left),
where interpretation of the data is quite uncertain, with Fig. 13 (right)
where separation of jet and non-jet contributions is accurately achieved.

Another important uncertainty relating to $\bar{p}_{t}$ data interpretation is
the consequence of event selection based on spherocity $S_{0}$. It is expected
that low spherocity will prefer “jetty” events and high spherocity will prefer
“isotropic” events. The structure of Fig. 13 (left) obscures how the
$\bar{p}_{t}$ vs $n_{ch}$ trend relates to jets whereas the corrected
structure of Fig. 13 (right) brings clarity. Given that clarification the
simplicity of Fig. 15 (left) leads to the correct interpretation of “…within
uncertainties the overall shape of the [$\bar{p}_{t}$ vs $n_{ch}$]
correlation…is not spherocity-dependent.” The correct interpretation is that
while jet production varies approximately quadratically with $n_{ch}$, and jet
production for SPD events increases $\approx 30$-fold relative to NSD $p$-$p$,
spherocity has almost no effect on the jet contribution to spectra,

##  VIII Discussion

In its introduction Ref. alicenewspec  asserts that a $p_{t}$ spectrum
“carries information of the dynamics of soft and hard interactions.” As noted
“The aim of [Ref. alicenewspec ] is to investigate the importance of jets in
high-multiplicity pp collisions and their contribution to charged-particle
production at low $p_{T}$.” It is proposed to “disentangle the energy and
multiplicity dependence” of $p_{t}$ spectra.” The TCM has been applied to
$p$-$p$ $p_{t}$ spectra over three orders of magnitude of $p$-$p$ collision
energy, and after fifteen years of development arguably extracts all available
information from $p_{t}$ spectra ppprd ; alicetomspec ; ppbpid . The energy
and multiplicity dependences of $p_{t}$ spectra are indeed factorizable, but
the energy dependence requires a large energy interval to identify systematic
variations accurately; the interval 5 to 13 TeV is too small to do so
alicetomspec ; jetspec2 . Given those observations it is instructive to
consider certain comments within Ref. alicenewspec  relative to TCM results.

###  VIII.1 Spectrum evolution with $\bf n_{ch}$

This topic mainly concerns Figs. 2 and 3 of Ref. alicenewspec  which present
ratios of spectra for different $n_{ch}$ classes, two energies and two event
selection criteria to minimum-bias INEL $>0$ spectra. It is noted that “the
features of the spectra…are qualitatively the same for both energies” and only
the 13 TeV result is further discussed. That strategy can be compared with
“disentangle the energy and multiplicity dependence.” If two things are
“qualitatively the same” then they are quantitatively dissimilar, i.e. the
difference is information carried by $p_{t}$ spectra.

Commenting on Figs. 2 and 3 “The [spectrum] ratios to the INEL $>0$ $p_{T}$
distribution exhibit two distinct behavior [sic].” In brief, at lower $p_{t}$
the spectrum ratios exhibit small $p_{t}$ dependence, but above 0.5 GeV/c the
ratios are strongly dependent on $n_{ch}$ and $p_{t}$. Referring to Fig. 2 is
the comment “…the $p_{T}$ spectra become harder as the multiplicity increases,
which contributes to the increase of the average transverse momentum with
multiplicity.” But the seemingly dramatic spectrum “hardening” in Fig. 2 does
not dominate $\bar{p}_{t}$($n_{ch}$) trends where the central issue is dijet
production as a function of $n_{ch}$ described comprehensively via the TCM
alicetommpt ; tommpt . The term “hardening” is ambiguous between increased jet
number with $n_{ch}$ and possible bias of jet fragment distributions fragevo .

In abstract and main text is the statement “The high-$p_{T}$ ($>4$ GeV/c)
yields of charged particles increase faster than the charged-particle
multiplicity, while the increase is smaller [i.e. less rapid than $n_{ch}$]
when we consider lower-$p_{T}$ particles.” In relation to its Fig. 6
describing charge integrals within specific $\Delta p_{t}$ intervals Ref.
alicenewspec  observes “Despite the large uncertainties, it is clear the data
show a non-linear [i.e. faster than $n_{ch}$] increase.” But the TCM has
provided an accurate quantitative picture of such trends for fifteen years as
noted above. The low-$p_{t}$ part of spectra ($<0.5$ GeV/c or $y_{t}\approx
2$) increases
$\propto\bar{\rho}_{s}\approx\bar{\rho}_{0}-\alpha\bar{\rho}_{s}^{2}$, i.e.
slower than $\bar{\rho}_{0}=n_{ch}/\Delta\eta$, whereas the higher-$p_{t}$
part of spectra (i.e. $>4$ GeV/c or $y_{t}\approx 4$) increases precisely
$\propto\bar{\rho}_{s}^{2}$, i.e. approximately quadratically with $n_{ch}$.
Again as noted, a characteristic aspect of certain observations in Ref.
alicenewspec  is confusion among several issues: (a) spectrum normalization
(or not), (b) jet production vs $n_{ch}$, and (c) various selection-bias
effects.

###  VIII.2 Manifestations of jets in $\bf p_{t}$ spectra

Given a primary goal of Ref. alicenewspec  – determination of the jet
contribution to particle production at lower $p_{t}$ – it is difficult to find
any responding result in the paper. Regarding spectra in relation to pQCD
“…the high-$p_{T}$ ($p_{T}>10$ GeV/c) particle production is quantitatively
well described by perturbative QCD (pQCD) calculations….” But no experimental
evidence is presented relating jet contributions at higher $p_{t}$, or jet
production in general, to hadron production at lower $p_{t}$. In contrast, the
TCM quantitatively isolates minimum-bias jet contributions to hadron
production over the entire $p_{t}$ acceptance as in Sec. III.3, and spectrum
hard components have been quantitatively related to pQCD via measured jet
energy spectra and fragmentation functions fragevo ; jetspec2 .

Reference alicenewspec  does acknowledge information derived from model fits
to spectra: “Commonly, the particle production is characterized by quantities
like integrated yields or any fit parameter of the curve extracted from fits
to the data, for example, the so-called inverse slope parameter [$T$]…” and
then emphasizes power-law exponent $n$ as relating to jet production. For
description of spectra at higher $p_{t}$ “the natural choice is fitting a
power-law function…to the invariant yield [$p_{t}$ spectrum?] and studying the
multiplicity dependence of the exponent ($n$) extracted from the fit.” One
conclusion – “the results [the $n(n_{ch})$ trend] using the two multiplicity
estimators [V0M and SPD] are consistent within the overlapping multiplicity
interval” – is problematic as argued below.

A mechanism for $n_{ch}$ dependence of exponent $n$ is conjectured as follows:
“In PYTHIA 8, it has been shown that the number of high-$p_{T}$ jets increases
with event multiplicity.” In fact, the exact multiplicity dependence of jet
production in inelastic $p$-$p$ collisions is reported in Ref. jetspec2  based
on event-wise-reconstructed jet measurements, not a Monte Carlo. The
conjecture continues: “…based on PYTHIA 8 studies, the reduction of the power-
law exponent [n] with increasing multiplicity [$n_{ch}$] can be attributed to
an increasing number of high-$p_{T}$ jets.” But for V0M event selection in
Fig. 9 (right) the exponent $n$ trend on $n_{ch}$ is consistent with a
constant (13 TeV) or even slight increase (5 TeV). The same jet population is
accessible to either selection criterion and jet number is strictly dependent
on $n_{ch}$ as demonstrated by Fig. 12. All that differs is the spectrum bias
at high $p_{t}$ induced by the selection method. Such arguments confuse basic
QCD jet production (with its well-established systematics) and manifestations
of event selection bias.

The dijet production rate for NSD $p$-$p$ collisions can be predicted from
measured jet cross sections, and such predictions can be compared
quantitatively with jet contributions to spectra and two-particle correlations
identified by their unique $n_{ch}$ dependence (e.g. as represented by the
TCM) fragevo . For 200 GeV NSD collisions 3% of events include a dijet per
unit of eta ppprd . For $\approx 10$ TeV NSD collisions 15% of events include
a dijet per unit of eta jetspec2 , where the quoted percentages are
$100(1/\sigma_{\text{NSD}})d\sigma_{\text{jet}}/d\eta$. For a typical range of
$p$-$p$ charge densities
$\bar{\rho}_{0}\in[\bar{\rho}_{0\text{NSD}},10\bar{\rho}_{0\text{NSD}}]$ the
dijet yield should increase by factor 100 due to the observed trend
$\bar{\rho}_{h}\propto\bar{\rho}_{s}^{2}$. 13 TeV $p$-$p$ collisions with SPD
$\bar{\rho}_{0}\approx 54$ should include 11 dijets per unit pseudorapidity on
average. “Jets” here assumes a minimum-bias jet spectrum wherein most jets
appear near 3 GeV (the effective jet spectrum mode).

###  VIII.3 Selection biases and QCD

The same minimum-bias INEL $>0$ event ensemble is partitioned into ten
multiplicity classes according to two selection criteria – V0M and SPD. The
V0M or “forward [on $\eta$] multiplicity estimator” is said to “minimize the
possible autocorrelations induced by the use of the midpseudorapidity
estimator.” The statement suggests that the V0M criterion should produce
substantially less bias. “The comparison of results obtained with these [V0M
and SPD] estimators allows to understand potential biases from measuring the
multiplicity and $p_{T}$ distributions in overlapping $\eta$ regions.” As with
statements about jet contributions to lower $p_{t}$ it is difficult to find
within Ref. alicenewspec  any such understanding. In fact “bias” as in that
sentence does not appear again in the paper.

For spectrum data plotted as ratios, as in Figs. 1 and 2 (b,d), selection bias
from V0M and SPD criteria appear quite different. However, the apparent large
high-$p_{t}$ bias for SPD involves a tiny fraction of all particles and even a
small fraction of jet fragments. When the two criteria are compared based on
significance as in Fig. 8 the SPD high-$p_{t}$ bias does not dominate the
structure. Ironically, V0M bias is at least as significant as SPD bias and the
$p_{t}$ dependence is similar, contradicting the expectation that
“autocorrelations” are minimized by disjoint $\eta$ intervals.

Particle production and QCD dynamics must be strongly correlated between V0M
and SPD acceptances. Hadron production at midrapidity depends in part on a
parton splitting cascade within each projectile proton that also contributes
hadrons at larger $\eta$. Thus, fluctuations in V0M must be strongly
correlated with fluctuations in SPD, albeit SPD has additional contributions
from parton scattering and fragmentation to jets. Figure 8 reveals that
fluctuations correlated between V0M and SPD are statistically dominant,
whereas fluctuations in low-energy jet formation play a less-significant role.

###  VIII.4 Ensemble $\bf\bar{p}_{t}$ vs spherocity

As part of a strategy to identify jet contributions at lower $p_{t}$ Ref.
alicenewspec  introduces spherocity $S_{0}$ (effectively an azimuthal
asymmetry measure): “The present paper reports a novel multi-differential
analysis aimed at understanding charged-particle production associated to
partonic scatterings with large momentum transfer and their possible
correlations with soft particle production.” But most of the jet-related hard
component arises from lower-energy partons jetspec2 ; fragevo , and most
fragments from any jet appear at low $p_{t}$ eeprd . “Transverse
spherocity…has been proven to be a valuable tool to discriminate between jet-
like and isotropic events….” That statement is based on material presented in
Ref. ortizspher  which is a study of various event-shape measures applied to
PYTHIA simulations. It is not clear from that study what spherocity
contributes to understanding real $p$-$p$ collisions.

“Studying observables as a function of spherocity reveals interesting
features.” Relative to the $\bar{p}_{t}$ vs $n_{ch}$ trend for the INEL $>0$
ensemble average as a reference the trend for “isotropic” events (high
$S_{0}$) “stays systematically below” the reference, whereas “…for jet-like
events” (low $S_{0}$) the $\bar{p}_{t}$ trend is consistently higher. The
following observation is interesting: “Moreover,…the overall shape of the
correlation [$\bar{p}_{t}$($n_{ch}$)], i.e. a steep linear rise below
$dN_{ch}/d\eta=10$ followed by a less steep but still linear rise above, is
not spherocity-dependent [emphasis added].”

That qualitative description relates to a confusing data trend arising from
questionable analysis variables. The data trend for uncorrected
$\bar{p}_{t}^{\prime}(n_{ch})$ includes the misleading dependence expressed by
Eq. (12) and hard-component suppression at lower $n_{ch}$ exhibited in Fig. 8.
In contrast, the approximate linear rise of corrected $\bar{p}_{t}$ vs
$\bar{\rho}_{s}$ in Fig. 13 (right) and Fig. 15 (left) is equivalent to the
linear rise of integrated yields with $\bar{\rho}_{s}$ in Fig. 12. In other
words, the TCM structure of Eq. (VI.1) is equivalent to the TCM structure of
Eq. (VI.1). Jets control the slope of the linear trend over a large $n_{ch}$
interval: in the absence of jets there would be no linear rise, i.e. the slope
of the trend is a measure of jet production. By observing that $\bar{p}_{t}$
vs $n_{ch}$ trends in Fig. 13 (left), equivalent to trends in Fig. 15 (left),
are “not spherocity-dependent” Ref. alicenewspec  admits that $S_{0}$ does
little to control jet production in $p$-$p$ collisions. That the high-$S_{0}$
and low-$S_{0}$ trends appear respectively systematically below and above the
reference in Fig. 7 of Ref. alicenewspec  is a consequence of $S_{0}$ biasing
the nonjet spectrum soft component as noted in Sec. VI.3.

##  IX summary

This article reports a study of high-statistics $p_{t}$ spectra from 5 TeV and
13 TeV $p$-$p$ collisions at the large hadron collider. This study is based on
the two-component (soft + hard) model (TCM) of hadron production near
midrapidity. The spectrum data and additional spectrum analysis were reported
by the ALICE collaboration with the nominal goal to estimate jet contributions
to $p_{t}$ spectra, especially at lower $p_{t}$. Part of the motivation was a
response to recent claims of “collective” behavior (flows) in small
(asymmetric?) collision systems. The principal data presentation was $p_{t}$
spectra for various event classes in ratio to an ensemble average over all
collision events.

The TCM has been demonstrated over a number of years to be a valuable tool for
isolating distinct hadron production mechanisms in A-B collisions. In
particular, hadron contributions to spectra from jet-related production (hard
component) are accurately distinguished (at the percent level) from nonjet
production (soft component). The signature element of the TCM for $p$-$p$
collisions is a quadratic relation between the hard component and soft
component as inferred from spectrum data.

A major goal for the ALICE study was characterization of selection bias
resulting from sorting collision events into classes according to particle
yields in two different pseudorapidity $\eta$ intervals denoted by SPD
(midrapidity) and V0M (forward rapidity). If event classes are simply compared
to an ensemble average the present study finds that three issues are
consequently confused: (a) spectrum normalization, (b) the jet contribution to
$p$-$p$ spectra as it varies in known ways with multiplicity $n_{ch}$ and (c)
the selection bias in question. This study demonstrates that ratios of data
spectra to TCM spectra remove issues (a) and (b) and expose selection bias (c)
to direct study.

One aspect of the present study is introduction of the Z-score statistical
measure: Whereas spectrum ratios tend to visually exaggerate deviations at
higher $p_{t}$ and suppress those at lower $p_{t}$ Z-scores provide a measure
of the statistical significance of data-model deviations. Data-model
deviations at high $p_{t}$ that seem to dominate ratios are actually only
modestly significant whereas deviations at lower $p_{t}$ are much more
significant and quite similar for V0M and SPD event classifications.

The present analysis reveals that selection bias has two main manifestations
depending on the event selection criterion. The spectrum soft component
appears unaffected by selection method. The V0M and SPD selection methods both
bias the lower-$p_{t}$ parts of the jet-related hard component (fragment
distribution) similarly. For lower charge multiplicities $n_{ch}$ the peaked
hard component is shifted to lower $p_{t}$, resulting in spectrum suppression
at higher $p_{t}$ (above the peak mode) and enhancement at lower $p_{t}$
(below the mode). However, the high-$p_{t}$ tails of spectra are negligibly
affected by V0M selection but strongly affected by SPD selection: with
increasing $n_{ch}$ SPD spectrum tails become increasingly harder (smaller
power-law exponent). The combined effects suggest that selection bias responds
to fluctuations in jet production in different ways depending on the relevant
$\eta$ acceptance.

Ensemble-mean $\bar{p}_{t}$ data as a function of $n_{ch}$ were obtained for
different values of spherocity $S_{0}$, a measure of the azimuth asymmetry of
the vector $\vec{p}_{t}(\phi)$ distribution. The intent was to bias events
according to their “jettiness,” a lower $S_{0}$ value expected to prefer
events with more or more-frequent jets. The published $\bar{p}_{t}$ data are
biased because of the incomplete detector $p_{t}$ acceptance (lower limit at
0.15 GeV/c) making data interpretation more difficult. In the present analysis
the $\bar{p}_{t}$ data are corrected and transformed to a TCM configuration
with fixed soft-component contribution and hard component varying with
$n_{ch}$ in a manner that should reveal the jet contribution to $\bar{p}_{t}$.
This study concludes that the main affect of $S_{0}$ selection on
$\bar{p}_{t}$ vs $n_{ch}$ trends is bias of the $\bar{p}_{ts}$ soft component.
The effect on the hard component (i.e. jet contribution) is barely detectable.
Given $S_{0}$ as an azimuth asymmetry measure, its lower values tend to
suppress events where the soft component includes more high-$p_{t}$ particles,
thus reducing its ensemble-mean $\bar{p}_{ts}$.

In summary, the TCM is observed to be a necessary and sufficient description
of $p$-$p$ $p_{t}$ spectra and arguably represents all information carried by
spectrum data for unidentified hadrons. The TCM hard component has been
quantitatively related to the properties of event-wise reconstructed jets.
Determination of an accurate TCM for isolated spectra (rather than ratios)
over a broad range of event multiplicities and and event selection criteria as
in the present study establishes an accurate and efficient representation of a
large volume of spectrum data. The contribution of jets to $p$-$p$ $p_{t}$
spectra is accurately determined at the percent level and the consequences of
several forms of event selection are isolated.

##  Appendix A Previous 13 $\bf TeV$ data TCM

A previous TCM analysis of 13 TeV $p$-$p$ spectra was reported in Ref.
alicetomspec . That study emphasized multiplicity and energy dependence of the
spectrum hard component in the context of QCD theory and jet measurements. The
13 TeV $p$-$p$ spectrum data as reported in Ref. alicespec  were quite limited
as to range of event multiplicity – see the 13 TeV solid triangles and open
circles in Fig. 17 (left) – and the presentation format was based on ratios of
spectra from three $n_{ch}$ classes to a minimum-bias INEL $>0$ reference. It
was required therefore to develop new techniques to isolate the hard and soft
components for the three $n_{ch}$ classes based on analysis of 200 GeV
spectrum data for which the TCM structure is well-established ppprd . In this
appendix relevant results from Ref. alicetomspec  are reviewed to provide
context for the present study of a much more extensive sample of 13 TeV
$p$-$p$ spectrum data.

###  A.1 Spectrum TCM multiplicity dependence

In Sec. III TCM model functions are held fixed for all $n_{ch}$ classes and
for both V0M and SPD. Systematic data biases are then revealed in a precise
way. However, in Ref. alicetomspec  the TCM hard component was parametrized to
accommodate the $n_{ch}$ dependence of the hard-component shape for 13 TeV SPD
spectra. In this subsection those results (for 200 GeV as well as 13 TeV) are
reviewed and the quality of the data description is evaluated. The 200 GeV
data are equivalent to Ref. alicenewspec  SPD with $|\eta|<1.0$

Figure 17 (left) shows variation of two 200 GeV $\hat{H}_{0}(y_{t})$
parameters with $\bar{\rho}_{s}$ (lower solid and dashed curves) that provides
accurate description of spectrum ratios above the hard-component mode for all
multiplicity classes. Optimized TCM parameters follow simple $\bar{\rho}_{s}$
trends

$\displaystyle 2/q$ $\displaystyle=$ $\displaystyle
0.373+0.0137(\bar{\rho}_{s}/\bar{\rho}_{s,ref})~{}~{}\text{(lower solid)}$
(16) $\displaystyle\sigma_{y_{t}}$ $\displaystyle=$ $\displaystyle
0.385+0.09\tanh(\bar{\rho}_{s}/4)~{}~{}~{}\text{(lower dashed)}.$

The nominal parameter values for the 200 GeV fixed $\hat{H}_{0}(y_{t})$ model
are represented by the dotted and dash-dotted lines (corresponding to
parameter values for $\bar{\rho}_{s}/\bar{\rho}_{s,ref}\approx 2$). Variation
of two parameters in combination serves to broaden the hard-component model
above the mode toward higher $y_{t}$. The saturation of $\sigma_{y_{t}}$ at
larger $\bar{\rho}_{s}$ is a consequence of increasing $2/q$. The transition
on $\hat{H}_{0}(y_{t})$ from Gaussian to exponential form then moves back
toward the mode and the exponential/power-law tail increasingly dominates the
higher-$p_{t}$ structure. $\bar{\rho}_{s,ref}$ values are noted in the Fig. 17
caption.

The corresponding parameter trends for 13 TeV inferred from spectrum ratios in
the earlier analysis of Ref. alicetomspec  are shown by the upper solid
triangles and open circles. New values for $2/q$ (upper solid dots) are
inferred in the present study via log derivative applied directly to 13 TeV
SPD $y_{t}$ spectra via Eq. (7) (upper line). Those results correspond
accurately with $n$ values plotted in Fig. 9 (right) obtained with Eq. (7)
(lower line) demonstrating the correspondence $n\approx q+1.8$. The $2/q$ data
are well described by

$\displaystyle 2/q$ $\displaystyle=$ $\displaystyle
0.472+0.021(\bar{\rho}_{s}/\bar{\rho}_{s,ref})~{}\text{(upper solid)}.$ (17)

Previous 13 TeV $2/q$ values derived from spectrum ratios in Ref. alicetomspec
(triangles) differ substantially from those in the present study and
demonstrate the difficulty of deriving accurate results from limited spectrum
ratios.

Figure 17:  Left: TCM hard-component parameters varying with $n_{ch}^{\prime}$
or $\bar{\rho}_{s}$ for a revised TCM. 200 GeV solid and dashed curves through
parameter data are defined by Eqs. (16). The 13 TeV open circles and solid
triangles are as reported in Ref. alicetomspec . The upper solid dots and
solid line are updates determined in the present analysis, derived from more
extensive $p$-$p$ data. $\bar{\rho}_{s,ref}=2.45$ for 200 GeV NSD $p$-$p$
collisions and 5.8 for 13 TeV NSD collisions. The factor 2 in $2/q$ permits
greater plot sensitivity. The mean-value energy trend for $\sigma_{y_{t}}$ is
shown in Fig. 20 (left) and for $1/q$ is shown in Fig. 19 (left). Right:
Variation of the Gaussian width below the hard-component mode
$\sigma_{y_{t}-}$ (solid points) for $n=2$-7 that accommodates data in that
$y_{t}$ interval. The Gaussian width above the mode $\sigma_{y_{t}+}$ (open
points) is included for comparison. The curves are defined in the text. The
correlation of two trends is notable.

Figure 17 (right) shows variation of the 200 GeV hard-component width required
to accommodate data below the mode in the form $1/\sigma_{y_{t}-}^{2}$ (solid
points). The solid curve through points is
$13.5\tanh[(\bar{\rho}_{s}-3.1)/5]$. Also included is the trend for the width
above the mode from the left panel and Eq. (16) (lower) plotted as
$1/\sigma_{y_{t}+}^{2}$ (open points and dashed curve respectively)
demonstrating correlation of the two trends. The two widths become equal near
$\bar{\rho}_{s}\approx 5$, or $\bar{\rho}_{s}/\bar{\rho}_{s,ref}\approx 2$ in
Fig. 17 (left) where the hard-component model is then, near its mode,
approximately symmetric as in Refs. ppprd ; ppquad .

Figure 18 (left) summarizes the revised 200 GeV TCM hard-component model for
six multiplicity classes. The hard component for the lowest $p$-$p$
multiplicity class $n=1$ is typically severely distorted due to selection bias
(see Sec. III.3 for example).

Figure 18:  Left: Evolution of the 200 GeV hard-component model over six
multiplicity classes that exhausts all information in high-statistics spectrum
data from Ref. ppquad . Right: Spectrum hard components for SPD event
selection and for nine multiplicity classes of 13 TeV $p$-$p$ collisions. The
order of line styles is reversed compared to other figures in this article to
match the order in the left panel, and the order of index $n$ is also reversed
from the present study.

Figure 18 (right) shows 13 TeV SPD spectrum hard components (curves) appearing
in Fig. 4 (d). SPD event selection based on particle yields within
$|\eta|<0.8$ is consistent with event selection for 200 GeV data based on
yields within $|\eta|<1$. Line types for this panel are reversed in order
compared to other figures in this study to match the convention in the left
panel. As noted, the hard component for the lowest multiplicity class (here
$n=1$, not 10 as in the present study) is not shown since there is very little
jet contribution to those events due to strong selection bias. Given those
minor differences the event-selection biases for 200 GeV and 13 TeV $p$-$p$
$p_{t}$ spectra are remarkably similar. The similarity at higher $p_{t}$ is
consistent with the solid lines in Fig. 17 (left).

###  A.2 Spectrum TCM collision-energy dependence

Figure 19 (left) shows soft-component exponents in the form $1/n$ inferred
from spectrum data for three collision energies (solid points) at the SPS,
RHIC and LHC. The solid curve is an algebraic hypothesis based on variation of
the soft component due to conjectured Gribov diffusion gribov . Low-$x$ gluons
result from a virtual parton splitting cascade within projectile nucleons
whose mean depth on $x$ is determined by the collision energy. Each step of
the cascade adds transverse-momentum components in a random-walk process. The
depth of the cascade is proportional to $\ln(s/s_{0})$, and
$\sqrt{s_{0}}\approx 10$ GeV is inferred from dijet systematics anomalous ;
jetspec2 . Given the properties of a random walk and with $1/n$ as a measure
of transverse-momentum excursions wilk  its trend is estimated as
$\propto\sqrt{\ln(\sqrt{s}/\text{10 GeV})}$ (solid curve). The open circles at
0.9, 2.76 and 7 TeV are interpolations of the Lévy exponent to $n=9.82$, 8.83
and $8.16$ respectively.

Figure 19:  Left: Measured Lévy exponents for three collision energies (solid
points). The curve is a fit by eye of the function
$A\sqrt{\ln(\sqrt{s}/\text{10 GeV})}$ (with $A=0.0475$) motivated by the
possibility of Gribov diffusion controlling the growth of transverse momentum
for low-$x$ partons (gluons) gribov . Open symbols are interpolations at
energies relevant to the study in Ref. alicetomspec . Right: Hard-component
exponents $q$ determined by analysis of $p$-$p$ spectrum data (solid points)
from Ref. ppquad  and Ref. alicetomspec . The solid curve is based on a jet-
spectrum parametrization in Ref. jetspec2  that also describes ensemble-
mean-$p_{t}$ hard-component energy variation alicetommpt . Open points are
interpolations and extrapolation relevant to Ref. alicetomspec .

Figure 19 (right) shows inverse values (solid points) of exponents $q=5.15$
for 200 GeV and $q=3.65$ for 13 TeV plotted vs quantity $\Delta
y_{max}\equiv\ln(\sqrt{s}/\text{6 GeV})$ that is observed to describe the
energy trend for jet spectrum widths $\propto\Delta y_{max}$ from NSD $p$-$p$
collisions assuming a jet-spectrum low-energy cutoff near 3 GeV jetspec2  (see
Fig. 5 of Ref. jetspec2  for a direct comparison with measured jet spectra).
The inverse $1/q$ effectively measures the hard-component peak width at larger
$y_{t}$. The relation $1/q\propto\Delta y_{max}$ (solid line) is expected
given that the $p$-$p$ $p_{t}$-spectrum hard component can be expressed as the
convolution of a fixed $p$-$p$ fragmentation-function ensemble with a
collision-energy-dependent jet spectrum fragevo , and the jet-spectrum width
trend has the same dependence jetspec2 . The vertical hatched band indicates
an inferred cutoff to dijet production from low-$x$ gluon collisions near 10
GeV. That the same relation applies to the ensemble $\bar{p}_{t}$ hard
component was established in Ref. alicetommpt . $1/q$ for $q=3.80$ for 7 TeV,
$q=4.05$ for 2.76 TeV and $q=4.45$ for 0.9 TeV (open circles) are
interpolations.

Figure 20 (left) shows NSD TCM hard-component model parameters $\bar{y}_{t}$
and $\sigma_{y_{t}}$ (points) vs collision energy. The solid points are
derived from data. The open points are interpolations or extrapolations
derived from the inferred or predicted trends in the figure (curves). The
trends for $\bar{y}_{t}$ and $\sigma_{y_{t}}$ are consistent with straight
lines. Whereas $\sigma_{y_{t}}$ increases by 50% the upper limit on
$\bar{y}_{t}$ variation is five percent (hatched band) and $\bar{y}_{t}$ may
not actually vary significantly over over three orders of magnitude of
collision energy. The trend for $\bar{y}_{t}$ is consistent with a fixed lower
bound on the underlying jet spectrum near 3 GeV, also nearly independent of
collision energy fragevo ; jetspec2 .

Figure 20:  Left: TCM NSD hard-component parameters derived from spectrum data
(solid points). Open circles are interpolations or extrapolations relevant to
Ref. alicetomspec . The solid lines are fits to data. The dashed and dash-
dotted curves related to $\alpha(\sqrt{s})$ are described in Ref. alicetomspec
. Right: Survey of spectrum hard components over the currently accessible
energy range from threshold of dijet production (10 GeV) to LHC top energy (13
TeV). The curves are determined by parameters in Table 2 except for the 200
GeV fine solid curves determined also by the $\sigma_{y_{t}}$ and $q$ trends
in Fig. 17 (left). The points are from Refs. ppquad  (200 GeV) and alicespec
(13 TeV).

Figure 20 (right) shows the TCM for quantity
$H(p_{t};E)/\bar{\rho}_{s}(E)\approx\alpha(E)\bar{\rho}_{s}(E)\hat{H}_{0}(p_{t};E)$
measuring the spectrum hard component per soft-component yield corresponding
to dijet production per participant low-$x$ gluon. The two dotted curves are
for 0.9 and 2.76 TeV and the dashed curve is for 7 TeV. Isolated hard
components rather than spectrum ratios clarify spectrum energy evolution and
its relation to dijet production. The hard-component mode on $p_{t}$ is near
0.5 GeV/c ($y_{t}\approx 2$) whereas the mode on $y_{t}$ is near 2.7
($p_{t}\approx 1$ GeV/c).

The predictions for six collision energies (curves) derived from parameter
values in Table 2 have been compared to data from four energies (13, 7, 0.9
and 0.2 TeV ) alicetomspec . The 17.2 GeV extrapolation indicates no
significant jet contribution to yields and spectra at that energy and explains
why no excess $p_{t}$ fluctuations were observed at the SPS na49fluct ;
tomaliceptfluct . However, evidence for SPS jets is visible in 17 GeV azimuth
correlations as a more sensitive detection method ceres . The 200 GeV summary
(thin solid) includes parametric variation of
$\hat{H}_{0}(y_{t};q,\sigma_{y_{t}},n_{s})$ for six multiplicity classes as
described in Sec. A.1. Corresponding data (solid points) represent NSD $p$-$p$
collisions. The overall result is a comprehensive and accurate description of
dijet contributions to $p_{t}$ spectra vs $p$-$p$ collision energy over three
orders of magnitude.

###  A.3 Spectrum TCM parameter summary

Table 2 summarizes NSD $p$-$p$ TCM parameters for a broad range of energies.
The entries are grouped as soft-component parameters $(T,n)$, hard-component
parameters $(\bar{y}_{t},\sigma_{y_{t}},q)$, hard-soft relation parameter
$\alpha$ and soft density $\bar{\rho}_{s}$. Slope parameter $T=145$ MeV is
held fixed for all cases consistent with observations. Its value is determined
solely by a low-$y_{t}$ interval where the hard component is negligible. The
interpolated Lévy exponent $n$ values are derived from Fig. 19 (left) (open
circles). Interpolated hard-component $q$ values are derived from Fig. 19
(right) (open circles). $\bar{\rho}_{s}$ values are derived from the universal
trend $\bar{\rho}_{s}\approx 0.81\ln(\sqrt{s}/\text{10 GeV})$ inferred from
correlation and yield data. All 0.9 and 2.76 TeV values are predicted via
interpolation. All remaining (unstarred) numbers are obtained from spectrum
data.

Table 2: Spectrum TCM parameters for NSD $p$-$p$ collisions at several energies from Ref. alicetomspec . Starred entries are estimates by interpolation or extrapolation. Unstarred entries are derived from yield, spectrum or spectrum-ratio data.  $\sqrt{s}$ (TeV) | T(̇MeV) | $n$ | $\bar{y}_{t}$ | $\sigma_{y_{t}}$ | $q$ | $100\alpha$ | $\bar{\rho}_{s}$
---|---|---|---|---|---|---|---
0.0172 | 145 | 27 | 2.55∗ | 0.40∗ | 6.75∗ | 0.07∗ | 0.45
0.2 | 145 | 12.5 | 2.59 | 0.435 | 5.15 | 0.6 | 2.45
0.9 | 145 | 9.82∗ | 2.62∗ | 0.53∗ | 4.45∗ | 1.0∗ | 3.65
2.76 | 145 | 8.83∗ | 2.63∗ | 0.56∗ | 4.05∗ | 1.2∗ | 4.55
5.0 | 145 | 8.47∗ | 2.63∗ | 0.58∗ | 3.85∗ | 1.3∗ | 5.00
7.0 | 145 | 8.16∗ | 2.64 | 0.595 | 3.8∗ | 1.4∗ | 5.30
13.0 | 145 | 7.80 | 2.66 | 0.615 | 3.65 | 1.5 | 5.80

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