# Comparison of spectrum models as applied to single-particle $\bf p_{t}$
spectra from high-energy p-p collisions and their physical interpretations

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

###### Abstract

A parametrized mathematical model is required to extract the information
carried by transverse momentum $p_{t}$ spectra from high-energy nuclear
collisions and subject it to physical interpretation in terms of possible
hadron production mechanisms. It is essential that model elements correspond
accurately to relevant mechanisms. The importance of proper model construction
and implementation has increased with the emergence of claims for
“collectivity” (flows) associated with small collision systems (e.g. $p$-$p$
and $p$-Pb). Argument by analogy concludes that if flows demonstrate quark-
gluon plasma (QGP) formation in A-A collisions, as currently believed, then
QGP formation must also occur in small systems despite low particle densities.
A two-element spectrum model, denoted herein as the Bylinkin model, includes
an exponential element and a “power-law” element interpreted by the authors to
represent emission from a thermalized source and from jet production
respectively. Application of the Bylinkin model to various collision systems
has led to conclusions about achievement of thermalization and other
characteristics of nuclear dynamics. In connection with the Bylinkin model
there has emerged theoretical conjecture that the thermalization mechanism
signaled by the exponential element is hard processes interacting with quantum
entanglement within projectile protons. Predating the Bylinkin model is a two-
component (soft+hard) model (TCM) derived empirically from the evolution of
$p$-$p$ spectrum data with event multiplicity as a form of data compression.
The TCM has been applied to many collision systems and hadron species from
which a systematic description of high-energy nuclear collisions has emerged.
The Bylinkin model can be seen as a limiting case of the TCM model functions
that is unsuited to represent underlying production mechanisms. The present
study provides detailed comparisons of the two models for a variety of
situations and contrasts two very different data interpretations that result.

††preprint: version 1.3

##  I Introduction

Inferred presence of collectivity (flows) within high-energy collisions
involving small systems (e.g. $p$-$p$, $p$-A collisions) since commencement of
LHC operations in 2010 has been interpreted to support claims of quark-gluon
plasma (QGP) formation in small systems nagle . The claims are based on
certain data features in spectra and angular correlations including
“hardening” of spectra with increasing event multiplicity $n_{ch}$ and hadron
mass that appear similar to spectrum trends in A-A collisions where QGP
formation is considered to be established alicepppid . Such claims are
counterintuitive based on conventional understanding of QCD and thus merit
careful examination of arguments and evidence in their favor.

Reference alicepppid  observes that at low $p_{t}$ “…collective phenomena are
observed in…[$p$-$p$, $p$-Pb and Pb-Pb] collisions….” It asserts that “…in
order to describe bulk particle production in A-A collisions, one usually
relies on hydrodynamic and thermodynamic modeling….” Spectrum variation that
may be associated with radial expansion (flow) is “….studied in the context of
the Boltzmann-Gibbs Blast-Wave [BW] model.” Parameter values derived from BW
model fits to $p_{t}$ spectra are interpreted as indicating thermal
temperatures and flow velocities.

If QGP formation in small collision systems is accepted based on
interpretation of certain collision data then the problem emerges how a
thermalized dense medium, assumed as a precursor to QGP formation, is achieved
in small low-density systems. One possibility involves the role of quantum
entanglement in high-energy collisions. A theoretical conjecture proposes that
entanglement might be manifested in the entropy of the hadronic final state of
deep-inelastic-scattering (DIS) data dimalev . The argument has been extended
to propose quantum entanglement as the mechanism for thermalization in small
collisions indicated by observation of certain spectrum features dimaentangle
.

Just as BW model fits to $p_{t}$ spectra have been interpreted to demonstrate
the presence of radial flow and to provide estimates of thermal temperature
and radial speed of a flowing medium, a spectrum model composed of two
functions has been interpreted to indicate thermalization based on one of its
elements. The model consists of combined exponential and power-law elements,
referred to here as the Bylinkin model after its principal author bylinkin2010
. As the model is described the presence or absence of its exponential
element, as required for description of collision data, determines whether a
collision system has thermalized in the Boltzmann sense of multiple particle
collisions that tend to maximize entropy.

The Bylinkin model has been applied to a number of collision systems (e.g.
bylinkin2010 ; bylinkin2014nucphb ; bylinkinanom ) with inference of
parameters $T_{th}$ for the exponential element and $T_{h}$ and $N$ for the
power-law element. Parameters $T_{th}$ and $T_{h}$ are observed to be linearly
correlated, which coupling has been interpreted to support the hypothesis of
thermalization via quantum entanglement pajares . The ratio of amplitudes for
exponential and power-law elements exhibits markedly different behavior for
different collision systems, and the exponential element appears to be absent
for proton spectra from $p$-$p$ and Au-Au collisions bylinkinanom .

Just as application of the BW model to spectra can be questioned tommodeltests
the appropriateness and interpretability of the Bylinkin model should be
closely examined, especially given the significance of inferences derived from
it. In this study the Bylinkin model is compared to a two-component
(soft+hard) model (TCM) applied to many collision systems with considerable
success ppprd ; alicetommpt ; ppquad ; ppbpid . In contrast to the Bylinkin
model the TCM is not based on a priori assumptions. It is instead derived
empirically from evolution of data spectra over a wide range of collision
systems and energies alicetomspec  and can be viewed as a form of lossless
data compression tommodeltests . In what follows the Bylinkin model is placed
in the context of model development over several decades, model-model and
model-data comparisons are presented, and the relation of $p_{t}$ spectrum
models to the question of thermalization is reviewed. As a general conclusion
(a) the Bylinkin model is a particular case of the more-general TCM, (b)
neither its exponential nor power-law elements is required by data and (c) its
parameters are not physically interpretable as they are intended. Given those
results the scenario that quantum entanglement leads to thermalization or even
the presence of thermalization within collisions itself, if based on that
model, can be questioned.

This article is arranged as follow Section II briefly summarizes the history
of $p_{t}$ spectrum models applied to data from high-energy collisions,
including combinations of model elements. Section III provides a detailed
study of data-model and model-model comparisons. Section IV describes
derivation of a TCM for densities on $p_{t}$ and $\eta$ empirically without a
priori assumptions as a form of data compression. Section V contrasts the
predictivity of models based on their ability to describe new data based on
extrapolation from previous analysis. Section VI considers the relation of
thermalization and entanglement to spectrum models. Sections VII and VIII
present discussion and summary.

##  II Two-component spectrum models

This section provides a brief history of $p_{t}$ spectrum model development
for high-energy nuclear collisions, describes empirical derivation of a two-
component model for $p$-$p$ collisions based on evolution of spectrum shape
with event multiplicity $n_{ch}$ and finally describes a two-element model
based on certain a priori assumptions.

###  II.1 $\bf p_{t}$ spectrum model development

The structure of $p_{t}$ spectra near midrapidity in high-energy hadron-hadron
and hadron-nucleus collisions has long been a subject of interest. Soon after
commencement of the National Accelerator Lab (NAL, now FNAL), fixed-target
$p$-A experiments revealed unexpectedly high particle yields at high $p_{t}$,
i.e. spectra falling off much slower than the anticipated exponential trend
cronin10 ; cronin0 . The mechanism(s) for transport of a major fraction of
projectile momentum to transverse momentum space far from projectile rapidity
were sought theoretically in the context of newly-formulated perturbative QCD
(pQCD).

As a complementary effort various empirical spectrum models have been
proposed. With initial publication of its S$p\bar{p}$S data ua1power  the UA1
collaboration (1982) proposed an empirical function

$\displaystyle E\frac{d^{3}\sigma}{dp^{3}}$ $\displaystyle\propto$
$\displaystyle\frac{p_{t_{0}}^{n}}{(p_{t_{0}}+p_{t})^{n}}\leftrightarrow\frac{1}{(1+p_{t}/p_{t_{0}})^{n}},$
(1)

sometimes referred to as a power-law or Hagedorn function, which visually
appears to transition from an exponential at lower $p_{t}$ to a power law at
higher $p_{t}$ hagedorn . One theoretical response to observed spectrum shape
evolution with event multiplicity was the suggestion that a quark-gluon plasma
or QGP is formed at such high collision energies vanhove ; hagedorn . In Ref.
pancheriqgp  the QGP conjecture is acknowledged along with one from Ref. jacob
: “…an energy threshold has been crossed such that many low-$x$ partons have
now enough energy to undergo hard scattering and produce several mini-jets, of
a few GeV each, which fragment independently from one another.” Reference
pancheriqgp  concludes: “…the multiplicity dependence of inclusive $p_{t}$
spectr[um shapes] reported by UA1 can probably be explained by invoking the
persistent high multiplicity associated with low energy (few GeV) jet events.”

That concept was followed up by theoretical proposal of a two-component model
of hadron production in high-energy collisions pancheri  (1985) of the form

$\displaystyle\frac{d\sigma}{dndp_{t}}$ $\displaystyle=$
$\displaystyle\frac{d\sigma_{0}}{dn}P_{0}(p_{t})+\frac{d\sigma_{1}}{dn}P_{1}(p_{t}),$
(2)

where the first term applies to soft or nonjet processes and the second
represents a contribution from “low-$x$ QCD jets” or so-called minijets.
$d\sigma_{x}/dn$ represent cross section fractions associated with event
multiplicity $n$, and $P_{x}(p_{t})$ are unit-normal model functions. That
theoretical model is then formally equivalent to TCM Eq. (IV.1) below.

An earlier empirical model was introduced (1978) to describe dimuon spectra
from $p$-A collisions yohqgauss  (FNAL) with the form

$\displaystyle E\frac{d^{3}\sigma}{dp^{3}}$ $\displaystyle\propto$
$\displaystyle\frac{1}{[1+(p_{t}/p_{t_{0}})^{2}]^{n}}.$ (3)

The same model has since been used to describe higher-mass hadron spectra
alijpsi ; phenixqgauss . That model is formally equivalent to the so-called
power-law element of the Bylinkin model discussed below.

More recently, a spectrum model intended to describe thermodynamic systems
with incomplete equilibration, associated with so-called Tsallis statistics
tsallis  (1988), has become popular

$\displaystyle\frac{d^{2}\sigma}{p_{t}dp_{t}dy_{z}}$ $\displaystyle\propto$
$\displaystyle\frac{1}{(1+m_{t}/Tn)^{n}},$ (4)

where $m_{t}=\sqrt{p_{t}^{2}+m_{0}^{2}}$ is transverse mass, $T$ is
interpreted as a thermodynamic temperature and $1/n\leftrightarrow q-1$ is
interpreted as a measure of incomplete equilibration. In a nuclear physics
context $1/n>0$ may be attributed to hard processes depending on jet
parameters. The formal similarity between Eq. (4) and Eq. (1) is notable.

A comment on notation: The TCM hard-component exponential tail is determined
by parameter $q$ as is the exponent of the Tsallis function above. In this
text parameter $q$ is reserved for TCM hard component $\hat{H}_{0}(y_{t})$ and
exponent $n$ for soft component $\hat{S}_{0}(m_{t})$. Transverse rapidity is
defined by $y_{t}=\ln((m_{t}+p_{t})/m_{0})$. Symbol $N$ is reserved for the
Bylinkin power-law element while $n$ is also used for miscellaneous exponents
such as in Eq. (1).

Single-element models such as Eqs. (1), (3) and (4) can be described as
monolithic and may be based on a priori assumptions including a thermalization
process (via multiple scattering) as central to HE collisions. As such,
monolithic models do not share the attributes of the TCM proposed
theoretically in Ref. pancheri  wherein separate jet and nonjet hadron
production mechanisms are acknowledged and addressed by distinct approaches.

###  II.2 Empirical two-component model or TCM

As an alternative to experimental or theoretical a priori conjecture one may
analyze spectrum data systematics in an attempt to identify what mathematical
model is required by data. That process can be viewed as a form of lossless
data compression tommodeltests : What is the minimum representation (in terms
of algebraic structure and parameter number) that can fully represent all
information in particle data to the limits of statistical uncertainty?

Reference ppprd  (2006) describes such a procedure, having as its central goal
an exhaustive study of the evolution of spectrum shapes with event
multiplicity $n_{ch}$. There are no a priori assumptions about spectrum
structure or theoretical interpretation. The study has two principal outcomes:
(a) spectra are composed of two distinct parts with different functional forms
independent of event multiplicity $n_{ch}$ or nearly so, and (b) the particle
yield associated with one part varies as the square of the yield for the other
part. The second property provides an accurate method to separate the parts
and characterize their functional forms. That approach is consistent with Eq.
(2).

Figure 1 provides an illustration from Ref. ppprd . The left panel shows
$y_{t}$ spectra (points) for ten $p$-$p$ multiplicity classes. Empirically,
all data spectra are observed to coincide at lower $y_{t}$ if rescaled by
soft-component multiplicity $n_{s}\approx n_{ch}-\alpha n_{ch}^{2}$ for some
$\alpha\approx 0.01$. As plotted the spectra are displaced upward from each
other by successive factors 40 relative to the lowest spectrum

Figure 1:  Left: $y_{t}$ spectra from ten charge-multiplicity classes of 200
GeV $p$-$p$ collisions (points) compared to fixed reference
$\hat{S}_{0}(y_{t})$ (solid curves) ppprd . Right: Running integrals
$N(y_{t})$ of normalized (and extrapolated) $y_{t}$ spectra in the left panel
for ten multiplicity classes (solid curves) compared to running integral
$N_{0}(y_{t})$ of fixed reference $\hat{S}_{0}(y_{t})$ (dash-dotted curve).
$\hat{n}_{ch}\approx n_{ch}/2$ is uncorrected event multiplicity. $y_{t}$ = 2
corresponds to $p_{t}\approx 0.5$ GeV/c and $y_{t}$ = 3.75 to $p_{t}\approx 3$
GeV/c

Figure 1 (right) shows running integrals of the ten rescaled spectra in the
left panel. A running integral may enhance a signal (spectrum shape) over
statistical noise. $N_{0}(y_{t})$ is an asymptotic limit. Different behavior
is observed in each of $y_{t}$ intervals A, B and C. In interval C the
integrals are nearly constant, and those constant values increase
approximately linearly with $n_{s}$ relative to $N_{0}$. The main shape
variation lies within interval B, effectively as the running integral of a
peaked spectrum component (hard component) with amplitude $\propto n_{s}$.

The resulting TCM has been applied to many collision systems and to both
unidentified hadrons ppprd ; alicetommpt ; alicetomspec ; tommpt  and
identified hadrons ppbpid ; pidpart1 ; pidpart2 ; transport . As a result, a
self-consistent system of simple parameter variation with A-B collision
system, hadron species and collision energy has been established. The TCM is
thereby a formally predictive model (as opposed to theoretical prediction).
With the algebraic properties of the two components determined, corresponding
physical mechanisms have been established a posteriori: projectile hadron
dissociation (soft component) and low-$x$ gluon large-angle scattering and
fragmentation to jets (minijets, hard component).

###  II.3 Bylinkin two-element model

Subsequent to Ref. ppprd  (2006) an alternative “two-component” model has been
proposed in Ref. bylinkin2010  (2010) consisting of an exponential element and
a “power-law” element, where the quotes question the terminology

$\displaystyle\frac{d\sigma}{p_{t}dp_{t}}$ $\displaystyle=$ $\displaystyle
A_{th}\exp[-(m_{t}-m_{0})/T_{th}]+\frac{A_{h}}{\left(1+\frac{p_{t}^{2}}{T_{h}^{2}N}\right)^{N}}.~{}~{}~{}$
(5)

That model is described in Ref. bylinkinanom  as “a modified Tsallis-type
function.” The two elements are described in Ref. bylinkinprd  thus: “The
power-like component [second element] is mainly originated from a relatively
large $p_{t}$ domain, that is from the mini-jet fragmentation, while the
exponential part accounts for some ‘thermalization’ caused by the final state
rescattering in the parton cloud and the ‘hadron gas’ formed by the
secondaries.”

There are several issues with that model before data are encountered. In this
text an “element” of a model is distinguished from a “component” of physical
hadron production with corresponding model representation. The second element
of Eq. (5) is, in a Tsallis context, a $q$-Gaussian with low-$p_{t}$ limiting
form $\exp(-p_{t}^{2}/T_{h}^{2})$ centered at zero $p_{t}$ and high-$p_{t}$
power-law tail $\propto p_{t}^{-2N}$. The relation between this “power-law”
term and Eq. (3) employed to describe entire spectra is notable. The Bylinkin
power-law element straddles two production mechanisms modeled as distinct by
the TCM described in the previous subsection – projectile dissociation and jet
production.

In material below it will be established that several conclusions associated
with the Bylinkin model conflict with data properties. Among them: the
low-$p_{t}$ part of a spectrum is described by an exponential that represents
thermalization, possibly by an exotic process; the thermalized portion is
present for pions but not other hadrons; a jet contribution is described by a
$q$-Gaussian; two “temperatures” characterizing those elements are linearly
coupled; and certain data statistics vs event multiplicity and collision
energy follow power-law trends.

##  III Data-model-model comparisons

In this section several issues relating to the Bylinkin model are discussed in
comparison with various data formats and the TCM. They include evolution of
model shapes with hadron mass, evolution of power-law parameters with
collision energy and event $n_{ch}$, evolution of soft and hard charge
densities with event $n_{ch}$ and collision energy, and variation of ensemble-
mean $\bar{p}_{t}$ inferred from the two models with $n_{ch}$ and collision
energy.

###  III.1 Bylinkin model properties vs TCM

This subsection responds to descriptions of the Bylinkin model in Refs.
bylinkin2010 ; bylinkinanom ; bylinkphen . Several assumptions associated with
the Bylinkin model are notable: It is observed that with increasing $p_{t}$,
spectra transition from exponential (Boltzmann-like) to power law
“traditionally interpreted as an onset of the perturbative QCD regime of
hadron production. … These features…are found to be universal for any type of
colliding particles [emphasis added]. Therefore, it is tempting to find one
universal smooth functional form…” to model $p_{t}$ spectra. “…the parameter
$T_{h}$ is a QCD analogy to a temperature in classical thermodynamics.” “…the
physical origin of the observed $T_{th}$-$T_{h}$ correlation is not quite
clear….”

The model elements are interpreted thusly: “As required by the perturbative
QCD the spectrum of particles produced in parton-parton interactions is
described by a power law distribution. The rest bulk of irradiated hadrons
represents a quasi-thermolized [sic] hadronic gas produced with a
characteristic temperature $T_{th}$. The spectrum of hadrons in this gas has
the Boltzman [sic] exponential shape. … only the inclusive spectra of charged
particles produced in pure baryonic collisions require a substantial
contribution of the Boltzman-like [sic] exponential term.” “The difference in
size [i.e. relative amplitude] of the exponential [element]…is mainly
responsible for the difference in the spectra shapes…” bylinkin2010 .

In the several papers presenting the Bylinkin model as applied to spectrum
data the plotted spectra are almost always minimum-bias (MB). Such spectra
have a minimal jet contribution (i.e. are dominated by the soft component),
and challenges from substantial spectrum evolution with event multiplicity
$n_{ch}$ are not addressed.

Figure. 2 (left) shows the two elements of the Bylinkin model presented as
curves E (exponential) and $q$-G (power law) vs pion transverse rapidity
$y_{t}$. The parameter values are taken from Ref. bylinkin2010  Figs. 3 and 8
(central values). As noted, the nominal power-law element has the form of a
$q$-Gaussian – a Gaussian with power-law tail. The curve labeled G is the
Gaussian limiting case with $1/N\rightarrow 0$. Since the “power-law” term is
associated with jet production this model asserts that jet fragments extend
down to zero $p_{t}$ and are maximum at that point, conflicting with the
popular notion that jet production is a “high $p_{t}$” phenomenon. The $T_{h}$
parameter controls the position on $y_{t}$ of the apparent shoulder of the
model, and $N$ controls the slope of the high-$p_{t}$ tail.

Referring to its Fig. 3 Ref. bylinkin2010  comments “The most surprising
feature of the new parameterization (2) is a strong [linear] correlation
between the parameters $T_{th}$ and $T_{h}$.” But that is not surprising if
the two model elements strongly overlap at lower $p_{t}$ and neither one is
well suited to model the targeted hadron production process. In essence, the
two model elements have a conflict of interest that is clearly demonstrated by
the trivial relation between two parameters. In contrast, soft and hard model
parameters of the TCM have unique properties clearly related to their
respective targeted spectrum components. Note changes $n\rightarrow N$,
$T_{e}\rightarrow T_{th}$ and $T\rightarrow T_{h}$ have been made in this text
to minimize confusion.

Figure 2:  Left: Bylinkin model elements: E is the exponential element, $q$-G
the power-law element ($q$-Gaussian) and G its asymptotic limit. Right: TCM
soft component $\hat{S}_{0}(m_{ti})$ for several species from pions to
J$/\Psi$ illustrating mass dependence. E is the exponential limit of the pion
soft component.

Figure 2 (right) shows TCM soft component $\hat{S}_{0}(m_{ti})$ for several
hadron species $i$ of increasing mass from pions to $J/\Psi$. Dashed curve E
is the exponential limit of pion $\hat{S}_{0}(m_{t})$, again for
$1/n\rightarrow 0$. The $\hat{S}_{0}(m_{ti})$ mass dependence has two
consequences: (a) the shoulder moves to higher $p_{t}$ leaving a nearly-flat
interval below it and (b) the slope at higher $p_{t}$ increases (i.e. the
effective slope parameter $T$ decreases). Those trends can both be understood
by the relation $(m_{ti}-m_{0i})/T=m_{0i}[\cosh(y_{ti})-1]/T$. Property (a)
arises from the $\cosh(y_{ti})$ term where $y_{ti}=\ln[(m_{ti}+p_{t})/m_{0i}]$
is the proper transverse rapidity for species $i$ including its mass
dependence. Property (b) arises because the effective slope parameter is
$T/m_{0i}$.

The Bylinkin model functions have no such mass dependence. To accommodate
spectra for different hadron species and collision conditions model parameters
$T_{th}$, $T_{h}$ and $N$ plus amplitudes $A_{th}$ and $A_{h}$ must vary
substantially. For more-massive hadrons the flat interval below the shoulder,
increasing in width with hadron mass, reduces or excludes the Bylinkin
exponential term as noted in Refs. bylinkinanom ; bylinkphen . Parameter
$T_{h}$, effectively the width $\sigma$ of the $q$-Gaussian, attempts to
follow shifts of the $\hat{S}_{0}(m_{ti})$ shoulder to higher $p_{t}$, and
growth of the jet-related hard component also at higher $p_{t}$, with
increasing event $n_{ch}$. If the exponential term is ruled out for baryons
and some collision systems as reported in Refs. bylinkinanom ; bylinkintherm
Bylinkin parameters $T_{h}$ and $N$ alone must accommodate both
$\hat{S}_{0}(m_{ti})$ evolving with hadron mass and $\hat{H}_{0}(y_{t})$
evolving with hadron species, collision system and energy.

###  III.2 Power-law element parameter evolution

This subsection responds to descriptions of Bylinkin model parameter
variations in Refs. bylinkin2010 ; bylinkin2015 ; bylinkphen . Bylinkin
parameter trends are compared directly with those from TCM analysis of $p$-$p$
spectra at several collision energies.

Figure 3 (left) shows Bylinkin power-law exponent $N$ (open triangles) from
Fig. 3 of Ref. bylinkin2015  plotted as $1/2N$. Those values are compared to
TCM hard-component exponent $q$ plotted as $1/(q+2)$ (squares) and TCM soft-
component exponent in the form $1/n$ (dots, triangles) from Ref. alicetomspec
. This plot makes clear that the Bylinkin power-law model approximates by its
tail the exponential tail (on $y_{t}$) of the TCM hard component and is quite
different from the power-law tail of the TCM soft component. The combination
$q+2$ (with $q$ on $y_{t}$) approximates the power law (on $p_{t}$) of the TCM
hard-component tail.

Referring to Ref. bylinkin2015  Fig. 3, decrease of $N$ with $\sqrt{s}$ (i.e.
increase of $1/2N$) is “…related to the fact that the probability to produce a
high-$p_{t}$ mini-jet should grow with $\sqrt{s}$.” Actually, it is likely not
the rate of jet production (probability to produce) but the shape of the
spectrum hard component (“hardening”) that changes with $\sqrt{s}$ reflecting
evolution of the underlying jet energy spectrum with increasing $p$-$p$
collision energy fragevo ; jetspec2 .

Referring to Fig. 2 of Ref. bylinkin2015  “Note that the parameter [$N$] shows
a growth [decrease of $1/2N$] with pseudorapidity that is explained by higher
thermalization of the spectra….” More likely, the trend is due to the dramatic
difference between soft- and hard-component $\eta$ distributions such that at
higher $\eta$ spectra are dominated by the soft component (see Fig. 7). From
Ref. bylinkin2015  Fig. 2 $\eta=0$ corresponds to $1/2N\approx 0.14$ whereas
for $\eta=3$ $1/2N\approx 0.11$, then see the TCM values from solid curves at
left corresponding to $\sqrt{s}=630$ GeV per Ref. bocquet .

The energy trend for TCM $\hat{S}_{0}(m_{t})$ exponent $n$ in the left panel
appears consistent with the concept of Gribov diffusion gribov ; gribov3 ;
gribov2  such that within a parton splitting cascade there is random walk of
parton $p_{t}$ from generation to generation leading to hardening of the soft-
component tail with increased $\sqrt{s}$ or depth of the splitting cascade.

Figure 3:  Left: Bylinkin power-law exponent $N$ (open triangles) from Ref.
bylinkin2015  compared to TCM soft-component exponent $n$ (lower solid curve
and points) and hard-component exponent $q$ (upper solid curve and points)
from Ref. alicetomspec  vs collision energy. Right: Correlated evolution of
Bylinkin power-law parameters $T$ and $N$ for $p$-$p$ collisions vs collision
energy (solid) and Au-Au collisions vs centrality (dashed) from Ref.
bylinkin2010 .

Figure 3 (right) shows Bylinkin power-law $T_{h}^{2}$ vs $N$ trends from Fig.
8 of Ref. bylinkin2010  for $p$-$p$ collisions (solid) varying with collision
energy and 200 GeV Au-Au collisions (dashed) varying with centrality. The
power-law exponent is plotted as $1/2N$ to be compatible with the TCM hard-
component power-law tail on $p_{t}$ with effective exponent $1/(q+2)$.
Quantity 1/exponent measures the slope of the power law. The hatched region
represents $\gamma$-$\gamma$ and $\gamma$-$p$ results from the same figure in
Ref. bylinkin2010 .

Those trends can be simply interpreted: The $p$-$p$ trend corresponds to a
large increase in the jet contribution to $p$-$p$ spectra with substantial
hardening of the distribution tail as demonstrated in Fig. 3 (left) of Ref.
tommpt . Thus, Bylinkin parameters $T_{h}^{2}$ and $1/2N$ must both increase
together substantially. The peripheral end (bottom) of the Au-Au curve should
correspond to 200 GeV $p$-$p$, and $1/2N$ is thus expected to have a value
near 0.13 per the left panel. One expects a large increase in $T_{h}^{2}$ with
Au-Au centrality, but the corresponding decrease in $1/2N$ is consistent with
high-$p_{t}$ suppression associated with jet quenching. The hatched band
indicates that $\gamma$-$\gamma$ and $\gamma$-$p$ spectra are modeled by a
$q$-Gaussian with small width and hard power-law tail. Compare with Fig. 12.

###  III.3 Particle densities $\bf\bar{\rho}_{x}$ vs p-p event $\bf n_{ch}$
and $\bf\sqrt{s}$

This subsection responds to particle density variation with changing event
conditions inferred from Bylinkin model fits to $p$-$p$ $p_{t}$ spectra.
Reference bylinkinprd  reports particle densities vs event multiplicity
$n_{ch}$ and collision energy $\sqrt{s}$. For both $n_{ch}$ and $\sqrt{s}$
trends fits to spectrum data with the Bylinkin model return soft and hard
particle densities $\bar{\rho}_{sb}$ and $\bar{\rho}_{hb}$. The $n_{ch}$ trend
is derived from hadron spectra for 200 GeV $p$-$p$ collisions presented in
Ref. ppprd . It is notable that spectrum data from Ref. ppprd  are employed
but there is no acknowledgment of first detailed derivation of the TCM
reported therein.

Regarding particle-density dependence on event $n_{ch}$ Ref. bylinkinprd
asserts that “One can notice that the contribution from the power-like
component (mini-jets) [$\bar{\rho}_{hb}$] grows faster than that from the
‘exponential’ one [$\bar{\rho}_{sb}$]. … This fact confirms that partly the
increase of $N_{ch}$ is due to a larger $E_{t}$ of mini-jets produced.” To the
contrary, in Ref. ppprd  it is established that the $p$-$p$ spectrum hard
component shape, which reflects MB jet energy fragevo ; jetspec2 , is
independent of event $n_{ch}$ to good approximation. The relative rates of
increase with $n_{ch}$ reflect a quadratic relation between hard and soft
particle densities which is a central discovery of Ref. ppprd . It is thus
informative to compare Bylinkin model trends with TCM results.

Figure 4 (left) shows particle densities $\bar{\rho}_{sb}$ and
$\bar{\rho}_{hb}$ (dash-dotted) inferred with the Bylinkin model vs corrected
event multiplicity $n_{ch}$. The same trends presented in Fig. 5 of Ref.
bylinkinprd  include quantity $N_{ch}$ which is actually the uncorrected
$\hat{n}_{ch}$ from Ref. ppprd  where $n_{ch}\approx 2\hat{n}_{ch}$. Those
trends can be compared with TCM trends $\bar{\rho}_{s}$ and $\bar{\rho}_{h}$
(solid) predicted by the TCM relation
$\bar{\rho}_{h}\approx\alpha\bar{\rho}_{s}^{2}$ (for $p$-$p$ collisions) with
$\alpha\approx 0.006$ for 200 GeV ppprd . Their sum $\bar{\rho}_{0}$ is
indicated by the uppermost solid line. The dashed lines extrapolate linearly
the $\bar{\rho}_{sb}$ and $\bar{\rho}_{hb}$ trends for low $n_{ch}$ and
represent 25% and 75% of the total represented by the Bylinkin sum (uppermost
dash-dotted). This result suggests that the Bylinkin power-law element
includes 1/4 of the actual data soft component as a bias. Relative to those
linear trends the curvatures of $\bar{\rho}_{sb}$ and $\bar{\rho}_{hb}$ do
approximately match the TCM versions.

Figure 4:  Left: Bylinkin exponential $\bar{\rho}_{sb}$ and power-law
$\bar{\rho}_{hb}$ particle densities (dash-dotted) vs total particle density
$\bar{\rho}_{0}$ bylinkinprd  compared to equivalent trends for TCM
$\bar{\rho}_{s}$ and hard $\bar{\rho}_{h}$ (solid). The dashed lines are
linear extrapolations of Bylinkin trends at lower densities. Right: Bylinkin
(dash-dotted) and TCM (solid) particle densities vs collision energy
$\sqrt{s}$. The solid dots represent NSD $\bar{\rho}_{0}$ for 200 GeV and 5
TeV. The dotted line represents the collision energy for the left panel.

Figure 4 (right) shows Bylinkin-model charge densities $\bar{\rho}_{sb}$ and
$\bar{\rho}_{hb}$ (dash-dotted) vs collision energy $\sqrt{s}$ as reported in
Fig. 2 of Ref. bylinkinprd . The curves represent

$\displaystyle\bar{\rho}_{sb}(n_{ch},s)$ $\displaystyle=$
$\displaystyle\bar{\rho}_{sb}(n_{ch})(s/s_{0})^{0.15}$ (6)
$\displaystyle\bar{\rho}_{hb}(n_{ch},s)$ $\displaystyle=$
$\displaystyle\bar{\rho}_{hb}(n_{ch})(s/s_{0})^{0.25}$

repeating Eqs. (12) and (13) of Ref. bylinkinprd , with $\sqrt{s_{0}}=200$ GeV
and $\bar{\rho}_{xb}(n_{ch})$ corresponding to NSD values 0.3 and 2.1 in the
left panel. The TCM predictions (solid) include the trend
$\bar{\rho}_{s}\approx 0.8\ln(\sqrt{s}/10~{}\text{GeV})$ alicetomspec  and
$\bar{\rho}_{h}\approx\alpha(\sqrt{s})\bar{\rho}_{s}^{2}$ with
$\alpha(\sqrt{s})\in[0.006,0.017]\approx 0.01$ here for simplicity. The TCM
trend $\bar{\rho}_{0}(\sqrt{s})$ provides a good description of NSD $p$-$p$
values from 200 GeV to 13 TeV. Solid dots represent NSD $\bar{\rho}_{0}$
values for 200 GeV and 5 TeV. Note that this $\bar{\rho}_{s}$ trend for gluon
production does not include substantial quark-related production at lower
energies.

###  III.4 Mean values $\bf\langle p_{t}\rangle_{x}$ vs p-p event $\bf n_{ch}$
and $\bf\sqrt{s}$

This subsection responds to ensemble-mean $\bar{p}_{t}$ variation on event
multiplicity $n_{ch}$ and collision energy $\sqrt{s}$ inferred from fits to
$p$-$p$ $p_{t}$ spectra with the Bylinkin model as reported in Ref.
bylinkinprd . Those results are compared to corresponding trends from the TCM.

Figure 5 (left) shows ensemble-mean contributions $\langle p_{t}\rangle$xb
(dash-dotted lines) inferred from the two elements of the Bylinkin spectrum
model as reported in Fig. 4 of Ref. bylinkinprd  for 200 GeV $p$-$p$ collision
data taken from Ref. ppprd . Also shown are $\langle
p_{t}\rangle$${}_{xm}\leftrightarrow\bar{p}_{txm}$ trends for soft and hard
components of the corresponding TCM (lower and upper dashed bands) and total
ensemble-mean $\langle p_{t}\rangle$ (solid curve) as reported in Ref. ppprd .
The last is evaluated as

$\displaystyle\langle p_{t}\rangle$ $\displaystyle=$
$\displaystyle\frac{\bar{\rho}_{s}\bar{p}_{tsm}+\bar{\rho}_{h}\bar{p}_{thm}}{\bar{\rho}_{s}+\bar{\rho}_{h}}$
(7)

with the $\bar{p}_{txm}$ derived from invariant TCM model functions. Densities
$\bar{\rho}_{s}$ and $\bar{\rho}_{h}$ are as shown in Fig. 4 (left).

The Bylinkin model result for $\langle p_{t}\rangle$hb can be explained in
terms of the “power-law” element from which it is derived. Because the latter
is essentially a $q$-Gaussian with centroid at zero $p_{t}$ the inferred mean
values must be well below $\bar{p}_{thm}$ for the TCM hard component, a
Gaussian on $y_{t}$ with mode near 1 GeV/c. The small increase of $\langle
p_{t}\rangle$hb is due to variation of Bylinkin parameter $T_{h}$
($q$-Gaussian width) with $n_{ch}$ as in Fig. 3 of Ref. bylinkin2010 ,
presumably in response to relative increase of jet production.

Figure 5:  Left: Bylinkin exponential $\langle p_{t}\rangle_{sb}$ and power-
law $\langle p_{t}\rangle_{hb}$ means vs total particle density
$\bar{\rho}_{0}$ (dash-dotted) bylinkinprd  compared to trends $\langle
p_{t}\rangle_{xm}$ for TCM soft- and hard-component model functions (hatched
bands). The solid curve is the combined TCM $\langle p_{t}\rangle$ prediction.
Right: Bylinkin exponential $\langle p_{t}\rangle_{sb}$ and power-law $\langle
p_{t}\rangle_{hb}$ means vs collision energy $\sqrt{s}$ (dash-dotted)
bylinkinprd  compared to TCM trends (solid).

Figure 5 (right) shows collision-energy $\sqrt{s}$ dependence of soft and hard
$\langle p_{t}\rangle$ components (dash-dotted) from the Bylinkin model as
presented in Fig. 3 of Ref. bylinkinprd . The $\langle p_{t}\rangle$hb trend
is reported as $\propto(\sqrt{s})^{0.07}$. Also presented are $\langle
p_{t}\rangle$x trends from the TCM, with the $\langle p_{t}\rangle$h trend
(solid line) taken from Fig. 3 (right) of tommpt  and the $\langle
p_{t}\rangle$s trend based on the usual TCM $\hat{S}_{0}(m_{t})$ for pions and
the exponent $1/n(\sqrt{s})$ energy trend from Fig. 3 (left). The resulting
values for $\langle p_{t}\rangle$s are slightly high because no model for the
low-$p_{t}$ resonance contribution is included. Reference tommpt  explains the
rise in $\langle p_{t}\rangle$h with $\sqrt{s}$ as due to corresponding
evolution of the spectrum hard component (jet fragment distribution) resulting
from variation of the underlying jet energy spectrum with collision energy.

##  IV Models and data compression

This section describes differential methods to obtain optimized models for
density distributions on $p_{t}$ and $\eta$ that represent lossless data
compression: the input data are reduced to a minimal set of functional forms
and parameters that retain almost all information carried by the data but do
not rely on a priori assumptions. A further constraint is that the functional
forms and their parameters must be a posteriori interpretable, that is,
directly and quantitatively comparable with QCD theory.

###  IV.1 Spectrum TCM with (almost) no assumptions

This subsection briefly summarizes the process of inferring TCM model
functions from 200 GeV $p$-$p$ $p_{t}$ spectra. Reference ppprd  (2006)
reported a two-component model of $p_{t}$ spectra from 200 GeV $p$-$p$
collisions. The model structure, inferred empirically from spectrum data, is
represented by

$\displaystyle\bar{\rho}_{0}(p_{t},n_{s})$ $\displaystyle\approx$
$\displaystyle d^{2}n_{ch}/p_{t}dp_{t}d\eta$ $\displaystyle=$ $\displaystyle
S(p_{t},n_{s})+H(p_{t},n_{s})$ $\displaystyle=$
$\displaystyle\bar{\rho}_{s}\hat{S}_{0}(p_{t})+\bar{\rho}_{h}\hat{H}_{0}(p_{t},n_{s}),$

where $\bar{\rho}_{x}=n_{x}/\Delta\eta$ within $\eta$ acceptance $\Delta\eta$
and $n_{s}$ is used as an event index. The main feature is two distinct
components (hadron production processes) each of which factorizes charge
density and $p_{t}$ dependence. Of central importance is discovery of the
relation $\bar{\rho}_{h}\propto\bar{\rho}_{s}^{2}$. Based on that system it is
possible to infer the functional forms of $\hat{S}_{0}(p_{t})$ and
$\hat{H}_{0}(p_{t},n_{s})$ by simple difference methods.

Figure 6 (left) shows spectra $\bar{\rho}_{0}(p_{t})$ as densities on $p_{t}$
plotted vs pion transverse rapidity $y_{t}$ for ten event classes $n$ from 13
TeV $p$-$p$ collisions. Spectra have been rescaled to the form
$\bar{\rho}_{0}(p_{t})/\bar{\rho}_{s}$ where soft particle density
$\bar{\rho}_{s}$ is derived from measured charge density $\bar{\rho}_{0}$ as
the root of the relation
$\bar{\rho}_{0}\approx\bar{\rho}_{s}+\alpha(\sqrt{s})\bar{\rho}_{s}^{2}$, and
$\alpha(\sqrt{s})\approx 0.017$ has been determined previously from systematic
analysis of several collision systems. See Fig. 16 (left) of Ref. alicetomspec
.

Figure 6:  Left: Pion spectra for ten event classes from 13 TeV $p$-$p$
collisions (solid) rescaled to the form
$\bar{\rho}_{0}(m_{t},n_{s})/\bar{\rho}_{s}$ vs pion transverse rapidity
$y_{t}$ pppid . The dashed curve is soft component $S_{0}^{\prime}(m_{t})$
defined by Eq. (27). Right: Successive spectrum differences defined by Eq. (9)
revealing data hard components. The dashed curve is TCM hard-component model
$\hat{H}_{0}(y_{t})$. Hard-component data and model are densities on $y_{t}$.

Figure 6 (right) shows normalized data hard components $H_{n}(p_{t})$ (dotted
curves) derived from full spectra by the following procedure. Based on Eq.
(IV.1) the ratio
$\bar{\rho}_{0,n}(p_{t})/\bar{\rho}_{s,n}\approx\hat{S}_{0}(p_{t})+H_{n}(p_{t})/\bar{\rho}_{s,n}$,
where $H_{n}(p_{t})$ here represents a data hard component. If ratio
$H_{n}(p_{t})/\bar{\rho}_{h,n}$ is invariant or slowly varying with event
class $n$ (as observed for various systems) it can be inferred from rescaled
spectra by the following difference procedure

$\displaystyle H_{n}(p_{t})/\bar{\rho}_{h,n}$ $\displaystyle\approx$
$\displaystyle\frac{\bar{\rho}_{0,n}(p_{t})/\bar{\rho}_{s,n}-\bar{\rho}_{0,n-1}(p_{t})/\bar{\rho}_{s,n-1}}{\bar{\rho}_{h,n}/\bar{\rho}_{s,n}-\bar{\rho}_{h,n-1}/\bar{\rho}_{s,n-1}},$
(9)

where $\hat{S}_{0}(p_{t})$ cancels in the numerator. By assumption
$H_{n}(p_{t})/\bar{\rho}_{h,n}\approx H_{n-1}(p_{t})/\bar{\rho}_{h,n-1}$ and
factors out of the numerator difference. The result is dotted curves in the
right panel confirming assumed invariance of $H_{n}(p_{t})/\bar{\rho}_{h,n}$.
The corresponding TCM hard-component fixed model $\hat{H}_{0}(p_{t})$ is shown
by the dashed curve. The two inferred TCM model functions are defined by

$\displaystyle\hat{S}_{0i}(m_{ti})$ $\displaystyle=$
$\displaystyle\frac{A_{i}}{[1+(m_{ti}-m_{i})/T_{i}n_{i}]^{n_{i}}},$ (10)

as density on $m_{t}$ for hadron species $i$ and

$\displaystyle\hat{H}_{0}(y_{t})$ $\displaystyle\approx$ $\displaystyle
B\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 higher $y_{t}$ -- the tail}$

as density on $y_{t}$ converted to $m_{t}$ by Jacobian $y_{t}/m_{t}p_{t}$.

It should be emphasized that the only untested (in this procedure) assumption
is the value of $\alpha$ for 13 TeV $p$-$p$ collisions that has been derived
from separate studies alicetomspec . The single parameter value permits
identification of $\hat{S}_{0}(m_{t})$ as the dashed curve in the left panel,
the asymptotic limit of rescaled data spectra for $n_{ch}\rightarrow 0$, and
identification of $\hat{H}_{0}(y_{t})$ as the dashed curve in the right panel
that describes inferred data hard components.

###  IV.2 $\bf\eta$ density TCM with (almost) no assumptions

This subsection responds to Ref. bylinkin2014nucphb  which examines the role
of quarks in nuclear collisions and wherein the thermal (exponential) element
of the Bylinkin model is “associated with the baryon valence quarks and a
quark-gluon cloud coupled to the valence quarks. Those partons preexist long
time before the interaction and could be considerer as being a thermalized
statistical ensemble.” The emphasis in Ref. bylinkin2014nucphb  is on $\eta$
distributions. The text below briefly summarizes the process of inferring TCM
model functions from 200 GeV $p$-$p$ $\eta$ density distributions without a
priori assumptions as in Ref. ppquad .

A difference procedure as in Eq. (9) can be used to derive a TCM for particle
densities on pseudorapidity $\eta$. Given a 2D particle density on
$(p_{t},\eta)$ for some event class corresponding to particle multiplicity
$n_{ch}$ and denoted by $\rho_{0}(p_{t},\eta;n_{ch})$, a 1D density on $p_{t}$
$\rho_{0}(p_{t};n_{ch},\Delta\eta)$ must be obtained by averaging over some
$\eta$ acceptance $\Delta\eta$ to obtain a $p_{t}$ spectrum TCM described per
Eq. (IV.1) by

$\displaystyle\bar{\rho}_{0}(p_{t};n_{ch},\Delta\eta)$ $\displaystyle\approx$
$\displaystyle\bar{\rho}_{s}(n_{ch},\Delta\eta)\hat{S}_{0}(p_{t})$
$\displaystyle+$
$\displaystyle\bar{\rho}_{h}(n_{ch},\Delta\eta)\hat{H}_{0}(p_{t},n_{s}),$

where $\bar{\rho}_{x}(n_{ch},\Delta\eta)\equiv n_{x}/\Delta\eta$ with $x=s$,
$h$ and bars indicate averages over $\Delta\eta$. That model, consistent with
results from Ref. ppprd , describes corrected and extrapolated $p_{t}$
spectra. For a $p_{t}$-integral study of densities on $\eta$ one may integrate
$\rho_{0}(p_{t},\eta;n_{ch})$ over $p_{t}\in[0,\infty]$ to obtain

$\displaystyle\rho_{0}(\eta;n_{ch})$ $\displaystyle\approx$
$\displaystyle\rho_{s0}(n_{ch})S_{0}(\eta)+\rho_{h0}(n_{ch})H_{0}(\eta),$ (13)

where $S_{0}(\eta)$ and $H_{0}(\eta)$ are TCM model functions to be determined
as described below. Averaging Eq. (13) over some acceptance $\Delta\eta$
symmetric about $\eta=0$ gives

$\displaystyle\bar{\rho}_{0}(n_{ch},\Delta\eta)$ $\displaystyle=$
$\displaystyle\rho_{s0}(n_{ch})\bar{S}_{0}(\Delta\eta)+\rho_{h0}(n_{ch})\bar{H}_{0}(\Delta\eta)$
(14) $\displaystyle\equiv$
$\displaystyle\bar{\rho}_{s}(n_{ch},\Delta\eta)+\bar{\rho}_{h}(n_{ch},\Delta\eta),$

with mean values
$\bar{\rho}_{x}(n_{ch},\Delta\eta)=\rho_{x0}\bar{X}_{0}(\Delta\eta)$. The
second line of Eq. (14) is then consistent with integrating Eq. (IV.2) over
$p_{t}$. The TCM of Eq. (13) can be rewritten

$\displaystyle\rho_{0}(\eta;n_{ch},\Delta\eta)$ $\displaystyle\approx$
$\displaystyle\bar{\rho}_{s}(n_{ch},\Delta\eta)\tilde{S}_{0}(\eta;\Delta\eta)$
$\displaystyle+$
$\displaystyle\bar{\rho}_{h}(n_{ch},\Delta\eta)\tilde{H}_{0}(\eta;\Delta\eta),$

with $\bar{\rho}_{s}(n_{ch},\Delta\eta)$ and
$\bar{\rho}_{h}(n_{ch},\Delta\eta)$ inferred from measured $n_{ch}$ based on
an assumed value for parameter $\alpha$ as discussed in Sec. IV.1 above. The
$\tilde{X}_{0}(\eta;\Delta\eta)$ (denoted by a tilde) have mean value 1 over
acceptance $\Delta\eta$ by definition. Averaging Eq. (IV.2) over $\eta$
acceptance $\Delta\eta$ should then be consistent with integrating Eq. (IV.2)
over $p_{t}$.

Figure 7 (a) shows uncorrected densities $\rho_{0}^{\prime}(\eta;n_{ch})$
rescaled by particle density soft component $\bar{\rho}_{s}$ for seven event
classes. The data have been efficiency corrected by factor $\lambda(\eta)$ but
there is a substantial asymmetry on $\eta$ due to different readout chamber
efficiencies in the two TPC end caps. Figure 7 (b) shows data from panel (a)
corrected by factor $1+g(\eta)$ that removes the asymmetry.

The full procedure for defining an $\eta$-density TCM is recursive: The
formula immediately below is applied to obtain data hard components from which
a hard-component model $\tilde{H}_{0}(\eta;\Delta\eta)$ is inferred. That
model is then combined with data $\rho_{0}(\eta;n_{ch})$ to obtain soft
components from which soft-component model $\tilde{S}_{0}(\eta;\Delta\eta)$ is
obtained.

To derive hard-component model function $\tilde{H}_{0}(\eta;\Delta\eta)$ the
same model-independent subtraction technique as for $p_{t}$ spectra in Sec.
IV.1 is employed. The results of that first step are not shown here but may be
found in Ref. ppquad . Data differences are determined as

$\displaystyle\tilde{H}_{0n}(\eta;\Delta\eta)$ $\displaystyle\equiv$
$\displaystyle\frac{\rho_{0}(\eta)_{n}/\bar{\rho}_{s,n}-\rho_{0}(\eta)_{n-1}/\bar{\rho}_{s,n-1}}{\bar{\rho}_{h,n}/\bar{\rho}_{s,n}-\bar{\rho}_{h,n-1}/\bar{\rho}_{s,n-1}},$
(16)

where index $n\in[1,7]$ represents seven multiplicity classes. As noted, the
difference in the numerator cancels common term
$\tilde{S}_{0}(\eta;\Delta\eta)$ (defined below). For $n=1$ it is assumed that
$\rho_{0}(\eta)_{n-1}/\bar{\rho}_{s,n-1}\rightarrow\tilde{S}_{0}(\eta;\Delta\eta)$,
and
$\bar{\rho}_{h,n}/\bar{\rho}_{s,n}-\bar{\rho}_{h,n-1}/\bar{\rho}_{s,n-1}\rightarrow\bar{\rho}_{h,1}/\bar{\rho}_{s,1}$
(i.e. extrapolation to $n=0$ is assumed to be pure soft component and
$\bar{\rho}_{h,0}=0$). Given Eq. (IV.2) as TCM that expression represents a
common data hard component $H_{0}(\eta)/\bar{H}_{0}(\Delta\eta)$. The inferred
model function for $\Delta\eta=2$ is

$\displaystyle\tilde{H}_{0}(\eta;\Delta\eta)\equiv\frac{H_{0}(\eta)}{\bar{H}_{0}(\Delta\eta)}$
$\displaystyle=$ $\displaystyle 1.47\exp[-(\eta/0.6)^{2}/2],$ (17)

where the “normalized” model function is denoted by a tilde. The data hard-
component form is, within statistical errors, approximately independent of
$n_{ch}$ over an interval implying a 100-fold increase in dijet production,
and suggests that most of the hard-component yield (MB dijet fragments) falls
within the acceptance $\Delta\eta=2$.

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

Figure 7:  (a) Uncorrected $\eta$ densities within $\Delta\eta=2$ for seven
multiplicity classes of 200 GeV $p$-$p$ collisions ppquad . (b) Corrected
$\eta$ densities within $\Delta\eta=2$ for seven multiplicity classes. The
dashed curve is normalized soft-component model $\tilde{S}_{0}(\eta)\equiv
S_{0}(\eta)/\bar{S}_{0}(\Delta\eta)$ from Eq. (19). The solid curves are Eq.
(IV.2) with TCM elements defined in Eqs. (17) and (19). (c) Data soft
components inferred as in Eq. (18). The solid curve is normalized soft-
component model $\tilde{S}_{0}(\eta)$ as defined in Eq. (19). The dashed curve
is Eq. (21), the Jacobian from $y_{z}$ to $\eta$. (d) Data hard components
inferred as in Eq. (20). The solid curve is normalized hard-component model
$\tilde{H}_{0}(\eta)$ defined in Eq. (17). The data for $n=1$ (solid dots) are
significantly low compared to the common trend. The dash-dotted curve is an
estimate for the Bylinkin power-law model element reported in Ref.
bylinkin2014nucphb , with $\sigma_{h}\approx 1.46$ for 200 GeV data.

Figure 7 (c) shows data soft-component estimator

$\displaystyle\tilde{S}_{0}(\eta;\Delta\eta)$ $\displaystyle\equiv$
$\displaystyle\rho_{0}(\eta;n_{ch})/\bar{\rho}_{s}-(\bar{\rho}_{h}/\bar{\rho}_{s})\tilde{H}_{0}(\eta;\Delta\eta),$
(18)

with $\tilde{H}_{0}(\eta;\Delta\eta)$ as defined in Eq. (17). The inferred
soft-component model for $\Delta\eta=2$ (solid curve) is defined by

$\displaystyle\tilde{S}_{0}(\eta;\Delta\eta)\equiv\frac{S_{0}(\eta)}{\bar{S}_{0}(\Delta\eta)}$
$\displaystyle=$ $\displaystyle
1.09-0.18\exp[-(\eta/0.44)^{2}/2].~{}~{}~{}~{}~{}$ (19)

The form of the soft component also appears to be invariant over a large event
$n_{ch}$ interval. Small data deviations from the model are consistent with
statistical errors. A minimum at $\eta=0$ is expected given the Jacobian for
$\eta\leftrightarrow y_{z}$, where an approximately uniform distribution on
$y_{z}$ is expected within a limited $\Delta y_{z}$ acceptance. The observed
excursion appears too large as discussed below.

Figure 7 (d) shows data hard components estimated with an alternative method
assuming $\tilde{S}_{0}(\eta)$ is known

$\displaystyle\tilde{H}_{0}(\eta;\Delta\eta)$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}(\eta;n_{ch})/\bar{\rho}_{s}-\tilde{S}_{0}(\eta;\Delta\eta)}{\bar{\rho}_{h}/\bar{\rho}_{s}},~{}~{}~{}$
(20)

which substantially reduces statistical noise in the differences. The dashed
curve is the hard-component model $\tilde{H}_{0}(\eta;\Delta\eta)$ defined in
Eq. (17) confirming TCM self-consistency. The points for peripheral $n=1$
(solid dots) are low compared to the general trend, which may indicate that
the hard component for $n=1$ extends low enough on $y_{t}$ to be reduced by
low-$p_{t}$ tracking inefficiency. The results in Fig. 7 (c,d) indicate that
soft and hard densities $\bar{\rho}_{x}(n_{ch};\Delta\eta)$ depend
substantially on detector $\eta$ acceptance $\Delta\eta$. The dash-dotted
curve is an estimate for the Bylinkin power-law model element reported in Ref.
bylinkin2014nucphb , with $\sigma_{h}\approx 1.46$ for 200 GeV data. The
amplitude is adjusted for tangency to the data at maximum.

The Jacobian relating densities on $\eta$ and $y_{z}$ is

$\displaystyle J(\eta)$ $\displaystyle=$
$\displaystyle\frac{\cosh(\eta)}{\sqrt{a^{2}+\cosh^{2}(\eta)}}$ (21)

with $a=m_{0i}/p_{t}$. For pions $\langle p_{t}\rangle$ $\approx 0.45$ GeV/c
and $m_{0}\approx 0.14$ GeV/$c^{2}$ so $a\approx 1/3$. Assuming uniform (boost
invariant) density on rapidity $y_{z}$ within acceptance $\Delta\eta=2$ the
dashed curve on $\eta$ in panel (c) is then predicted. It is thus unlikely
that the data trend in panel (c) is simply a consequence of the Jacobian. A
possible source of the large excursion, transport of hadrons from soft to hard
component that conserves species identities and follows statistical-model
predictions, is described in Ref. transport , and see the discussion in Sec.
V.2.

##  V Spectrum Models and predictivity

This section demonstrates that the TCM describes high-mass hadron spectra in
$p$-$p$ (and other) collisions accurately and reveals new aspects of hadron
production. The TCM for higher-mass hadrons is predicted based on trends
determined from lower-mass hadrons. The Bylinkin model in contrast has only
one effective element (power law) to describe such spectra, and the model
parameters are not predicted or interpretable within the context of the
Bylinkin model itself. Parameter trends may be interpreted in the larger
context of the TCM.

###  V.1 13 TeV p-p collisions and multistrange hadrons

Figure 8 shows spectrum data (points) for neutral kaons (left) and Omegas
(right) from 13 TeV $p$-$p$ collisions reported in Ref. alippss , here as
densities on $p_{t}$ vs transverse rapidity $y_{t}$ with pion mass assumed.
The spectra are rescaled by powers of 10 according to $10^{10-n}$ where
$n\in[1,9]$ is the centrality class index and $n=1$ is most central. Solid
curves are full TCM parametrizations ppsss . Dashed curves are TCM soft
components in the form $z_{si}\bar{\rho}_{s}\hat{S}_{0i}(m_{t})$. Soft-
component parameters for all hadrons except pions are $T\approx 200$ MeV and
$n\approx 14$. Variation of $\hat{S}_{0i}(m_{t})$ with hadron species is
entirely due to mass dependence of $m_{t}$ – see Fig. 2 (right). Data$-$soft-
component differences represent minimum-bias jet contributions modeled by hard
component $z_{hi}\bar{\rho}_{h}\hat{H}_{0i}(p_{t})$ with $\hat{H}_{0i}(y_{t})$
(as density on $y_{t}$) parameters $\bar{y}_{t}=2.7$, $\sigma_{y_{t}}=0.58$
and $q=3.7$ for kaons and $\bar{y}_{t}=3.0+0.55x(n_{s})$,
$\sigma_{y_{t}}=0.50$ and $q=4.6$ for Omegas. Factors $z_{xi}$ are fractions
($\leq 1$) of soft and hard densities $\bar{\rho}_{s}$ and $\bar{\rho}_{h}$
for unidentified hadrons determined previously ppbpid , and
$x(n_{s})=\bar{\rho}_{h}/\bar{\rho}_{s}$.

Figure 8:  $p_{t}$ spectrum data (points) for strange hadrons: Left: Neutral
kaons from Ref. alicepppid . Right: Omegas from Ref. alippss . Solid curves
represent the PID spectrum TCM. Dashed curves are TCM soft components in the
form $z_{si}(n_{s})\bar{\rho}_{s}\hat{S}_{0}(y_{t})$. The dotted curve at
right is a TCM hard component in the form
$z_{hi}(n_{s})\bar{\rho}_{h}\hat{H}_{0}(p_{t})$. The dash-dotted curves are
Bylinkin power-law model elements with three parameters adjusted to best
describe data for each of two event classes in either panel.

Also shown are two dash-dotted curves in each panel based on the Bylinkin
model power-law element as it appears in Eq. (5). The parameters $T_{h}$ and
$N$ and constant multiplier $A_{h}$ are freely adjusted to best describe the
data. For central ($n=1$) and peripheral (n = 5 or 9) data respectively the
$T_{h}$ parameter values are 0.66 and 0.48 GeV/c for kaons and 1.44 and 1.04
GeV/c for Omegas. Exponent $N$ values are 2.8 for kaons and 3.8 for Omegas
with no significant variation with event class. While that three-parameter
model does describe spectra for higher-mass hadrons reasonably well,
interpretation of parameter values is problematic. Note that the Tsallis model
assumes $T$ is a thermodynamic temperature. Within $p$-$p$ collisions do
different hadron species in different event classes experience different
temperatures?

In contrast, the TCM is simply interpreted in terms of standard QCD processes
and measured jet properties. Figure 2 (right) demonstrates that the apparent
shoulder of soft-component $\hat{S}_{0}(m_{t})$ extends to higher $p_{t}$ with
increasing hadron mass, accounting for some of the $T_{h}$ variation in the
Bylinkin power-law model (i.e. from kaons to Omegas). Variation of $N$
corresponds to variation of fragmentation functions with hadron species as
noted in Ref. ppbpid . In addition, jet production increases quadratically
with increasing event $n_{ch}$ which further drives the apparent data shoulder
to higher $p_{t}$ because of soft- and hard-component interaction, thus
explaining variation with event class for $T_{h}$ but lack thereof for $N$.

###  V.2 Omega hard/soft ratios and hadron mass

Systematic TCM analysis of a variety of collision systems and hadron species
leads to new insights into collision dynamics. In particular, analysis of PID
spectra including multistrange hadrons reveals close correlations between soft
and hard components that relate to the statistical model of hadron production.
Those result demonstrate that soft and hard production are quantitatively
described and predicted by the TCM, and the relation between the two is also
established. Such results demonstrate what is missed by single-element or
monolithic spectrum models whose parameters are uninterpretable.

In Refs. ppbpid ; pidpart1 ; pidpart2 ; pppid  soft and hard fractions
$z_{si}(n_{s})$ and $z_{hi}(n_{s})$ (where $n_{s}$ is the soft particle
multiplicity used as an event index and $i$ denotes hadron species) noted in
the previous subsection are inferred from 13 TeV $p$-$p$ and 5 TeV $p$-Pb PID
spectra. From the algebraic forms inferred from data, ratios
$\tilde{z}_{i}(n_{s})\equiv z_{hi}(n_{s})/z_{si}(n_{s})$ are observed to be
approximately independent of $n_{s}$ and linearly proportional to hadron mass
irrespective of baryon identity or strangeness. In Ref. transport  that
information is combined to determine that density ratios
$\bar{\rho}_{si}/\bar{\rho}_{0i}$ and $\bar{\rho}_{hi}/\bar{\rho}_{0i}$ are
simply related. The result has major implications for the spectra of more-
massive hadrons.

Figure 9 (left) shows results from Ref. transport . The curves correspond to
predicted ratios $\bar{\rho}_{si}/\bar{\rho}_{0i}$ and
$\bar{\rho}_{hi}/\bar{\rho}_{0i}$ vs hard/soft ratio
$x(n_{s})=\bar{\rho}_{h}(n_{s})/\bar{\rho}_{s}(n_{s})\propto\bar{\rho}_{s}$
for kaons (dotted) and Omega baryons (solid) from 13 TeV $p$-$p$ collisions.
The single control parameter for those trends is ratio $\tilde{z}_{i}\propto
m_{0i}$. This plot illustrates how low-$p_{t}$ behavior
[$\bar{\rho}_{si}(n_{s})=z_{si}(n_{s})\bar{\rho}_{s}$] predicts high-$p_{t}$
behavior [$\bar{\rho}_{hi}(n_{s})=z_{hi}(n_{s})\bar{\rho}_{h}$], for example
enabling correction of proton inefficiencies in Refs. pidpart1 ; pppid . It
also illustrates the very strong dependence of hard/soft ratios on hadron
mass. Hadron production mechanisms for jet and nonjet components of the final
state are closely correlated so as to preserve the total hadron number or
fraction for each species consistent with statistical-model predictions
statmodel .

The hatched bands correspond to values of hard/soft ratio $x(n_{s})$ for 13
TeV $p$-$p$ event classes 1 (0.28, high $n_{ch}$) and 5 (0.052, low $n_{ch}$)
for Omega spectra. For event class 5 more than two-thirds of Omegas are nonjet
(projectile proton fragments) while for event class 1 three quarters of Omegas
are jet fragments, thus representing a dramatic change in hadron production
while still maintaining approximately the same total Omega fraction
$z_{0i}^{*}\approx 0.0004$. The actual fractions $z_{0i}(n_{s})$ used are
consistent with $p$-$p$ Omega fraction data in Fig. 6 (right) of Ref. ppsss .

Figure 9:  Left: Hard- and soft-component yield ratios
$\bar{\rho}_{si}/\bar{\rho}_{0i}$ and $\bar{\rho}_{hi}/\bar{\rho}_{0i}$ for 13
TeV $p$-$p$ collisions from Ref. transport  providing a prediction (curves)
for Omega baryons based on extrapolation of a mass trend for lower-mass
hadrons. The hatched bands correspond to hard/soft ratio $x(n_{s})$ values for
Omega spectra from $p$-$p$ event classes 1 (high $n_{ch}$) and 5 (low
$n_{ch}$). The curves are labeled with numerators on the $y$ axis. Right: PID
spectra (points) for Omega baryons for event classes 1 and 5 from 13 TeV
$p$-$p$ collisions ppsss  plotted per the y-axis label vs pion transverse
rapidity $y_{t}$. The solid curves are TCM spectra rescaled per the $y$-axis
label. The dashed and dotted curves are corresponding soft and hard TCM model
functions.

Figure 9 (right) shows Omega spectra (points) for $p$-$p$ event classes 1 and
5 in the form $(1/z_{0i}\bar{\rho}_{0})d^{2}n_{chi}/dp_{t}dy_{z}$ without the
usual additional factor $1/p_{t}$, better indicating what fraction of
particles lies within a given $p_{t}$ interval. In that format the TCM spectra
each integrate on $p_{t}$ to 1. The error bars are published total systematic
uncertainties. The solid curves are TCM spectra appearing in Fig. 8 (right)
rescaled in the same way. The dashed curves are soft components with $T=200$
MeV and $n=14$, and the dotted curves are corresponding TCM hard components
(jet fragment distributions) with $\bar{y}_{t}=3.03,~{}3.15$,
$\sigma_{y_{t}}=0.5$ and $q=4.6$ ($\sim 1/p_{t}^{6.6}$).

As in the left panel the integrated hard component is three times the
integrated soft component for event class 1, and data points lie essentially
entirely on the hard component. Thus, jets dominate Omega production from
$p$-$p$ collisions for event class $n=1$. That is why in Fig. 8 (right) the
monolithic $q$-Gaussian for $n=1$ (dash-dotted) lies well above the TCM
(solid) at low $y_{t}$ whereas it lies significantly below the TCM for $n=5$.
The TCM for Omega spectra is based on predictions from Ref. transport  where
it is determined that fractional coefficient $z_{0i}$ controls integrated
yields from entire hadron spectra, including jet contributions at high
$p_{t}$, while the TCM accurately describes dramatic changes in differential
spectrum structure in terms of two fragmentation processes.

###  V.3 Ensemble-mean $\bf\bar{p}_{t}$ systematics

This subsection responds to ensemble $\langle p_{t}\rangle$ or $\bar{p}_{t}$
results reported in Ref. bylinkinprd , in particular regarding its Fig. 6, and
in Ref. bylinkin2016b . The Bylinkin $\langle p_{t}\rangle$ trends vs
$\bar{\rho}_{0}$ and $\sqrt{s}$ are discussed in Sec. III.4 above. Those
trends are then combined to produce curves for $\langle p_{t}\rangle$ vs
$n_{ch}$ at several collision energies in Fig. 6 of Ref. bylinkinprd . The
paper concludes that “Accounting for mini-jet contribution the $\langle
p_{t}\rangle$ should increase with $n_{ch}$ since another way to enlarge
multiplicity is to produce mini-jets with a larger $E_{t}$ (and thus a larger
multiplicity in jet-fragmentation).” Although the composite Bylinkin model
curves are reasonably close to data the individual elements of the Bylinkin
model are problematic as illustrated in Figs. 4 and 5 above. The former
demonstrates that Bylinkin charge densities exhibit strong cross talk between
soft and hard elements vs $n_{ch}$, and the energy dependence deviates
strongly from measured total charge densities on collision energy. The latter
demonstrates that power-law $\bar{p}_{t}$ is intermediate between TCM soft and
hard components because of strong overlap between two Bylinkin elements.

Reference bylinkin2016b  focuses on $p_{t}$ spectra for baryons where it
concludes that the exponential (soft) Bylinkin element is not required by
data, so only the Bylinkin power-law function is implemented. Three models are
applied to data: an isolated exponential (QGSM), a Tsallis function and the
Bylinkin power-law element. It is asserted that “these three approaches have
the exponential-like behavior in the low-$p_{t}$ region…” But the power-law
element is a Gaussian at low $p_{t}$. Apparently any model may be applied to
determine $\bar{p}_{t}$ for baryons “…since high-$p_{t}$ particles…do not give
a big contribution to the mean value.” That remark conflicts with the
conclusion “The enhancement of power-low [sic] contributions into the spectra
at high $p_{t}$’s [sic] causes the change of low $p_{t}$ exponential slopes,
so that [$\langle p_{t}\rangle$] are growing with energy.” Those statements
conflict with TCM results as shown below.

In Fig. 5 (right) above, the Bylinkin $\langle
p_{t}\rangle_{hb}\propto(\sqrt{s})^{0.07}$ trend bylinkinprd  is intermediate
between TCM hard and soft components. As noted, the power-law model confuses
two production mechanisms. The TCM hard-component trend (upper solid) is
$\langle p_{t}\rangle_{h}\propto(\sqrt{s})^{0.05}$. In Ref. bylinkin2016b  the
trend for baryons in its Fig. 2 is given as $\langle p_{t}\rangle_{hb}\propto
s^{0.055}$ (confirmed from that figure in the present study, note $s^{x}$ vs
$(\sqrt{s})^{x}$) which is said to be the same as the trend for pions in Ref.
bylinkinprd , presenting an unresolved contradiction.

Reference bylinkin2016b  also considers the hadron mass-dependence of
$\bar{p}_{t}$ trends. Based on its Fig. 3 it concludes that “These [$\langle
p_{t}\rangle$] are also growing linearly [with mass] if [one is analyzing
different baryon spectra based on masses].” Note that statements in Ref.
bylinkin2016b  are based on proton spectra for $p_{t}\in[0.35,2]$ GeV/c, a
restricted interval. The simple statement conceals more-detailed relationships
revealed by TCM analysis. An example is provided by study of $\bar{p}_{t}$ for
multistrange hadrons below.

The TCM form for the conventional $\langle p_{t}\rangle$ or $\bar{p}_{t}$
ratio

$\displaystyle\bar{p}_{t}^{\prime}\equiv\frac{\bar{P}_{t}^{\prime}}{n_{ch}^{\prime}}$
$\displaystyle\approx$
$\displaystyle\frac{\bar{p}_{ts}+x(n_{s})\bar{p}_{th}(n_{s})}{\xi+x(n_{s})},$
(22)

where $x(n_{s})\equiv n_{h}/n_{s}\approx\alpha\bar{\rho}_{s}$ is the ratio of
hard-component to soft-component yields ppprd  and $\xi\leq 1$ accounts for a
low-$p_{t}$ spectrum acceptance cutoff, conflates two simple TCM trends.
Alternatively, the ratio

$\displaystyle\frac{n_{ch}^{\prime}}{n_{s}}\bar{p}_{t}^{\prime}\approx\frac{\bar{P}_{t}}{n_{s}}$
$\displaystyle=$
$\displaystyle\bar{p}_{ts}(\sqrt{s})+x(n_{s})\bar{p}_{th}(n_{s},\sqrt{s})$
(23)

where $n_{ch}^{\prime}=\xi n_{s}+n_{h}$, provides a simple and convenient
basis for testing the TCM hypothesis precisely. Note that $\bar{p}_{ts}$ and
$\bar{p}_{th}$ depend on collision energy per Fig. 5 (right). That dependence
is left implicit in what follows.

Equation (23) describes TCM $\bar{p}_{t}$ trends for unidentified hadrons. For
identified hadrons of species $i$ from $p$-A collisions ($\nu\geq 1$) the
corresponding description is

$\displaystyle\frac{\bar{P}_{ti}}{n_{si}}$ $\displaystyle=$
$\displaystyle\frac{n_{chi}}{n_{si}}\bar{p}_{ti}\approx\bar{p}_{tsi}+\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})\,\bar{p}_{thi}(n_{s}).~{}~{}~{}$
(24)

Factors $\tilde{z}_{i}(n_{s})x(n_{s})$ have been defined above ,
$\nu(n_{s})\equiv 2N_{bin}/N_{part}$ (binary collisions per participant pair)
is a geometry parameter for $p$-A and A-A collisions, and soft- and hard-
component mean values $\bar{p}_{tsi}$ and $\bar{p}_{thi}(n_{s})$ are obtained
from TCM models $\hat{S}_{0i}(y_{t})$ and $\hat{H}_{0i}(y_{t},n_{s})$, where
$\hat{H}_{0i}(y_{t},n_{s})$ may be slowly varying with $n_{s}$ pidpart2 ;
pppid .

Figure 10 (left) shows $\bar{p}_{t}$ data for neutral kaons, Lambdas, Cascades
and Omegas in a conventional $\bar{p}_{t}$ format equivalent for PID data to
Eq. (22). The solid symbols are for 13 TeV $p$-$p$ collisions, the open
symbols for 5 TeV $p$-Pb collisions. The line marked NSD is the NSD
$\bar{\rho}_{0}=n_{ch}/\Delta\eta$ value for 13 TeV $p$-$p$ collisions.

Figure 10:  Left: Ensemble-mean $\bar{p}_{t}$ data for 13 TeV $p$-$p$
collisions from Fig. 5 of Ref. alippss  (solid dots) and for 5 TeV $p$-Pb
collisions (open circles) from Ref. aliceppbpid . Open squares (connected by
dashed curves) are $p$-$p$ $\bar{p}_{t}$ values inferred from TCM spectra in
Ref. ppsss . Dash-dotted curves are $p$-Pb $\bar{p}_{t}$ TCM trends from Refs.
ppbpid ; pidpart2 . Right: Data and curves from (a) transformed to the TCM
format of Eq. (24).

Figure 10 (right) shows the same data and curves transformed to the format of
Eq. (24) by multiplication with factor
$1+\tilde{z}_{i}(n_{s})x(n_{s})\nu(n_{s})$ ($\nu\rightarrow 1$ for $p$-$p$
data and $x(n_{s})=\alpha\bar{\rho}_{s}$). In the format of Eq. (24) $p$-$p$
and $p$-Pb data are directly comparable, and the close correspondence between
the two systems (note open circles vs solid dots) is evident, whereas that is
not the case at left. The solid lines are a TCM with fixed hard components.
The dashed curves and open boxes include $p$-$p$ hard components
$\bar{p}_{thi}(n_{s})$ that vary systematically with event class per
$x(n_{s})$ to accommodate data ppsss .

The slopes in the right panel are determined by product
$\tilde{z}_{i}(n_{s})\bar{p}_{thi}(n_{s})$ per Eq. (24). While
$\bar{p}_{thi}(n_{s})$ may vary by a few tens of percent from one hadron
species to another (and not monotonically with mass) it is hard/soft fraction
ratio $\tilde{z}_{i}(n_{s})$ that varies directly proportional to hadron mass
$m_{0i}$ (cf. Sec. V.2), leading to factor 30 increase in the product (slope)
from pions to Omegas. Thus, it is mass dependence of jet number, not hard-
component shape, that dominates $\bar{p}_{t}$ variation with hadron species.
The $\bar{p}_{t}$ mass trend reported in Fig. 3 of Ref. bylinkin2016b  may be
compared with values at the dotted line of Fig. 10 (left). In particular, the
values for pions and kaons in Ref. bylinkin2016b  seem to have large
systematic errors.

To summarize from Figure 10 (right), all hadron species require distinct soft
and hard spectrum components inferred from data as in Fig. 6. The Bylinkin
exponential mimics the former to a greater or lesser extent depending on
hadron mass and jet/nonjet ratio. $\bar{p}_{t}$ soft component $\bar{p}_{tsi}$
is mass dependent because of required evolution of the $\hat{S}_{0}(m_{t})$
model function with hadron mass as exhibited in Fig. 2 (right). Higher-mass
hadrons, i.e. baryons, require distinct model representation and make large
contributions to $\bar{p}_{t}$ as demonstrated above.

##  VI Thermalization mechanisms

Inference of “collective phenomena” in small collision systems since 2010 has
generated interest in possible mechanisms to achieve thermalization in such
systems. The Bylinkin spectrum model and interpretation of its exponential
element as an indicator for thermalization plus a proposal of quantum
entanglement as a mechanism for thermalization have drawn significant
attention.

###  VI.1 Thermalization within small collision systems

As noted, a chronic problem for the nuclear physics community over the past
decade has been appearance in small ($p$-$p$, $p$-A) collision systems at the
LHC of “flow-like” phenomena interpreted as such for A-A collisions hydro .
Either those phenomena are not manifestations of flows, which jeopardizes QGP
formation claims for A-A collisions, or a dense medium is generated even in
smaller systems whose gradients then drive the claimed flows. Resolution of
that dilemma motivates introduction of ad hoc mechanisms that either rely on
final-state particle rescattering in a low-density environment to produce a
thermalized high-density medium or invoke some more-exotic strategy. The
narrative outlined below appears to fall into the latter category, relying on
the Bylinkin-model exponential element to signal thermalization.

In Ref. bylinkintherm  a novel thermalization mechanism is based on the
Bylinkin spectrum model and its response to various experimental situations in
terms of the perceived presence or absence of its exponential element. The
argument proceeds with “The [Bylinkin model] thermal component is due to the
effective event horizon introduced by the confining string…. The slope
[$T_{th}$] of the soft component of the hadron spectrum … is determined by the
saturation momentum that drives the deceleration in the color field….” Those
statements are associated with an Unruh effect conjectured to arise from
acceleration in a vacuum, possibly related to Hawking radiation from black
holes. Thermal emission of hadrons is linked to deceleration by color fields
(strings) between partons.

The argument continues with “The hard component is well understood as
resulting from the high momentum transfer scattering of quarks and gluons, and
their subsequent fragmentation. The ‘soft’ one is ubiquitous in high energy
collisions and has the appearance of the thermal spectrum – but its origin
remains mysterious to this day. …while in nuclear collisions one may expect
thermalization to take place, it is hard to believe that thermalization can
occur in … DIS or $e^{+}$-$e^{-}$ annihilation. … The universal thermal
character of hadron [$p_{t}$] spectra and abundances in all high energy
processes…begs for a theoretical explanation.” The paper concludes with “We
hope that our analysis sheds some new light on the origin of the thermal
component in hadron production. … the soft hadron production [Bylinkin
exponential] is a consequence of the quantum evaporation from the event
horizon formed by deceleration in longitudinal color fields.”

That narrative is based on several assumptions, (a) that the Bylinkin-model
exponential term is demonstrably required by spectrum data, (b) that it
actually represents hadron production from a thermalized emitter and (c) that
there are no alternative interpretations of spectrum data available, thus
requiring a novel explanation.

Assumption (a) fails as demonstrated in previous sections: see Figs. 4 and 5
for Bylinkin model parameters that deviate strongly from trends required by
data and other figures where the exponential element is excluded. Assumption
(b) fails as demonstrated for example by Fig. 16 (left) where hadron emission
from $e^{+}$-$e^{-}$ collisions (points) exhibits the same soft-component
$\hat{S}_{0}(m_{t})$ (solid curve) as that from $p$-$p$ and $p$-A collisions
that strongly diverges from a pure exponential (dotted curve). That difference
indicates that the emitting system is not thermalized. $\hat{S}_{0}(m_{t})$
exponent $1/n>0$ signals that the emitting system (parton splitting cascade)
includes a noise source in the form of Gribov diffusion gribov  equivalent to
a random walk in transverse phase space of successive parton generations,
deeper with increasing energy. Diffusion in configuration space leads to
increasing inelastic cross sections gribov3 . Diffusion in momentum space
leads to increase of $\hat{S}_{0}(m_{t})$ parameter $1/n$ in Fig. 3 (left)
gribov2 .

Assumption (c) collides with decades of work to understand, theoretically and
experimentally, the role of QCD in the context of high-energy nuclear
collisions. The theoretical basis for the TCM was already enunciated in Ref.
pancheri  (1985): The low-$p_{t}$ part or soft component, what the Bylinkin
exponential element is nominally intended to describe, “…does not show any
multiplicity dependence….” (referring to $P_{0}(p_{t})$ in Eq. (2)),
consistent with TCM $\hat{S}_{0}(m_{t})$ but not with the strongly varying
$T_{th}$ of the Bylinkin exponential element. Referring to UA1 measurements,
that detector “measures not only the hard parton scattering process but also
the debris which result from the breaking up of the [projectile] proton when a
hard parton is emitted [emphasis added] pancheri .” The splitting cascade
resulting in “debris” from projectile nucleon dissociation is the same process
evident in Fig. 16 below.

Aside from the issues raised above there is one of consistency in assigning
the Bylinkin model exponential element to indicate system thermalization via
various means according to various conditions. In Ref. bylinkin2014nucphb  it
is stated that “Those partons preexist long time before the interaction and
could be considered as being a thermalized statistical ensemble. The hadrons
[what species?] from this source are distributed presumably according to the
Boltzmann-like exponential…distribution….” In Ref. bylinkin2015  there appears
“The exponential term in this model is associated with thermalized production
of hadrons by valence quarks and a quark-gluon cloud coupled to them.” Such
scenarios contrast with that proposed in Ref. bylinkintherm  relying on “the
effective event horizon” also noted above.

Problems arise for a scenario of hadrons emerging from a thermal source when
the Bylinkin exponential is limited to pions only: In Ref. bylinkinanom  “…the
spectra of charged kaons and protons…shows no room for the contribution of the
exponential term” with the comment “…only pions require a sizable exponential
term in their…spectra. This anomaly observed in pion production didn’t find a
consistent explanation yet.” In Ref. bylinkphen  “…the $\pi$ production in
$pp$ collisions is dominated by a release of quasi-thermalized particles,
while the spectra of heavier [hadrons] are dominated by pQCD like production
mechanisms, leaving a relatively small room for thermalized particle
production.” Apparently, pions arise from one production mechanism while more-
massive hadrons arise from another. Referring to spectrum ratios in Ref.
bylinkphen  “the observed significant difference in the exponential
contributions to the hadron spectra implies the difference in the hadron
production mechanisms rather than an artifact of the fit procedure [emphasis
added].”

###  VI.2 Thermalization and entanglement

Reference dimaentangle  seeks to account for apparent thermalization in small
collision systems (e.g. $p$-$p$ collisions): “…we test the hypothesis about
the link between quantum entanglement and thermalization in high-energy
collisions.” Evidence for thermalization is inferred from a specific $p_{t}$
spectrum model: “…we address the origin of the apparent thermalization in
high-energy collisions that is usually inferred from the presence of the
exponential component [of the Bylinkin model] in the [$p_{t}$] distributions
of produced particles and the thermal abundances of the hadron yields
[emphasis added].” The latter phrase refers to observed agreement between data
and statistical-model predictions for hadron species abundances statmodel .
Inferred thermalization has usually been attributed to particle multiple
scattering: “The emergence of the thermal features in a high energy proton-
proton collision is surprising, as the number of secondary interactions in
this process is relatively low and does not favor thermalization through
conventional final-state interaction mechanisms.” An alternative mechanism is
sought.

The authors invoke quantum entanglement: “…we will investigate the possibility
that the apparent thermalization in high energy collisions is achieved during
the rapid ‘quench’ induced by the collision due to the high degree of
entanglement inside the…colliding protons. … Since a high-energy collision can
be viewed as a rapid quench of the entangled partonic state, it is thus
possible that the effective temperature inferred … can depend on the momentum
transfer….” Based on spectrum analysis with the Bylinkin model “We confirm the
relation between the effective temperature [$T_{th}$] and the hard scattering
scale [$T_{h}$] observed at lower energies, and find that it extends even to
the Higgs boson production process.”

Within the context presented by Ref. dimaentangle  the observed close coupling
of $T_{th}$ and $T_{h}$ is viewed as supporting a conjecture, but from a
standpoint of general data modeling any close correspondence of parameters
from distinct model elements signals redundancy. The two TCM model components,
derived empirically from data trends, are completely independent in that
regard. Such parameter independence indicates, as an experimental finding,
that there are two distinct hadron production mechanisms.

###  VI.3 Entanglement and hadron-hadron collisions

Reference dimaentangle  analyzes data spectra from several collision systems
with the Bylinkin model for requirement or not of the exponential model
element: a hadron spectrum from nondiffractive 13 TeV $p$-$p$ collisions, a
dimuon spectrum from the process $\gamma$-$\gamma\rightarrow\mu^{+}$-$\mu^{-}$
for 13 TeV $p$-$p$ collisions, and Higgs boson decays to $\gamma$-$\gamma$ and
to four leptons. Reference dimaentangle  first applies the Bylinkin model to
inelastic $p$-$p$ collisions to confirm that exponential (“thermal”) and
power-law (“hard”) model elements combined are able to describe spectrum data.
That result is interpreted to confirm thermalization in such collisions.

Figure 11 (left) shows MB unidentified-hadron data from 13 TeV $p$-$p$
collisions (points). The dash-dotted curves include a $q$-Gaussian labeled
$q$-G and an exponential labeled E representing the two elements of the
Bylinkin spectrum model as they appear in Fig. 2 of Ref. dimaentangle . The
Bylinkin model Eq. (5) (solid, sum of two elements) describes the MB data
reasonably well. The exponential element is determined by
$T_{e}\leftrightarrow T_{th}=0.17$ GeV which for this figure is adjusted to
0.145 GeV for better agreement with data, consistent with TCM analyses. Power-
law parameter $T_{h}=0.72$ GeV/c determines the shoulder position or width of
the $q$-Gaussian. Parameter $N=3.1$ determines the power-law exponent as
$1/p_{t}^{2N}\rightarrow 1/p_{t}^{6.2}$ illustrated by the dotted line. The
dotted curve is TCM $\hat{S}_{0}(m_{t})$ with slope parameter $T=0.145$ GeV
and exponent $n=6$ displaced down from data for visibility to demonstrate that
a single function can describe those data. This comparison demonstrates that
for MB data the Bylinkin model is sufficient, but not necessary. Monolithic
model $\hat{S}_{0}(m_{t})$ performs just as well.

Figure 11:  Left: Minimum-bias unidentified-hadron spectrum from 13 TeV
$p$-$p$ collisions (points) and Bylinkin model elements exponential E and
power-law ($q$-Gaussian) $q$-G (dash-dotted) with the sum (solid). A TCM soft-
component model $\hat{S}_{0}(m_{t})$ (dotted) with $n=6$ (matching $N\approx
3$) is included for comparison. Right: Identified-pion spectra from 13 TeV
$p$-$p$ collisions (points) for event classes $n=1$ (high $n_{ch}$) and $n=9$
(low $n_{ch}$). The spectra have been rescaled by TCM soft-component particle
density $\bar{\rho}_{s}$. Model elements $\hat{S}_{0}(m_{t})$ (dotted) and
$x(n_{s})\hat{H}_{0}(y_{t})$ (dashed) are not fitted to individual spectra and
constitute predictions. The full TCM (solid) includes a resonance contribution
at low $p_{t}$ (see Fig. 17).

Figure 11 (right) shows pion spectrum data (open circles) for low ($n=9$) and
high ($n=1$) event multiplicity 13 TeV $p$-$p$ collisions. The same data
appear in Fig. 6 (left). The published spectrum data are presented in the form
$\bar{\rho}_{0}(m_{t})=d^{2}n_{ch}/p_{t}dp_{t}dy_{z}$ rescaled to
$\bar{\rho}_{0}(m_{t})/\bar{\rho}_{s}$ for pions, with $\bar{\rho}_{s}$ the
root of $\bar{\rho}_{0}=\bar{\rho}_{s}+\alpha\bar{\rho}_{s}^{2}$ and
$\alpha\approx 0.017$ for 13 TeV $p$-$p$ alicetomspec . Since NSD
$\bar{\rho}_{s}\approx 5.8$ for 13 TeV is near factor $2\pi$ the unrescaled MB
hadron spectrum at left is similar to $\hat{S}_{0}(m_{t})$ (dashed) at right.
TCM parameters for $\hat{S}_{0}(m_{t})$ are slope parameter $T=0.145$ GeV and
Lévy exponent $n=8.0$ alicetomspec  and for $\hat{H}_{0}(y_{t})$ (defined as a
density on $y_{t}$) are mode $\bar{y}_{t}=2.45$, width above the mode
$\sigma_{y_{t}}=0.58$ and exponential parameter $q=3.8$ pppid .
$\hat{H}_{0}(y_{t})$ is here transformed to a density on $p_{t}$. The value
for $q$ is thus related to the $p_{t}$ power law as $2N\approx q+2$ because
the Jacobian from density on $p_{t}$ to density on $y_{t}$ is
$m_{t}p_{t}/y_{t}$ involving in effect a factor $p_{t}^{2}$ at higher $p_{t}$.

The spectrum rescaling $\bar{\rho}_{0}(m_{t})/\bar{\rho}_{s}$ ensures that all
spectra coincide at lower $p_{t}$ where the hard/soft model-function ratio is
$\ll 1$. Note that NSD $\bar{\rho}_{s}(\sqrt{s})$ is consistently defined from
13 TeV down to 64 GeV and is not a fit parameter, nor is parameter
$\alpha(\sqrt{s})$. Likewise, exponent $n(\sqrt{s})$ is determined by the
expression in Fig. 3 (left). $\hat{H}_{0}(y_{t})$ parameters for each hadron
species (here pions) are also determined by previous studies and are not
available as fit parameters for individual spectra.

The right panel demonstrates that variation of pion spectrum shape with
increasing event multiplicity consists entirely of different admixtures of two
fixed functions where the admixture (i.e. two fractions) is predicted based on
the relation $\bar{\rho}_{h}\approx\alpha(\sqrt{s})\bar{\rho}_{s}^{2}$ or
$x(n_{s})\equiv\bar{\rho}_{h}/\bar{\rho}_{s}\approx\alpha(\sqrt{s})\bar{\rho}_{s}$.
Those two functional forms, determined empirically from data ppprd , can be
seen as “eigenfunctions” for $p$-$p$ spectrum description: Any $p$-$p$
spectrum for a given collision energy may be expressed as a linear combination
of the two. In contrast, parameters $T_{th}$, $T_{h}$ and $N$ of the two
elements of the Bylinkin model must change substantially with changing
collision conditions (e.g. $n_{ch}$). Their values are then not interpretable
and the Bylinkin model is excluded. The TCM is shown to be sufficient and
necessary – thus statistically equivalent to $p$-$p$ data.

###  VI.4 Entanglement and exotic systems

Having concluded that Bylinkin-model fits to hadron spectra from MB 13 TeV
$p$-$p$ collisions demonstrate the existence of an exponential element, and
therefore evidence for thermalization, Ref. dimaentangle  considers a scenario
where thermalization is not expected via entanglement – a dimuon spectrum from
the process $\gamma$-$\gamma$ $\rightarrow\mu^{+}$-$\mu^{-}$. It is remarked
that “such diffractive events are expected to have a suppressed thermal
(exponential) component. … As the presence of the thermal component in this
approach is the consequence of the entanglement, we expect it to be absent in
diffractive events.” It is notable that events accepted require that both
protons remain intact. Thus, the statement “…these diffractive processes allow
an analysis of the fragmentation of a high energy proton in an intense
electromagnetic field produced by the other proton” seems contradictory.
Presence or absence of a soft component (what the Bylinkin exponential
responds to) depends on dissociation or not of projectile protons.

Figure 12 (left) shows a dimuon $p_{t}$ spectrum (points) adapted from Fig. 3
of Ref. dimaentangle  representing the process $\gamma$-$\gamma$
$\rightarrow\mu^{+}$-$\mu^{-}$ for 13 TeV $p$-$p$ collisions. Data and curves
are plotted in the format of density on $p_{t}$ vs pion transverse rapidity
$y_{t}$ with zero lower bound that gives improved access to lowest $p_{t}$.
The dash-dotted curve is the Bylinkin model with parameters $T_{h}=0.45$ GeV
and $N=1.55$ adjusted here to best describe the data. Also shown is TCM model
$\hat{S}_{0}(m_{t})$ (solid) with pion $m_{t}$ and parameters $T=0.145$ GeV
and $n=3.2$ that should correspond to Bylinkin $N=1.6$. The dashed curve is
$\hat{S}_{0}(m_{t})$ with parameters $T=0.245$ GeV and $n=3.3$. The latter
demonstrates that a function with exponential limiting case is not ruled out
by the available data. It should also be noted that projectile protons here
remain intact: there is no dissociation and therefore no parton splitting
cascade hadronizing to a TCM soft component.

Figure 12:  Left: Dimuon $p_{t}$ spectrum (points) from Fig. 3 of Ref.
dimaentangle  (points) representing the process
$\gamma$-$\gamma\rightarrow\mu^{+}$-$\mu^{-}$ for 13 TeV $p$-$p$ collisions.
The dash-dotted curve is the Bylinkin power-law model element ($q$-Gaussian)
denoted by $q$-G. The solid and dashed curves are TCM model function
$\hat{S}_{0}(m_{t})$ with parameters as described in the text. The lower
dotted line represents the power law for hadrons from 13 TeV $p$-$p$
collisions. Right: Higgs boson $p_{t}$ spectrum from
$H\rightarrow\gamma$-$\gamma$ (points). The dash-dotted curves are Bylinkin
model elements with parameters adjusted to best accommodate data. The solid
curve is TCM model function $\hat{S}_{0}(m_{t})$ with parameters as described
in the text.

In contrast to expectations for $\gamma$-$\gamma\rightarrow\mu^{+}$-$\mu^{-}$,
spectra for Higgs bosons decaying to $\gamma$-$\gamma$ or to four leptons are
expected to include an exponential component reflecting thermalization if
entanglement is involved. Applying the Bylinkin model to a Higgs spectrum in
its Fig. 4, Ref. dimaentangle  concludes that “there clearly are both the hard
scattering (power law) and thermal (exponential) components in the [$p_{t}$
spectrum] similarly to…” that for their Fig. 2.

Figure 12 (right) shows the same Higgs boson spectrum (points) from
$H\rightarrow\gamma$-$\gamma$. The dash-dotted curves are elements of the
Bylinkin model (as they appear in Ref. dimaentangle  Fig. 4) where the
exponential element E has $T_{th}=10$ GeV and the power-law element $q$-G has
$T_{h}=53$ GeV and $N=1.6$ (i.e. $n=3.2$). The dotted curve is the power-law
element alone with $T_{h}=15$ GeV and $N=1.42$ displaced upward for
visibility. The solid curve is TCM model $\hat{S}_{0}(m_{t})$ with pion mass
for $m_{t}$ and with parameters $T=5$ GeV and $n=3.2$. These data are not of
sufficient quality to distinguish among three models and cannot establish the
presence or absence of a separate exponential function. The same may be said
for Fig. 5 of Ref. dimaentangle  representing $H\rightarrow 4l$. One may ask
what sense does a “temperature” value $T_{th}\approx 15$ GeV convey and what
does $T_{h}\approx 53$ GeV (“hard scale”) mean?

In response to conjecture in Ref. dimalev  that entanglement might lead to
thermalization and related predictions – “We propose the entanglement entropy
as an observable that can be studied in deep inelastic scattering” – Ref. h1
(H1) responds “…in order to further understand the role of colour confinement
in high energy collisions, it has been suggested that quantum entanglement of
partons could be an important probe of the underlying mechanism…. The
predictions [derived] from the entropy of gluons are found to grossly disagree
with the hadron entropy obtained from the multiplicity measurements presented
here, and therefore the data do not support the basic concept of equality of
the parton and hadron entropy with the current level of theory development.”

In Ref. iskander  quantum entanglement and thermal behavior are further probed
with neutrino-induced reactions. The basis for analysis is again the Bylinkin
model and the perceived presence or absence of its exponential element in fits
to spectra. A contrast is made between (a) interactions where only part of a
nucleon is involved (termed differential scattering) and (b) where the entire
target nucleon or nucleus is involved (termed coherent scattering). Case (a)
is represented by $\bar{\nu}_{\mu}+N\rightarrow\mu^{+}+\pi^{0}+X$ and (b) is
represented by $\bar{\nu}_{\mu}+A\rightarrow\mu^{+}+\pi^{-}+A$

Figure 13 (left) shows data (points) adapted from Fig. 2 of Ref. iskander  for
case (a). The dash-dotted curves represent the elements of Eq. (5) with
parameters $T_{th}=0.145$ GeV, $T_{h}=0.8$ GeV and $N=2.2$ adjusted to best
reproduce the Ref. iskander  Fig. 2 results. The solid curve is
$\hat{S}_{0}(m_{t})$ with parameters $T=0.145$ GeV and $n$ = 4.6. The two
models are indistinguishable and describe the data reasonably well. However,
the Bylinkin model requires five fitted parameters whereas
$\hat{S}_{0}(m_{t})$ requires only two, since $T=0.145$ is predicted for a
pion spectrum. There is no possibility from these data to demonstrate that
either the $q$-Gaussian or exponential is required. If the Bylinkin
exponential were replaced by $\hat{S}_{0}(m_{t})$, whose limiting case for
$1/n\rightarrow 0$ is indeed an exponential, it is likely that a $\chi^{2}$
fit would return zero amplitude for the $q$-Gaussian, and Bayesian analysis
would reject the composite Bylinkin model based on needless complexity. There
is thus no evidence in these data for a required exponential element signaling
thermalization.

Figure 13:  Left: Data (points) adapted from Fig. 2 of Ref. iskander  for the
process $\bar{\nu}_{\mu}+N\rightarrow\mu^{+}+\pi^{0}+X$. The dash-dotted
curves represent elements of Eq. (5). The solid curve is TCM soft-component
model $\hat{S}_{0}(m_{t})$. Parameter values are given in the text. Right:
Data (points) adapted from Fig. 3 of Ref. iskander  for the process
$\bar{\nu}_{\mu}+A\rightarrow\mu^{+}+\pi^{-}+A$. The dash-dotted curve is the
power-law element of Eq. (5). The solid curve is TCM soft component
$\hat{S}_{0}(m_{t})$. The dashed curve is TCM hard component
$\hat{H}_{0}(y_{t})$ transformed to a density on $m_{t}$.

Figure 13 (right) shows data (points) adapted from Fig. 3 of Ref. iskander
for case (b). $q$-Gaussian (dash-dotted) parameters are $T_{h}=0.8$ GeV and
$N=2.2$ ($n=4.4$). TCM soft component $\hat{S}_{0}(m_{t})$ (solid) parameters
are $T=0.145$ GeV and $n=4.6$. Hard component $\hat{H}_{0}(m_{t})$ (dashed)
parameters are $\bar{y}_{t}=2.65$, $\sigma_{y_{t}}=0.9$ and $q=2.5$ (i.e.
$n=4.5$). Recall that $\hat{H}_{0}(y_{t})$ is a Gaussian on $y_{t}$ with
exponential tail transformed here to a density on $m_{t}$ via Jacobian
$y_{t}/m_{t}p_{t}$. Data prefer the TCM hard component to the Bylinkin
$q$-Gaussian. Data appear to reject a soft component which is reasonable since
once again the projectile hadrons remain intact in the final state and there
is thus no parton splitting cascade.

###  VI.5 Relation between thermal and hard scales

Reference pajares  addresses the issue of possible thermalization via
entanglement within high-energy $p$-$p$ collisions, emphasizing an inferred
relation between thermal temperature $T_{th}$ and “hard scale” $T_{h}$. The
conjectured thermalization process is described thus: “In a high-energy
collision a hard parton interaction produces a rapid quench of the entangled
partonic state [of projectile nucleons] and thus the characteristic effective
temperature [$T_{th}$ e.g. inferred from the Bylinkin exponential model
element] can depend on the energy scale of the hard process… [parameter
$T_{h}$ of the Bylinkin $q$-Gaussian].” In order to establish the relation
between $T_{th}$ and $T_{h}$ “we…perform an extensive study of the energy and
multiplicity dependence of the hard and soft scales in $pp$ collisions….” The
study asserts that “…the relation between both scales is determined
approximately by the inverse of the normalized fluctuations of the number of
partons of the initial wave function or, equivalently, of the normalized
fluctuations of the hard scale.” Results of that study are examined here in
context of the TCM.

Figure 14 (left) repeats Fig. 3 (left) with the addition of values $k$ from
Table I of Ref. pajares  plotted as $1/2k$ (open squares) to be compatible
with the other data. Open triangles represent parameter $N$ taken from Fig. 3
of Ref. bylinkin2015 . Those data are obtained via fits of Eq. (5) to charged-
hadron spectra from MB or NSD $p$-$p$ collisions. The solid points are derived
from the TCM applied to LHC, RHIC, SPS and NAL (fixed-target) $p$-$p$ spectra.
Displacement of open points from the upper solid curve (TCM hard component)
arises from the Bylinkin $q$-Gaussian performing double duty: not only
describing the jet contribution, peaked near 1 GeV/c, but also part of the
nonjet soft component, thus pulling the $1/2N$ or $1/2k$ values down
corresponding to a softer tail.

Figure 14:  Left: Model exponents plotted as reciprocals. TCM hard component
$1/(q+2)$ and TCM soft component $1/n$ from Ref. alicetomspec  (solid points)
compared with Bylinkin $1/2N$ from Ref. bylinkin2010  and equivalent parameter
$1/2k$ from Ref. pajares  (open points). Right: TCM hard-component parameters
$\bar{y}_{t}$ (mode) and $\sigma_{y_{t}}$ (width above the mode) from Ref.
alicetomspec  vs Bylinkin power-law parameter $T_{h}$ ($q$-Gaussian width)
from Ref. pajares .

Figure 14 (right), adopted from Ref. alicetomspec , presents energy evolution
of TCM hard-component $\hat{H}_{0}(y_{t})$ parameters indicating that the peak
mode $\bar{y}_{t}$ is essentially independent of collision energy while the
peak width above the mode $\sigma_{y_{t}}$ increases linearly according to
$\ln(\sqrt{s})$ reflecting evolution of underlying jet spectra. Solid points
are from RHIC and LHC while open points at lowest energies are from NAL fixed-
target spectra tomnmf . The open triangles are Bylinkin-model power-law
parameter $T_{h}$ obtained from Table I of Ref. pajares . The systematic
upward displacement of the last arises from the difference in required width
$T_{h}\leftrightarrow\sigma_{h}$ of a $q$-Gaussian centered at zero $p_{t}$ vs
that for a peaked distribution centered near 1 GeV/c. Otherwise, the energy
trends agree within uncertainties.

Figure 15 (left) responds to Eq. (4) of Ref. pajares , a proposed linear
relation between “thermal” parameter $T_{th}$ and “hard” parameter $T_{h}$
said to arise from “rapid quench of the entangled partonic state” as noted
above. Open squares represent $T_{h}$ and $k$ values from Table I of Ref.
pajares  plotted as $T_{h}/X$ where $X=k-1,~{}k$ or $k+1$ to illustrate the
consequences of choosing one form over others. Bylinkin exponential parameter
$T_{th}$ increases significantly with energy, but that increase can be
attributed to increased jet contribution to spectra at higher $p_{t}$ (see
below). Soft-component slope parameter $T_{\text{TCM}}=0.145\pm 0.005$ GeV for
unidentified hadrons is represented by the hatched band – no significant
energy dependence. Relations $T_{h}/X$ on the other hand strongly increase
with collision energy; the mechanism should be clear from Fig. 14: strong
increase of jet production. No reason is given for the specific choice of Eq.
(4) in Ref. pajares , but it best accommodates the claimed relation.

$\bf k\leftrightarrow N$

Figure 15:  Left: Bylinkin-model exponential parameter $T_{th}$ and power-law
parameters $T_{pl}/X$ ($X$ representing three instances of power-law exponent
$k$) from Ref. pajares  vs collision energy. The hatched band represents TCM
soft-component $T\approx 0.145$ GeV for unidentified hadrons. Right: 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 systematic variation with energy of TCM parameters as in Refs.
alicetomspec ; tomnmf . The points are from Refs. tomnmf  (23.8 GeV), ppquad
(200 GeV) and alicespec  (13 TeV).

Figures 5 and 6 of Ref. pajares  show $T_{th}$ vs $T_{h}/(k+1)$ comparisons
for identified hadrons from $p$-$p$ and $p$-Pb collisions vs event
multiplicity $n_{ch}$. The figures suggest confirmation of Eq. (4) in Ref.
pajares  given apparent overlap of $T_{th}$ and $T_{h}/(k+1)$ error bands;
however, that can be questioned. TCM analysis of PID spectrum data indicates
that while pions consistently require $\hat{S}_{0}(m_{t})$ $T\approx 0.145$
GeV all heavier hadrons are well-described by $T\approx 0.200$ GeV independent
of $n_{ch}$. See Fig. 9 (right) for Omegas. That $T_{th}$ might increase
substantially with $n_{ch}$ and hadron mass could arise because the Bylinkin
exponential model element is poorly suited to describe data and must become
“flatter” with increasing mass and jet production (noted above). And in Refs.
bylinkinanom ; bylinkphen ; bylinkintherm  the exponential element and
$T_{th}$ are said to be required only for pions, which presents a complication
for Ref. pajares  Eq. (4).

The PID trends in Ref. pajares  can be explained as follows. The Bylinkin
$T_{h}$ parameter (width of a $q$-Gaussian) must increase substantially with
hadron mass because (a) the shoulder of the data soft component moves to
higher $p_{t}$ with mass as in Fig. 2 (right) and (b) the hard/soft fraction
ratio $\tilde{z}_{i}=z_{hi}/z_{si}$ increases linearly with mass as described
in Sec. V.2 and leads to dominance of jet fragments for higher-mass hadrons as
illustrated in Fig. 9. The Ref. pajares  $k$ parameter ($=N$ for Bylinkin)
also increases with hadron mass (the tail softens) again per Fig. 2 (right)
which is partly offset by the growth of jet production and hard component with
$n_{ch}$. Ratio $T_{h}/(k+1)$ confuses several effects and is uninterpretable.

Figure 15 (right) adopted from Refs. alicetomspec ; tomnmf  responds to Eq.
(9) of Ref. pajares  which relates to the statement: “…we devise also a
convenient and simple parametrization of the whole soft and hard [$p_{t}$]
spectrum for the full range of energies explored at RHIC and the LHC.” The
procedure starts with the Bylinkin power-law element, adds fluctuations and
obtains a single function with exponential behavior at low $p_{t}$ and power-
law at high $p_{t}$. The result is shown in its Fig. 8 as applied to several
presumably minimum-bias $p_{t}$ spectra where the jet contribution is minimal.
The functional form of TCM $\hat{S}_{0}(m_{t})$ alone as a monolithic model
(i.e. no jets) can do as well with much less complexity as demonstrated in
Fig. 11 (left).

What such approaches do not do is confront differential evolution of spectrum
shape with event $n_{ch}$ or centrality. The result for full spectrum
description is demonstrated in Figs. 6 and 11 (right) above where the
structure of the TCM is required by data. The TCM then provides the means to
isolate accurately the jet contribution as shown in Fig. 15 (right), including
results from low-energy fixed-target experiments (open squares) tomnmf .

Ironically, while the stated goal in Ref. pajares  is to establish a direct
connection between soft and hard components, in the end it states that “…the
exact form of the relation between the soft and hard sectors is nevertheless
unknown.” In contrast, the relation between TCM soft and hard components is
established accurately to within data statistical uncertainties. Each spectrum
component has its own unique structure, but they are connected by (a) the all-
important relation $\bar{\rho}_{h}\approx\alpha(\sqrt{s})\bar{\rho}_{s}^{2}$
that remains valid over the full range of collision energies and by (b) the
universality of a parton splitting cascade as viewed perpendicular or parallel
to its axis, per Fig. 16.

##  VII Discussion

Interpretation of the Bylinkin exponential element as an indicator for
thermalization within small collision systems is a recurring issue (see Sec.
VI.1). “The ‘soft’ one [exponential element] is ubiquitous in high energy
collisions and has the appearance of the thermal spectrum – but its origin
remains mysterious to this day. …while in nuclear collisions one may expect
thermalization to take place, it is hard to believe that thermalization can
occur in…DIS or $e^{+}$-$e^{-}$ annihilation. … The universal thermal
character of hadron [$p_{t}$] spectra and abundances in all high energy
processes…begs for a theoretical explanation.” In this section several issues
are addressed: (a) the relation of the TCM soft component to dijets arising
from $e^{+}$-$e^{-}$ $\rightarrow$ $q$-$\bar{q}$ collisions, (b) conjectured
quantum entanglement as a mechanism for thermalization in small collision
systems vs parton splitting cascades in $p$-$p$ inelastic collisions and dijet
formation, (c) spectrum model design – a priori vs a posteriori and (d)
resonance contributions to pion spectra and so-called “anomalous behavior” of
thermalization as inferred from the Bylinkin model.

###  VII.1 Dijet properties and p-p soft component

If one accepts that hadron production via fragmentation of energetic leading
particles (hadrons or partons) is a dominant process in high-energy nuclear
collisions then the process $e^{+}$-$e^{-}$ $\rightarrow$ $q$-$\bar{q}$
$\rightarrow$ dijet should provide a particularly clear example. The
$e^{+}$-$e^{-}$ data presented below, $p_{t}$ and $p_{z}$ spectra relative to
the dijet axis, correspond to an event condition of single dijet production.
One should then expect no jet contribution to the $p_{t}$ spectrum (no three-
jet events). Within the Bylinkin model context that condition implies only the
exponential model element should play a role, no power-law element.

Figure 16 (left) shows an $e^{+}$-$e^{-}$ $p_{t}$ spectrum replotted vs pion
$y_{t}$ (derived from measured $p_{t}$ assuming all hadrons are pions as for
$y_{z}$ below). The points are published $dn_{ch}/dp_{t}$ data (Fig. 18a of
Ref. alephff ) (ALEPH) divided by $p_{t}$ with no other scaling. According to
Ref. alephff  that is the “charged particle momentum component transverse to
the sphericity [dijet] axis and projected into the event plane.” The solid
curve is soft component $\hat{S}_{0}(m_{t})$ with slope parameter $T=90$ MeV
and exponent $n=7.8$. The $e^{+}$-$e^{-}$ data are described within
uncertainties except for two points at 0.02 and 0.07 GeV/c. In comparison,
soft-component $\hat{S}_{0}(y_{t})$ for pions from 13 TeV $p$-$p$ collisions
has $T=145$ MeV and $n=8.5$ pppid .

Ironically, this result deviates dramatically from expectations within the
Bylinkin model context: For instance, “only the inclusive spectra of charged
particles produced in pure baryonic collisions require a substantial
contribution of the Boltzman[n]-like exponential term bylinkin2010 ” and “…the
‘thermal’ production expressed by the exponential term…is essential only for
$pp$\- collisions….bylinkin2014nucphb .” But the Bylinkin exponential element
is approximating, more or less, the TCM soft-component model in most settings.
The dash-dotted curves are the two Bylinkin model elements whose sum
approximates the solid curve. The Bylinkin “thermal” exponential has
$T_{th}=90$ MeV whereas the “power-law” element has $T_{h}=0.32$ GeV and
$N=3.3$. The conventional Bylinkin model interpretation implies that the
$p_{t}$ spectrum from $q$-$\bar{q}$ dijets includes a “hard component”
transverse to the dijet axis.

Figure 16:  Left: Hadron distribution on $p_{t}$ (perpendicular to the dijet
axis) from 91 GeV $e^{+}$-$e^{-}$ collisions alephff  plotted vs pion $y_{t}$.
The solid curve is TCM model $\hat{S}_{0}(m_{t})$ with parameters ($T,n$)
noted in the figure. The dash-dotted curves are Bylinkin exponential E and
$q$-Gaussian $q$-G representing two elements interpreted as thermal and hard
contributions. Their sum is dotted and lies on the solid curve. Right: Hadron
distribution on $y_{z}$ (along the dijet axis) from 91 GeV $e^{+}$-$e^{-}$
$\rightarrow$ $q$-$\bar{q}$ collisions. The curve is Eq. (VII.1) explained in
the text.

Figure 16 (right) shows a hadron distribution on $y_{z}$ from $e^{+}$-$e^{-}$
collisions reported by Ref. alephff . $y_{z}$ ($y_{s}$ in Fig. 16a of Ref.
alephff  with $s$ for “sphericity”) is derived from measured $p_{L}\rightarrow
p_{z}$ along the sphericity (dijet) axis and is based on assigning the pion
mass to all hadrons. The solid curve is a generalized $q$-Gaussian
distribution

$\displaystyle G(y_{z})$ $\displaystyle=$
$\displaystyle\frac{A}{[1+(|y_{z}|/\sigma_{y_{z}})^{\kappa}/n]^{n}}$
$\displaystyle\sim$ $\displaystyle
A\exp_{q}[-(|y_{z}|/\sigma_{y_{z}})^{\kappa}],$

with $\kappa=5$, $\sigma_{y_{z}}=3.35$, $n=4.2$ and $A=6.6$. The distribution
differs from a Gaussian in two ways: (a) the parameter value $\kappa=$ 5
rather than 2 leads to a more rectangular distribution and (b) parameter value
$n=$ 4.2 controls the distribution tails. The $q$ in “$q$-Gaussian” refers to
relation $q-1=1/n$. For $1/n\rightarrow 0$ (and $\kappa=2$) the distribution
reverts to a standard Gaussian form.

The ALEPH combination of $p_{t}$ and $p_{z}$ distributions in Fig. 16 provides
an alternative to fragmentation functions (FFs) presented as densities on
total momentum $p$ or momentum fraction $x_{p}=p/p_{jet}$ eeprd , the
resulting shape sometimes referred to as a “humpbacked plateau” humpback .
Whereas the density on $p_{z}$ is approximately uniform near the origin
($y_{z}$ = 0) the density on total momentum $p$ falls to zero at the origin,
hence the term “humpbacked.”

Equation (VII.1) can be seen as a generalization of the Bylinkin power-law
model element wherein $\kappa\rightarrow 2$ and $n\rightarrow N$. Just as the
Bylinkin exponential does not describe the $e^{+}$-$e^{-}$ $p_{t}$ spectrum,
where there is no jet contribution, the Bylinkin power law does not describe
the $p_{z}$ spectrum which is, in a sense, all jet contribution. In $p$-$p$
collisions the $p_{t}$ spectrum soft component is formally equivalent to Fig.
16 (left) described by TCM soft component $\hat{S}_{0}(m_{t})$ because
dissociation of projectile protons along the beam axis is formally equivalent
to the parton splitting cascades from $q$-$\bar{q}$ fragmentation. However,
the $p_{t}$ spectrum contribution from MB dijets is not equivalent to Fig. 16
(right) since (a) hadron $p_{t}$ relative to the beam axis is not well
correlated with individual jet axes and (b) there is a broad underlying jet
energy spectrum. Spectrum hard components derived empirically from data as in
Fig. 6 are shown to be equivalent to a convolution of measured FFs (on total
momentum $p$) and a measured jet energy spectrum fragevo . The result is very
different from the Bylinkin power-law model element.

###  VII.2 Entanglement vs parton splitting cascade

Arguments in Refs. dimalev ; dimaentangle ; bylinkintherm ; pajares ; iskander
advance a narrative in which apparent thermalization within high-energy
collisions, especially in small collision systems, may be explained via
quantum entanglement. For example, Ref. pajares  asserts that a parton hard
scatter may quench an entangled partonic state (within a projectile nucleon),
producing a correlation between Bylinkin exponential parameter $T_{th}$ and
power-law parameter $T_{h}$ as noted in Sec. VI.5. That is the most recent of
a series of steps in which (a) certain data features in A-A collisions (e.g.
jet quenching, elliptic flow) were interpreted to demonstrate quark-gluon
plasma (QGP) formation within a thermalized high-density medium, (b) similar
data features were observed in small collision systems (LHC 2010 et seq.) and
(c) arguments developed that QGP must also form in small collision systems.
The problem then emerges: whence thermalization in small low-density systems?

The Bylinkin spectrum model with its exponential element has emerged as a
pivotal actor: The narrative continues that if the exponential model element
seems required by spectra then thermalization has been achieved, presumably
via entanglement for small systems. Claims of QGP in small collision systems
are then validated. But it is also important to demonstrate where
thermalization has not been achieved, with absence of an exponential element.
Then see the several figures in Sec. VI.4.

What is missing from that history is an alternative narrative in which (a)
certain data features in A-A collisions may have been misinterpreted because
of incomplete understanding of elementary collisions whether in isolation or
as superposed within A-A collisions, (b) observation of similar features in
small collision systems motivates more-careful study and improved
understanding and (c) QCD manifestations in all collision systems are better
understood. Equal weight should be given to both narratives until
understanding and resolution are achieved.

Interpretation of TCM components has relied on the fundamental element of
parton splitting cascades and universality. The soft component is associated
with projectile nucleon dissociation (fragmentation via splitting cascade)
along the beam axis which should then be approximately equivalent to a
$q$-$\bar{q}$ dijet as in Fig. 16 (left). In neither case is thermalization
expected. The hard component is associated with MB dijet production from
large-angle scattering of low-$x$ gluons again followed by splitting cascades
to form jets. Measured $p$-$p$ spectrum hard components are quantitatively
predicted by convolution of measured jet energy spectra with measured
fragmentation functions per QCD factorization fragevo ; eeprd ; jetspec2 .

Products of quantum transitions averaged over an event ensemble tend to
produce distributions that might be attributed to classical thermalization by
extended rescattering. i.e. maximum-entropy distributions. As examples, soft
component $\hat{S}_{0}(m_{t})$ of hadron spectra describing projectile
dissociation, the beta distribution that describes parton fragmentation
functions eeprd  and the Gaussian distribution on jet rapidity that describes
jet energy spectra jetspec2 . In each case it is highly unlikely that entropy
is maximized in the Boltzmann sense to arrive at those distributions. One may
instead look to Feynman path integrals and the least action principle: an
ensemble of quantum transitions, given fixed constraints, may arrive at
maximum-entropy distributions on average. Observation of data structures
possibly associated classically with thermalization via multiple scattering
does not confirm thermalization in high-energy nuclear collisions.

###  VII.3 Spectrum model design: a priori vs data

Spectrum model design requires choices: (a) a priori argument based on
assumptions vs empirical derivation from systematic data trends, (b) number of
model elements and parameters, (c) interpretability in terms of comparison
with theory and previous experiment.

The choice of specific model elements (functions) is critical for an
interpretable model. Consider a space whose points represent possible
functions that might be combined to model $p_{t}$ spectra. An exponential
function (as in Bylinkin Eq. (5)) is effectively a point in a subspace spanned
by the TCM soft component $\hat{S}_{0}(m_{t})$ with free parameters $T$ and
$n$. As noted, $\hat{S}_{0}(m_{t})$ becomes exponential
$\exp[-(m_{t}-m_{0})/T]$ in the limiting case $1/n\rightarrow 0$. Similarly,
the $q$-Gaussian in Eq. (5) is a point in a subspace spanned by Eq. (VII.1)
with $\kappa\rightarrow 2$ and $\sigma_{y_{z}}\rightarrow T_{h}$. As noted
above, the corresponding structure in the $p_{t}$ spectrum of $p$-$p$
collisions is described by a peaked distribution with power-law tail and
different widths above and below its mode as determined empirically from data.
But that is just the TCM hard component $\hat{H}_{0}(y_{t})$ with its
parameters: mode $\bar{y}_{t}$, widths $\sigma_{y_{t}+}$ and $\sigma_{y_{t}-}$
and exponential tail parameter $q$. If $\bar{y}_{t}\rightarrow 0$,
$\sigma_{y_{t}+}\rightarrow T_{h}$ and $q+2\rightarrow N$ the result is
approximately Bylinkin $q$-Gaussian on $m_{t}$ (modulo transformation with
suitable Jacobian). The Bylinkin model is in effect the result of a choice of
specific points in a model space based on a priori assumptions.

Particular applications of the Bylinkin model seem to describe some data
adequately but other data may exclude the model. Figures 16 (left), 11 (left),
12 (right) and 13 (left) demonstrate that those data can be adequately
represented by $\hat{S}_{0}(m_{t})$ alone with at most two parameters whereas
Eq. (5) requires at least four including the amplitude ratio for exponential
to $q$-Gaussian components. Those data thus do not require a Bylinkin-model
exponential that might be connected to conjectured thermalization. Figures 6
and 11 (right) demonstrate (along with Ref. ppprd ) that evolution of data
spectrum shape with event multiplicity requires a hard component
$\hat{H}_{0}(y_{t})$ and therefore rejects the Bylinkin $q$-Gaussian.
Regarding the absence of an exponential model element, data in Fig. 12 (left)
are not determining whereas Fig. 13 (right) is inconsistent with a “soft
component” whatever the details. The TCM soft component represents fragments
from projectile hadron dissociation or leading-quark fragmentation in
$e^{+}$-$e^{-}$ as in Fig. 16. If no splitting cascade occurs then there is no
soft component. That does not seem relevant to the question of thermalization.

###  VII.4 Resonance contribution to pion spectra

In Ref. bylinkinanom  it is noted that as a result of PID spectrum fits with
the Bylinkin model the exponential (soft) element, attributed to
thermalization and prominently present in fits to pion spectra, is much lower
or even absent in fits to heavier hadron spectra, described as “anomalous
behavior.” As an explanation it is stated that “It is known that the vast
majority of pions in hadronic collisions is produced via multiple cascade
decays of the heavier hadronic resonances. This might result in a
transformation of some part of the initial power law spectrum of the produced
short-lived heavy hadrons to an exponential decay pion distribution.” Based on
TCM analysis it is possible to isolate resonance contributions to pion spectra
quantitatively as described below.

Figure 17 (left) shows pion data from 5 TeV $p$-Pb collisions (points) and
corresponding TCM (curves) including soft-component model
$\hat{S}_{0i}(m_{t})$ with predicted model parameters for that collision
system ppbpid . Those parameter values are required to describe data spectra
above $y_{t}$ = 2 ($p_{t}\approx 0.5$ GeV/c), but the pion data spectra rise
increasingly above the TCM $\hat{S}_{0i}(m_{t})$ below $y_{t}$ = 2.

Figure 17:  Left: Pion spectrum data (points) and TCM (curves) for seven
centralities of 5 TeV $p$-Pb collisions. The TCM soft component in this case
does not include factor $f(y_{t})$ described in the text below. Right: Ratios
of $p$-Pb data to TCM without factor $f(y_{t})$ applied to the soft component.
The bold dotted curve following data is defined by Eq. (26).

Figure 17 (right) shows the ratio of spectrum data to TCM as they appear in
the left panel for six centrality classes of 5 TeV $p$-Pb collisions.
Deviations from 1 at higher $p_{t}$ correspond to expected minor evolution of
spectrum hard components with $p$-Pb centrality. The bold dotted curve is
defined by the expression

$\displaystyle f(y_{t})$ $\displaystyle=$ $\displaystyle
2/\left\\{1+\tanh[(y_{t}-0.3)/1.2]\right\\}.$ (26)

For TCM pion spectra to accommodate the resonance contribution to data unit-
normal model function $\hat{S}_{0}(m_{t})$ is replaced by
$S_{0}^{\prime}(m_{t})$ (not normalized) defined by

$\displaystyle S_{0i}^{\prime}(m_{t})$ $\displaystyle=$ $\displaystyle
f(y_{t})\hat{S}_{0i}(m_{t}).$ (27)

In Fig. 11 (right) the solid curve at low $p_{t}$ is $S_{0i}^{\prime}(m_{t})$
(resonances) and the dotted curve is $\hat{S}_{0i}(m_{t})$ (none). Note that
$\hat{S}_{0i}(m_{t})$ for pions has soft-component ensemble-mean
$\bar{p}_{tsi}\approx 0.44$ GeV/c whereas modified model function
$S_{0i}^{\prime}(m_{t})$ has soft-component mean $\bar{p}_{tsi}\approx 0.40$
GeV/c due to the additional resonance contribution at lower $p_{t}$. Also note
$\hat{S}_{0}(m_{t})$ etc. is a density on $m_{t}$ or $p_{t}$ while $f(y_{t})$
is a function of pion $y_{t}$ defined in terms of particle $p_{t}$.

Quantity $[f(y_{t})-1]\hat{S}_{0i}(m_{t})$ represents a third component not
described by the basic TCM. One may conjecture that this third component is
the pion contribution from resonance decays predicted to appear below 0.5
GeV/c resonances . Figure 17 (right) indicates that the resonance yield scales
exactly proportional to the soft component within data uncertainties. Neither
the resonance contribution nor TCM soft component $\hat{S}_{0i}(m_{t})$
changes shape significantly with varying $p$-Pb centrality. Those results
suggest that the resonance contribution has nothing to do with jet production
or hard processes.

To summarize, the resonance contribution to pion spectra appears at very low
$p_{t}$ with a unique $p_{t}$ dependence that is not exponential (see Eq.
(26)). TCM soft component $\hat{S}_{0i}(m_{t})$ is present for all hadron
species, with shape evolution depending simply on hadron mass and collision
energy. That the Bylinkin exponential might be favored for pion spectra but
not more-massive hadrons results from a combination of the resonance
contribution at lower $p_{t}$ and movement of the data (and
$\hat{S}_{0}(m_{t})$) soft-component shoulder to higher $p_{t}$ with
increasing hadron mass. There is no “anomalous behavior.”

##  VIII Summary

This article considers two $p_{t}$ spectrum models applied to data from high-
energy $p$-$p$ collisions that are interpreted in quite different ways. A
Bylinkin model includes two elements: an exponential function on $m_{t}$ and a
“power-law” function on $p_{t}$ or $m_{t}$ (actually a Gaussian centered at
zero $p_{t}$ with power-law tail). The two elements are interpreted
respectively as representing hadron emission from a thermalized source and
from a hard-scattering process. In contrast, a two-component (soft + hard)
model (TCM) was inferred empirically from evolution with event multiplicity
$n_{ch}$ of $p_{t}$ spectra from 200 GeV $p$-$p$ collisions. The two
components were interpreted later as arising from projectile-nucleon
dissociation (soft) and large-angle scatter of low-$x$ gluons to jets (hard).

The Bylinkin model has been applied to several collision systems where the
apparent presence or absence of the exponential element in fits to data is
interpreted to indicate a thermalized particle source or not. Apparent
presence of thermalization has subsequently been coupled to conjectures of
quantum entanglement as a mechanism for that process. In contrast, the TCM has
been applied to many collision systems and several hadron species from which
has emerged a coherent picture of collision dynamics generally consistent with
conventional QCD theory.

In the present study a detailed comparison is made between the properties of
the Bylinkin model and the TCM, for instance trends for parameter variation vs
$n_{ch}$ and collision energy. Derivation of the TCM by finite differences is
presented as an example of lossless data compression wherein the model retains
all information carried by particle data. Predictivity of the TCM is
demonstrated for the case of high-mass hadron spectra (Cascades, Omegas) with
spectra predicted based on the mass dependence of lower-mass spectra (pions,
kaons, protons) further illustrated by corresponding ensemble-mean
$\bar{p}_{t}$ trends.

The following issues emerge from the present study:

(a) The Bylinkin exponential element does not include the “built-in” mass
dependence of TCM soft component $\hat{S}_{0}(m_{t})$ demonstrated in this
study. As a result, Bylinkin model parameters must vary substantially to
accommodate data, but in ways that are not easily interpretable.

(b) Results of Bylinkin model fits to data spectra for varying conditions
return parameter trends that strongly disagree with expectations. Examples
include soft and hard particle densities vs event $n_{ch}$ and collision
energies, and corresponding results for ensemble-mean $\bar{p}_{t}$.

(c) Given the assumed a priori constraint on Bylinkin model parameters the two
model elements compete to describe the same data. At lower $p_{t}$ exponential
$T_{th}$ and Gaussian width $T_{h}$ must accommodate soft-component mass
dependence and jet/nonjet ratio increasing with event $n_{ch}$ and collision
energy. At higher $p_{t}$ Gaussian exponent $N$ attempts to accommodate
different soft and hard power-law tails again depending on the jet/nonjet
ratio. Those issues are camouflaged to an extend if analysis is confined to
minimum-bias spectra where the jet contribution is minimal. In contrast, the
TCM is required to describe all data from any collision system.

(d) Bylinkin model elements are special cases of more-general functions. The
exponential element is TCM $\hat{S}_{0}(m_{t})$ with $1/n=0$ held fixed. The
“power-law” element is a Gaussian on $p_{t}$ with power-law tail in which the
centroid is fixed at zero $p_{t}$. A fit to data spectra with both of those
parameters freely varying is likely to return nonzero values for both that
closely approximate TCM results by which a broad range of spectrum data are
accurately described within statistical uncertainties.

(e) The energy dependence of TCM $\hat{S}_{0}(m_{t})$ exponent $n$ that
determines its deviation from an exponential is consistent with Gribov
diffusion as a fundamental aspect of parton splitting cascades and thus can be
extrapolated across a broad range of collision energies and systems. In
particular it describes the $p_{t}$ spectrum from $q$-$\bar{q}$ dijets
(relative to the dijet axis) within data uncertainties. Exponent $n$ (i.e.
$1/n$) measures departure from equilibrium for some systems. It is notable
that $n$ is independent of particle density, again to the limits of data
uncertainties.

(f) Given the above issues interpretation of the Bylinkin exponential as
signaling thermalization or not depending on model fit results is problematic.
TCM soft component $\hat{S}_{0}(m_{t})$ represents hadrons from longitudinal
fragmentation of a leading particle (projectile hadron or quark). If there is
no such fragmentation $\hat{S}_{0}(m_{t})$ is absent. But perceived absence of
the Bylinkin exponential in model fits (e.g. for more-massive hadron spectra)
does not imply absence of projectile-hadron dissociation.

Taken together, the results of the present study suggest that the Bylinkin
model is not a suitable model for $p_{t}$ spectra from high-energy nuclear
collisions. Physical interpretation of model parameter trends within that
model context alone appear quite difficult. In particular the model cannot
determine whether a particular data spectrum reflects thermalization of a
particle source.

## References

  * (1) J. L. Nagle and W. A. Zajc, Ann. Rev. Nucl. Part. Sci. 68, 211-235 (2018).
  * (2) S. Acharya et al. (ALICE), Eur. Phys. J. C 80, no.8, 693 (2020).
  * (3) D. E. Kharzeev and E. M. Levin, Phys. Rev. D 95, no.11, 114008 (2017).
  * (4) O. K. Baker and D. E. Kharzeev, Phys. Rev. D 98, no.5, 054007 (2018).
  * (5) A. A. Bylinkin and A. A. Rostovtsev, arXiv:1008.0332.
  * (6) A. A. Bylinkin and A. A. Rostovtsev, Eur. Phys. J. C 72, 1961 (2012).
  * (7) A. A. Bylinkin and A. A. Rostovtsev, Nucl. Phys. B 888, 65-74 (2014).
  * (8) X. Feal, C. Pajares and R. Vazquez, Phys. Rev. C 104, no.4, 044904 (2021).
  * (9) T. A. Trainor, arXiv:2107.10899.
  * (10) T. A. Trainor and D. J. Prindle, Phys. Rev. D 93, 014031 (2016).
  * (11) J. Adams et al. (STAR Collaboration), Phys. Rev. D 74, 032006 (2006).
  * (12) T. A. Trainor, J. Phys. G 47, no.4, 045104 (2020).
  * (13) T. A. Trainor, Phys. Rev. C 90, no. 2, 024909 (2014).
  * (14) T. A. Trainor, J. Phys. G 44, no. 7, 075008 (2017).
  * (15) J. W. Cronin et al. (E100), Phys. Rev. D 11, 3105-3123 (1975).
  * (16) D. Antreasyan, J. W. Cronin, H. J. Frisch, M. J. Shochet, L. Kluberg, P. A. Piroue and R. L. Sumner, Phys. Rev. D 19, 764-778 (1979).
  * (17) G. Arnison et al. (UA1), Phys. Lett. B 118, 167-172 (1982).
  * (18) R. Hagedorn, Riv. Nuovo Cim. 6N10, 1-50 (1983).
  * (19) L. Van Hove, Phys. Lett. B 118, 138 (1982).
  * (20) G. Pancheri, LNF-83-78-P.
  * (21) M. Jacob, Conf. Proc. C 830718, 787 (1983) CERN-TH-3693.
  * (22) G. Pancheri and Y. Srivastava, Conf. Proc. C 850313, 28 (1985).
  * (23) J. K. Yoh, S. W. Herb, D. C. Hom, L. M. Lederman, J. C. Sens, H. D. Snyder, K. Ueno, B. C. Brown, C. N. Brown and W. R. Innes, et al. Phys. Rev. Lett. 41, 684 (1978) [erratum: Phys. Rev. Lett. 41, 1083 (1978)].
  * (24) A. Adare et al. (PHENIX), Phys. Rev. Lett. 98, 232002 (2007).
  * (25) B. Abelev et al. (ALICE), Phys. Lett. B 718, 295-306 (2012) [erratum: Phys. Lett. B 748, 472-473 (2015)].
  * (26) C. Tsallis, J. Statist. Phys. 52, 479-487 (1988).
  * (27) T. A. Trainor, arXiv:1708.09412.
  * (28) T. A. Trainor, arXiv:2210.08419.
  * (29) T. A. Trainor, arXiv:2112.09790.
  * (30) T. A. Trainor, arXiv:2112.12330.
  * (31) A. A. Bylinkin and M. G. Ryskin, Phys. Rev. D 90, no.1, 017501 (2014).
  * (32) A. A. Bylinkin and A. A. Rostovtsev, arXiv:1203.2840.
  * (33) A. A. Bylinkin, D. E. Kharzeev and A. A. Rostovtsev, Int. J. Mod. Phys. E 23, no.12, 1450083 (2014).
  * (34) A. Bylinkin, N. S. Chernyavskaya and A. A. Rostovtsev, Eur. Phys. J. C 75, no.4, 166 (2015).
  * (35) T. A. Trainor, Phys. Rev. D 89, 094011 (2014).
  * (36) T. A. Trainor, Phys. Rev. C 80, 044901 (2009).
  * (37) G. Bocquet et al. (UA1-MIMI), CERN-PPE-94-47.
  * (38) Y. L. Dokshitzer and D. E. Kharzeev, Ann. Rev. Nucl. Part. Sci. 54, 487 (2004).
  * (39) M. L. Nekrasov, Particles 4, no.3, 381-390 (2021).
  * (40) M. L. Nekrasov, Mod. Phys. Lett. A 35, no.38, 2050314 (2020).