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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The conventional intensive ratio of extensive quantities

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

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

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

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

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

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

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